Hamel A. - Variational principles on metric and uniform spaces-Martin-Luther-Univ. (2005).pdf

Variational Principles on Metric and
Uniform Spaces
H a b i l i t a t i o n s s c h r i f t
zur Erlangung des akademischen Grades
Dr. rer. nat. habil.
vorgelegt der
Mathematisch–Naturwissenschaftlich–Technischen
Fakultät
der Martin-Luther-Universität Halle-Wittenberg
von
Herrn Dr. rer. nat. Andreas Hamel
geboren am 08.09.1965 in Naumburg (Saale)
Gutachter
1. Prof. Dr. Johannes Jahn, Erlangen-Nürnberg
2. Prof. Dr. Christiane Tammer, Halle-Wittenberg
3. Prof. Dr. Constantin Zălinescu, Iasi
Halle (Saale), 24.10.2005
urn:nbn:de:gbv:3-000009148
[http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000009148]

Contents
1 Introduction 5
2 Basic Framework 11
2.1 Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Conlinear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Semilinear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Order structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Ordered product sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 Power sets of ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.4 Ordered monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5 Ordered conlinear spaces . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.6 Ordered semilinear spaces . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.7 Historical comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Topological and uniform structures . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.2 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3.3 Completeness in uniform spaces . . . . . . . . . . . . . . . . . . . . . 54
2.3.4 The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.5 Conlinear spaces via topological constructions . . . . . . . . . . . . . 59
3 Order Premetrics and their Regularity 61
4 Variational Principles on Metric Spaces 65
4.1 The basic theorem on metric spaces . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.2 The basic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.3 Equivalent formulations of the basic theorem . . . . . . . . . . . . . 67
4.1.4 The regularity assumptions . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.6 Set relation formulation . . . . . . . . . . . . . . . . . . . . . . . . . 70
3

4 Contents
4.2 Results with functions into ordered monoids . . . . . . . . . . . . . . . . . . 73
4.2.1 Ekeland’s variational principle . . . . . . . . . . . . . . . . . . . . . 73
4.2.2 Kirk-Caristi fixed point theorem . . . . . . . . . . . . . . . . . . . . 76
4.2.3 Takahashi’s existence principle . . . . . . . . . . . . . . . . . . . . . 77
4.2.4 The flower petal theorem . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.5 An equilibrium formulation of Ekeland’s principle . . . . . . . . . . 79
4.2.6 Ekeland’s variational principle on groups . . . . . . . . . . . . . . . 81
4.3 Ekeland’s principle for set valued maps . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Power set of ordered monoids . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Ekeland’s principle for single valued Functions . . . . . . . . . . . . . . . . 86
4.5 Ekeland’s principle for real valued functions . . . . . . . . . . . . . . . . . . 87
4.6 Geometric variational principles in Banach spaces . . . . . . . . . . . . . . . 91
4.6.1 Results in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6.2 Results in locally complete locally convex spaces . . . . . . . . . . . 94
4.7 Minimal elements on product spaces . . . . . . . . . . . . . . . . . . . . . . 97
5 Partial Minimal Element Theorems on Metric Spaces 101
5.1 The basic theorem on metric spaces . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Results involving ordered monoids . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Results involving power sets of ordered monoids . . . . . . . . . . . . . . . . 104
5.4 Results involving linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Variational Principles on Complete Uniform Spaces 109
6.1 The basic theorem on complete uniform spaces . . . . . . . . . . . . . . . . 109
6.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.4 Set relation formulation . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1.5 Special cases of Theorem 24 . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Results with functions into ordered monoids . . . . . . . . . . . . . . . . . . 116
6.2.1 Ekeland’s principle over quasiordered monoids . . . . . . . . . . . . 116
6.2.2 Power sets of quasiordered monoids . . . . . . . . . . . . . . . . . . 119
6.2.3 Single valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 A partial minimal element theorem on complete uniform spaces . . . . . . . 121
7 Variational Principles on Sequentially Complete Uniform Spaces 123
7.1 The basic theorem with sequential completeness . . . . . . . . . . . . . . . . 123
7.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.1.4 Set relation ordering principle . . . . . . . . . . . . . . . . . . . . . . 125
7.2 The basic theorem on a product space . . . . . . . . . . . . . . . . . . . . . 127

Chapter 1
Introduction
DEAE IN AETERNUM INCOGNITAE.
The main goal of the present work is to give most general formulations of Ekeland’s
Variational Principle as well as of so-called Minimal Element Theorems on metric
and uniform spaces.
A minimal element theorem gives conditions for the existence of minimal elements of
an ordered set X or X × Y with respect to certain order relations. Ekeland’s variational
principle ensures the existence of minimal points for a (small) perturbation of a function
f : X → Y , where Y is supplied with an order relation.
We call both kinds of theorems simply Variational Principles since they have a
fundamental idea in common: to vary a certain point to obtain another one, not so far
away, with some useful extremality properties. Moreover, in several situations a minimal
element theorem turns out to be an equivalent formulation of a suitable Ekeland’s principle
and vice versa. A further object of this work is to find the right equivalent formulation in
each situation.
¿From a historical point of view, the story began with X being a topological linear
space (Lemma 1 in Phelps’ paper [101] from 1963) and a complete metric space (the varia-
tional principle, see Ekeland’s papers [28], [29], [30] from the beginning of the seventhies),
respectively, and Y = IR in both cases. Since the topology of metric spaces as well as of
topological linear spaces can be generated by a uniform structure, it is a natural idea to
look for a common formulation in uniform spaces. Such a formulation has already been
given by Brønsted in the paper [8] from 1974.
However, it turned out that there are two different approaches to the proof: The
first one is to assume that X is a complete uniform space and to work with nets instead
of sequences. As a rule, Zorn’s lemma (or a transfinite induction argument) has to be
involved in this case and the assumptions are stronger than in the metric case. Compare
Chapter 6 for this approach which is also the basic idea of the work of Nemeth [92], [93],
5

6 Chapter 1. Introduction
[94].
The second one is to find assumptions which allows to work with sequences even in
uniform spaces. Such assumptions essentially involve a scalarization, i.e., a real valued
function linking topological properties and properties of the order relation in question.
This approach is presented in Chapter 7 and it is shown that it yields a link between
Brønsted’s results [8] (he also used a scalarization technique) and recent results of Göpfert,
Tammer and Zălinescu [114], [47], [44] and even corresponding set valued variants as in
[50].
Using the latter approach, it is also possible to leave the framework of uniform spaces
and to work only on ordered sets. This has been done by Brézis and Browder in the
influential paper [6]. Subsequent generalizations can be found e.g. in [1], [67] and in
several papers by Turinici such as [119], [120], [121], [122], [123]. Results of this type are
out of the scope of this work, since it is restricted to the case in which the existence of
minimal elements essentially follows from completeness.
Of course, a minimal element theorem on an ordered set (X, X) can be applied to
a product set (X × Y, X×Y ) provided the corresponding assumptions are satisfied by
X × Y and X×Y . Results of this type can be found e.g. in Section 4.7. But in many
applications it is desirable to have different sets of assumptions for X and Y . Remaining in
the framework of the present thesis, i.e., X is assumed to be a complete metric or a uniform
space, the question is what assumptions are essential for Y to obtain a minimal element
theorem on X × Y and an Ekeland type theorem for functions f : X → Y , respectively.
The answer is: Algebraic and topological assumptions to Y , e.g., Y is assumed to be
a topological linear space, are not essential. To the authors opinion, this is one of the
more surprising results of the present investigation. Assumptions have to be put only
on the order relation X×Y which are satisfied even in cases in which Y is neither a
topological nor a linear space. The crucial assumption deals with decreasing sequences: If
{(xn, yn)}n∈IN is decreasing with respect to X×Y and {xn}n∈IN is convergent to x ∈ X,
then there must be y ∈ Y such that (x, y) X×Y (xn, yn) for all n ∈ IN. The importance of
this assumption has been figured out in [47], but it is also strongly related to assumption
(2) of Brézis–Browder’s Theorem 1 in [6].
This allows to obtain minimal element and Ekeland type theorems for example if Y is
the power set of a linear space. In fact, so called set valued Ekeland’s principles as recently
established by Chen, Huang and Hou in [11], [12], [13], Truong in [117] and Hamel and
Löhne in [50] are the main motivation to look for minimal element theorems on X × Y
with Y being more general than a linear space (compare [44]) or a topological Abelian
group (compare [93], [94]).
Following this path, it was possible to prove variational principles on (X × Y, X×Y )
and for f : X → (Y, Y ), respectively, under very mild assumptions concerning Y . Es-
pecially, Y can be assumed to be an ordered monoid. Since the power set of an ordered
monoid is an ordered monoid as well (with suitable order relations), this covers also set
valued variational principles.

7
This is the reason for the investigations of Chapter 2 of this thesis: The topic is the
structure of ordered monoids with special emphasis to those properties being invariant
under passing to power sets. Several details of this chapter are not new: For example, it
is well–known that the power set of a monoid is a monoid with respect to the Minkowski
operation. Also, the order relations for power sets of ordered sets are not new. But the
author is not aware of a thorough presentation of algebraic and order theoretic properties
of ordered monoids and their power sets together with their interrelations.
The present work contains five main chapters.
Chapter 2 deals with basic structures: algebraic, order and topological structures used
in the subsequent chapters. Mainly, the concepts of this chapter are developed in order to
formulate the variational principles.
However, there are several results not used in the subsequent chapters. They remained
in this text since they shed some light on basic ideas or illustrate the difference to widely
used concepts or may serve as a starting point for future developments: For example, this
applies for the results about the interrelations between the infimum and the set of minimal
points of a subset of an ordered set (W, ) on one side and the infima with respect to the
two canonical extensions of to the power set of W on the other side. Compare Section
2.2. The so–called domination property (lower external stability, cf. [85] in the context of
linear spaces) plays a pivotical role and the relationships between vector and set valued
optimization problems are turned up side down in some sense.
On the other hand, there are some concepts without an explicit definition like group
or complete metric space. In this cases, the definitions are very standard and the terms
are used with the same meaning in almost all text books on corresponding topics.
The basic algebraic structure is a commutative monoid, i.e., a commutative semigroup
with a neutral element. This seems to be a natural starting point since the power set of
a group is a monoid (with respect to the corresponding Minkowski operation) as well as
order completion of a group leads to an ordered monoid whereas the power set of a monoid
is a monoid again as well as the order completion of an ordered monoid. This means: The
monoidal structure is stable under passing to power sets and under order completion.
A new concept is that of a conlinear space introduced in Section 2.1. This concept
is more general than that of a linear space and it turns out that the power set as well
as the order completion of a linear space is a conlinear space. Moreover, a convex cone
in the classical sense (i.e., a subset of a linear space invariant under multiplication with
nonnegative real numbers and under addition) is also a conlinear space. On the other
hand, there are conlinear spaces which can not be identified with a cone as a subset of a
linear space.
It appears to me that the concept of a conlinear space might be a natural framework
to define and investigate convexity. Several initial clues supporting this idea can be found
in Section 2 of this thesis. Some elementary concepts and facts carry over from Convex
Analysis in linear spaces to a Convex Analysis in conlinear spaces, a topic under devel-

opment. Some more results in this direction can be found in the PhD thesis of A. Löhne
[83] and the diploma thesis of C. Schrage [109].
Section 2.2 contains order theoretic concepts, especially the definition and basic prop-
erties of the two canonical extensions 4, 2 of a quasiorder on a set W to the power
set b
P (W) (including ∅). These extensions are widely used in theoretical information sci-
ences. A thorough survey is the 1993 paper by C. Brink [7]. Our exposition emphasizes
on formulas for the infimum and supremum of subsets of b
P (W) with respect to 4 and
2, respectively. As mentioned above, there are close relationships between these extrema
and the sets of infimal and minimal points of W with respect to . This shed some new
light on inherent difficulties of vector optimization and provokes a surprising answer to
the question what we shall understand by a solution of an optimization problem with a
set valued objective function. However, this is not a main topic of this thesis.
The concepts connected with topological and uniform structures are introduced in
order to have as much freedom as possible to define order relations on uniform spaces
satisfying the regularity assumption of the minimal element theorem. This is motivated
by the fact that there are at least three different types of Ekeland type results on uniform
spaces concerning the order relation. Mizoguchi [86] used pseudometrics and Fang [33]
quasimetrics whereas Nemeth’s results in several papers involve so called cone valued
metrics. A few attempts have been made to unify these approaches, e.g. in [10] and [51].
The observation that all these approaches apply for different order relations, but for the
same class of spaces seems to be new.
Therefore, we collect four possibilities to introduce a uniform structure and show their
equivalence. Only two of them are quite standard.
In Chapter 4 variational principles on metric spaces are presented.
Although most of the results, especially the main Theorem 16, are special cases of
results of corresponding theorems on uniform spaces, we prefered to give direct proofs in
metric spaces. This is for several reasons: First, the proofs are in many cases simpler,
more direct, in some sense constructive and already contain the essential ideas. Secondly,
for the vast majority of applications especially the metric case is interesting and most
of the papers on variational principles deal with this case. And thirdly, the metric case
served as a blueprint for the sequential analysis in uniform spaces of Chapter 7.
The leading questions of this chapter (and also for the two subsequent chapters) are
the following: What are the indispensable ingredients for a proof of a variational principle?
What are the mathematical concepts lying at the bottom of the theory? Is it possible to
find a general sceme for all proofs of Ekeland type results?
The answer is as follows. The indispensable ingredients are a (complete) metric space,
a quasiorder with lower closed sections and a further link between topological properties
and the properties of the order. This linking assumption is called regularity of the order:
Decreasing sequences have to be asymptotic. The basic result, Theorem 16, contains just
these things. To the opinion of the author, all Ekeland type theorems on metric spaces
including vector and set valued variants can be proven by verifying the assumptions of the

9
basic theorem for a suitable order relation. This program is carried out in the remaining
part of the chapter producing almost all known results in the field – among them Ekeland’s
original result, the Kirk-Caristi fixed point theorem and the drop theorem as well as a lot
of new results especially for functions with values in ordered monoids and its power sets.
Let us note that several authors try to avoid using order relations explicitely while
proving Ekeland type results. For example, Ekeland itself in [31] gave such a proof.
However, the order relation is still present (and is sometimes called dynamical system as
in [4] and several papers of Isac) and therefore it seems to be adequate to start with
an order relation on a metric space. The idea for a proof of Ekeland’s principle using a
minimal element theorem on metric spaces can be traced back to the 1983 paper [22] of S.
Dancs et al. There are many authors who used the Dancs–Hegedüs–Medvegyev theorem
for proving one or another variant of Ekeland’s principle, see for example [59], [60], [61],
but the central importance of such a theorem seems to be new knowlegde as well as its
far reaching applicability and its equivalence to the other theorems in Section 4.1.
Concerning the meaning of the equivalence between different variational principles we
refer the reader to the discussion in Section 4.1. There are several papers presenting lists
of theorems being equivalent to Ekeland’s principle in some sense, e.g. [24], [98], [38], [96],
[97] and also the book [58] to mention a few. In Section 4.1 we present some theorems
being equivalent to the basic minimal element principle on metric spaces, Theorem 16. In
Section 4.2 results are presented being equivalent to Ekeland’s principle for functions with
values in ordered monoids and Section 4.5 contains a series of theorems being equivalent to
Ekeland’s original result (Theorem 1.1 of [30]) involving realvalued functions. Of course,
for each image space Y a corresponding list is possible but we do not focus on such
equivalence assertions for each type of Y .
In Chapter 5, a minimal element theorem is presented for a subset of a product set
X × Y , where X is a (complete) metric space. Similar results for X a (sequentially)
complete uniform space are contained in Section 6.3 and 7.2. The main new feature of
these results is again that Y is merely assumed to be a nonempty set. Only assumptions
to the order relation on X × Y appear. Therefore, Y can be chosen as the power set of a
linear space for example. This leads to a minimal element theorems on X × P (V ), where
P (V ) is the power set of a linear space V .
Chapter 6 deals with variational principles on complete uniform spaces without scalar-
ization. The development in this direction originates from I. Vályi [124] and A. B. Németh
[92]. We start with a basic minimal element theorem on complete uniform spaces, Theorem
24 and apply it in order to obtain a series of corollaries for various single and set valued
situations. Considering functions on uniform spaces with values in ordered monoids we
establish generalizations of the most recent results of Németh [93], [94].
In Chapter 7, a systematic treatment of situations is given in which a scalarization
function is present. Under this assumption we need to consider only sequentially complete
uniform spaces. A series of corollaries is presented involving more complicated scalarization

functions from step to step starting from continuous linear function (Y has to be a locally
convex space), going to sublinear functions of Tammer-Weidner type on linear spaces and
even on power sets of linear spaces. The starting point of these developments is Brønsted’s
paper [8], but we also obtain generalizations of results collected in the book [44] as well
as those of [51] and [50].
This thesis does not deal with applications of the presented theorems, not even appli-
cations in ”pure” mathematics such as ABB theorems, existence for solutions of vector
optimization problems, necessary optimality and approximate optimality conditions, fuzzy
metric space theory, geometry of Banach spaces, economical fixed point theory, to mention
a few main fields. We only remark that variational principles in the spirit of Phelps and
Ekeland turned out to be undispensable tools for recent developments in various fields
of mathematics. The history of necessary optimality conditions for nonsmooth optimal
control problems since Clarke’s pioneering work [17] may serve as a prominent example.

Chapter 2
Basic Framework
2.1 Algebraic structures
In this section, several algebraic structures are introduced forming the framework for the
theory of the next chapters. The goal is to replace the concept of a linear space by a
more general one. This is motivated on the one hand by the structure of the power set
P (V ), V denoting a real linear space and on the other hand by the algebraic properties of
IR∪{+∞}, IR∪{−∞} and IR∪{±∞}, respectively. The elementwise addition (Minkowski
sum) of two subsets of V does not satisfy the axiom of the existence of an inverse element.
The same phenomenon appears in IR ∪ {±∞}, for example: It does not make sense to
define (+∞) + (−∞) = 0 in most cases. Depending on the purpose, definitions like
(+∞)+(−∞) = +∞ and (+∞)+(−∞) = −∞ occur, called inf-addition and sup-addition,
respectively, in [106], Section 1.E.
2.1.1 Monoids
Let Y be a nonempty set and Y × Y the set of all ordered pairs of elements of Y . A
binary operation on Y is a mapping of Y × Y into Y .
Definition 1 Let Y be a nonempty set and ◦ a binary operation on Y . The pair (Y, ◦) is
called a monoid iff
(M1) ∀y1, y2, y3 ∈ Y : y1 ◦ (y2 ◦ y3) = (y1 ◦ y2) ◦ y3;
(M2) ∃θ ∈ Y ∀y ∈ Y : y ◦ θ = θ ◦ y = y.
A monoid is called commutative iff the relation ◦ is commutative, i.e.
(M3) ∀y1, y2 ∈ Y : y1 ◦ y2 = y2 ◦ y1.
A monoid is nothing else than a semigroup with a neutral element; hence all results on
semigroups apply also on monoids. The neutral element of a monoid is unique.
In this note, we only consider commutative monoids, even though several results may
be formulated in a more general framework.
11

12 Chapter 2. Basic Framework
Example 1 (i) A set consisting of three elements, say Y = {U, L, θ}, can be provided
with a monoidal structure by defining L◦L = L, U ◦U = U, θ ◦θ = θ, L◦U = U ◦L = U,
L ◦ θ = θ ◦ L = L, U ◦ θ = θ ◦ U = U. The axioms (M1), (M2) are easy to check by
noting that all expressions involving U produce U and expressions not containing U, but
L produce L. Thus the three elements are in a certain hierarchical order with respect to
the operation ◦: U dominates the two others, L dominates θ. Of course, (Y, ◦) is not a
group. This example will be of some importance later on.
(ii) The set Y = {0, 1, 2}, together with the operation y1◦y2 = min {y1, y2} can be identified
with the monoid in (i) by setting U = 0, L = 1, θ = 2. The neutral element is θ = 2. In
the same way, a monoidal structure on {−∞, 0, +∞} is obtaind by identifying U = +∞,
L = −∞ (inf-addition) and vice versa U = −∞, L = +∞ (sup-addition).
(iii) The set IRn ∪ {+∞} as well as IRn ∪ {−∞} can be made to an commutative monoid
by defining
x + (+∞) = +∞ + x = (+∞) and x + (−∞) = −∞ + x = (−∞) ,
respectively, for all x ∈ IRn and +∞+(+∞) = +∞ in the first case and −∞+(−∞) = −∞
in the second one.
Considering IRn ∪ {+∞, −∞} there are two main possibilities to extend the operation
+ to the case when both summands are non finite elements of IRn, namely,
(−∞) + (+∞) = (+∞) + (−∞) = +∞;
(−∞) + (+∞) = (+∞) + (−∞) = −∞.
Each of these possibilities leads to an commutative monoid. Later on, we shall discuss
some applications. The definition (−∞) + (+∞) = (+∞) + (−∞) = 0, at the first glance
more natural, does not produce a monoid since the associative law (M1) is violated.
(iv) The set of all nonempty subsets of the real line with respect to elementwise addi-
tion or multiplication is a commutative monoid. The neutral elements are {0} and {1},
respectively.
The last example can be generalized in order to produce new monoids. Let Y be a
nonempty set. We denote by P (Y ) the set of all nonempty subsets of Y and by b
P (Y ) the
set of all subsets of Y including the empty set ∅, i.e. b
P (Y ) = P (Y ) ∪ {∅}.
Let (Y, ◦) be a monoid. We define an operation on P (Y ) by
∀M1, M2 ∈ P (Y ) : M1 M2 := {y1 ◦ y2 : y1 ∈ M1, y2 ∈ M2} .
The operation can be extended to b
P (Y ) by
∀M ∈ b
P (Y ) : M ∅ = ∅ M = ∅.
This means, ∅ ∈ b
P (Y ) is defined to be a zero element in the sense of [19], p. 3. A zero
element of a commutative monoid is always unique.
The property of being a monoid is stable under passing to power sets.

2.1. Algebraic structures 13
Proposition 1 Let (Y, ◦) be a monoid. Then (P (Y ) , ) and

b
P (Y ) ,

are monoids
as well. In each case, the neutral element is Θ = {θ}. If (Y, ◦) is commutative, so are
(P (Y ) , ),

b
P (Y ) ,

.
Proof. Immediately from the definition.
Example 2 Let (Y, ◦) be a commutative monoid. For M ⊆ Y we define the (plus)-
indicator function belonging to M by
I+
M (y) :=
(
0 : y ∈ M
+∞ : y 6∈ M
.
Denote by I+ (Y ) the set of all functions f on Y such that f (Y ) ⊆ {0, +∞}, i.e., I+ (Y )
is the set of (plus)-indicator functions for subsets of Y . Defining

I+
M1
I+
M2

(y) := inf
n
I+
M1
(y1) + I+
M2
(y2) : y1 ◦ y2 = y
o
for M1, M2 ∈ b
P (Y ) one may see that (I+ (Y ) , ) is a commutative monoid. Since
M = M1 M2 ⇐⇒ I+
M = I+
M1
I+
M2
,
there is an isomorphism between

b
P (Y ) ,

and (I+ (Y ) , ).
Of course, a similar consideration is possible with I− (Y ) replacing +∞ by −∞ and
inf by sup.
We introduce further notation considering elements of monoids with special properties.
Definition 2 Let (Y, ◦) be a commutative monoid. An element y ∈ Y is called invertible
iff there exists an y0 ∈ Y such that
y ◦ y0
= y0
◦ y = θ.
The set of all invertible elements of (Y, ◦) is denoted by Yin.
Clearly, θ ∈ Y is always invertible. Moreover, (Yin, ◦) is a subgroup of the given monoid
being maximal in the sense that there is no other subgroup of (Y, ◦) containing all invert-
ibles and at least one more element. Therefore, the set Yin ⊂ Y is called the maximal
subgroup of the given monoid. Of course, (Y, ◦) is a group iff Y = Yin.
Note that in several textbooks on semigroups, e.g. [19], p. 21ff, invertible elements are
called units. Passing to power sets, the maximal subgroup of a monoid is invariant.
Proposition 2 Let (Y, ◦) be a commutative monoid with the maximal subgroup (Yin, ◦).
Then it is also the maximal subgroup of (P (Y ) , ) and

b
P (Y ) ,

, respectively, in the
sense that y ∈ Yin is identified with {y} ∈ P (Y ).

Proof. Let Y1, Y2 ∈ P (Y ) be invertible such that Y1 Y2 = {θ}. Then
∀y1 ∈ Y1, y2 ∈ Y2 : y1 ◦ y2 = θ
contradicting the uniqueness of inverse elements in groups if at least one of Y1, Y2 contains
more than one element. Concerning b
P (Y ), it suffices to note that, by definition of , ∅ is
not invertible.
Example 3 (i) The set Y := IR2
+ of all elements of IR2 with nonnegative components,
together with the usual vector addition, forms a commutative monoid with Yin =
n
(0, 0)T
o
.
(ii) The set Y :=
n
y = (y1, y2)T
∈ IR2 : y2 ≥ 0
o
, together with the usual vector addi-
tion, forms a commutative monoid with Yin =
n
(y1, 0)T
∈ IR2 : y1 ∈ IR
o
.
Proposition 3 Let (Y, ◦) be a commutative monoid and Ynin := Y Yin ∪ {θ} the set of
all noninvertible elements and θ. Then (Ynin, ◦) is a monoid as well.
Proof. Let y, y0 ∈ Ynin. Then y ◦ y0 is a noninvertible since otherwise there would be
a u ∈ Y such that (y ◦ y0) ◦ u = y ◦ (y0 ◦ u) = θ. Hence y is invertible contradicting the
assumption.
Definition 3 Let (Y, ◦) be a monoid. An element y ∈ Y is said to be idempotent iff
y = y ◦ y.
An idempotent element y 6= θ is called nontrivial.
Of course, an idempotent element is an element coinciding with all of its n-powers, i.e.,
∀n = 1, 2, . . . : y = yn
:= y ◦ . . . ◦ y
| {z }
n times
.
Proposition 4 Let (Y, ◦) be a commutative monoid and Yid ⊆ Y the set of all idempotent
elements. Then (Yid, ◦) is a commutative monoid as well.
Proof. Let y, y0 ∈ Y be idempotent elements. Then
y ◦ y0

◦ y ◦ y0

= (y ◦ y) ◦ y0
◦ y0

= y ◦ y0
,
i.e., the operation ◦ transfers idempotent elements into idempotent elements.
Proposition 5 Let (Y, ◦) be a commutative monoid and y ∈ Y be an idempotent element.
Then {y} is an idempotent element of P (Y ) and b
P (Y ).

Proof. Obvious.
The following proposition shows the difficulties connected with nontrivial idempotent el-
ements.
Proposition 6 A monoid with a nontrivial idempotent element can not be embedded in
a group.
Proof. Let (Y, ◦) be a monoid and (G, ) be a group such that Y ⊂ G and coincides
with ◦ on Y . Let y ∈ Y be a nontrivial idempotent element. Then there exists g ∈ G such
that y g = θ. This implies
θ = y g = (y ◦ y) g = y (y g) = y ◦ θ = y
contradicting the nontriviality of y.
An commutative monoid with the unique idempotent element θ can be embedded in a
group if and only if the cancellation property holds true, i.e. for three elements y, y1, y2
we have
y1 ◦ y = y2 ◦ y =⇒ y1 = y2.
A nontrivial idempotent element destroys the cancellation property, compare [19], p. 6, 1.
(b) and p. 34ff.
In Example 1, (i), (iii) we have seen that the monoid operation can be dominated by
certain elements. We give a precise definition of this property which is essentially due to
A. Löhne [82].
Definition 4 Let (Y, ◦) be a monoid. The subset Y1 ⊆ Y is said to dominate the subset
Y2 ⊆ Y , shortly Y1 Y2, iff
y1 ∈ Y1, y2 ∈ Y2 =⇒ y1 ◦ y2 ∈ Y1.
An element b
y ∈ Y is called the dominant element of Y with respect to ◦ iff {b
y} Y .
Proposition 7 Let (Y, ◦) be a commutative monoid and ŷ ∈ Y be a dominant element
different from θ. Then it is unique and an idempotent element.
Proof. The uniqueness is obvious. By definition of dominant elements, ŷ ◦ y = ŷ for all
y ∈ Y . Setting y = ŷ, the result follows.
Example 4 (i) Easy to check examples for monoides with dominant elements are given
in Example 1, (i) and (ii).
(ii) Considering Example 1, (iii), we denote by (IRn)M
the monoid (IRn ∪ {±∞} , +)
where the element +∞ is dominant. Likewise, (IRn)O
is (IRn ∪ {±∞} , +) where −∞ is
dominant. In case n = 1 we write IRM and IRO, respectively.

(iii) If (Y, ◦) is a monoid, the monoid

b
P (Y ) ,

contains the dominant element ∅.
(iv) Let N = {1, 2, . . .} be the set of positive integers and a◦b := max {a, b}. Then (N, ◦)
is a commutative monoid with neutral element 1 consisting only of idempotent elements.
Likewise, the set N ∪ {+∞} with the operation a ◦ b := min {a, b} is a commutative
monoid with neutral element +∞ and the nontrivial dominant element 1.
2.1.2 Conlinear spaces
The concept of a conlinear space generalizes the concept of a real linear (vector) space. In
this section, we start with monoids with a binary operation called addition and denoted
by +.
Definition 5 A set Y , together with an addition +, is said to be a (real) conlinear
space (Y, +) iff the following axioms are satisfied:
(C1) (Y, +) is a commutative monoid with neutral element θ.
(C2) There is mapping from IR+ × Y into Y , assigning t ≥ 0 and y ∈ Y the product
ty := t · y ∈ Y such that the following conditions are satisfied:
(i) ∀y ∈ Y, ∀s, t ≥ 0 : s · (t · y) = (st) · y;
(ii) ∀y ∈ Y : 1 · y = y;
(iii) ∀y ∈ Y : 0 · y = θ;
(iv) ∀t ≥ 0, ∀y1, y2 ∈ Y : t · (y1 + y2) = (t · y1) + (t · y2).
Note that the validity of the second distributive law (s + t) y = (sy)+(ty) is not required,
not even for s, t ≥ 0. Instead, we impose (C2, (iii)). This is the main difference to the
concept of (ordered) cones in [71], Section 1.1. As a consequence, a conlinear structure
is stable under passing to power sets whereas a cone in the sense of [71] is not. See
Proposition 10 below.
The following properties are easy to prove directly from the axioms.
Proposition 8 Let (Y, +) be a conlinear space. Then:
(i) If t ≥ 0, then tθ = θ.
(ii) If t 0, y1, y2 ∈ Y and ty1 = ty2, then y1 = y2.
(iii) If y ∈ Y {θ} and t 0, then ty 6= θ.
Proof. (i) We have tθ = t (0 · y) = (t0) · y = 0 · y = θ.
(ii) Multiplying the equality ty1 = ty2 by t−1 and using (C2, (i)) we obtain the result.
(iii) ty = θ would imply t−1 (ty) = t−1 · θ = θ which contradicts t−1 (ty) = t−1t

y =
1 · y = y 6= θ.
Note that t1y = t2y for y ∈ Y {θ} does not imply t1 = t2. An example is given below.
Let (Y, +) be a conlinear space. If Y 0 ⊆ Y and (Y 0, +) is itself a conlinear space, then
it is called a conlinear subspace of Y . A subset Y 0 ⊆ Y is a conlinear subspace if and
only if t ≥ 0, y, y1, y2 ∈ Y 0 imply ty ∈ Y 0 and y1 + y2 ∈ Y 0.

Example 5 (i) IR∪{+∞} is a conlinear space if the usual multiplication with nonnegative
real numbers is extended as follows:
∀t 0 : t · (+∞) = +∞ and 0 · (+∞) = 0.
Likewise, IR∪{−∞} can be supplied with a conlinear structure. These two conlinear spaces
can be decomposed into the linear space IR (see Definition 9 below) and the conlinear spaces
{0, +∞} and {0, −∞}, respectively.
(ii) (IRn)M
and (IRn)O
are conlinear spaces using the same conventions dealing with ±∞,
i.e.,
∀t 0 : t · (+∞) = +∞, t · (−∞) = −∞
and
0 · (+∞) = 0 · (−∞) = 0.
Similarly, {−∞, 0, +∞} can be supplied with a conlinear structure in two different ways.
Compare (ii) of Example 1.
Proposition 9 Let X be a nonempty set and (Y, +) be a conlinear space. Then the
set R (X, Y ) of all functions mapping X into Y is a conlinear space with respect to the
pointwise operations
(f1 ⊕ f2) (x) := f1 (x) + f2 (x) , x ∈ X (2.1)
(t · f) (x) := tf (x) , t ≥ 0, x ∈ X. (2.2)
Proof. By (2.1), (2.2) the expressions f1 ⊕ f2 and t · f are well-defined for f, f1, f2 ∈
R (X, Y ), t ≥ 0. Defining the neutral element θ in R (X, Y ) by
∀x ∈ X : θ (x) = θY
where θY is the neutral element of Y the axioms (C1) and (C2) of Definition 5 are easy
to check.
Let X be a nonempty set. With the definitions of Example 5, the following spaces can
be recognized as examples of conlinear spaces with respect to the corresponding pointwise
operations by means of Proposition 9: Since {0, +∞} and {0, −∞} are conlinear, the sets
I+
(X) := R (X, {0, +∞}) and I−
(X) := R (X, {0, −∞})
can be supplied with a conlinear structure according to (2.1) and (2.2). The same is true
for
R+
(X) := R (X, IR ∪ {+∞}) and R−
(X) := R (X, IR ∪ {−∞})
since IR ∪ {+∞} and IR ∪ {−∞} are conlinear and also for
RM
(X) := R X, IRM

and RO
(X) := R X, IRO

since IRM and IRO are conlinear.

Remark 1 Let f ∈ R+ (X). Then there are fV ∈ V (X) := {f ∈ R+ (X) : f (X) ⊆ IR}
and fI+ ∈ I+ (X) such that f = fV ⊕fI+ , f coincides with fV on dom f := {x ∈ X : f (x) ∈ IR}
and fI+ is uniquely determined, namely, fI+ = I+
dom f .
Both of (I+ (X) , ⊕) and (V (X) , ⊕) are conlinear spaces, the latter one is even linear
(see Definition 9 below).
Of course, an analogous consideration can be done for R− (X).
As it is the case for monoids, the property of being a conlinear space is stable under passing
to power sets. We define the product of α ≥ 0 and M ∈ P (Y ) by αM := {αy : y ∈ M}.
Concerning b
P (Y ), we define α · ∅ = ∅ for α 0 and 0 · ∅ = {θ}.
Proposition 10 Let (Y, +) be a conlinear space. Then (P (Y ) , ⊕) and

b
P (Y ) , ⊕

are
conlinear spaces as well.
Proof. We know from Proposition 1 that (P (Y ) , ⊕) and

b
P (Y ) , ⊕

are commutative
monoids with neutral element Θ = {θ}, hence axiom (C1) is satisfied. The properties (C2,
(i)) to (C2, (iv)) are easy to check.
Definition 6 Let (Y, +) be a conlinear space. An element y ∈ Y is said to be a cone iff
∀t 0 : ty = y.
A cone y 6= θ is called nontrivial. The set of all cones of Y is denoted by Yc.
This definition looks somehow unusual. Setting Y = P (IRn) for example, we rediscover
cones as subsets of the linear space IRn, see [105], p. 13. There are further objects being
cones in the sense of the above definition. For example, +∞ is a cone of IRM, compare
(ii) of Example 5. Note that a cone of a conlinear space is not necessarily an idempotent
element of the underlying monoid. This is since 2y 6= y + y in general.
Proposition 11 Let (Y, +) be a conlinear space. Then (Yc, +) is a conlinear space as
well.
Proof. It already suffices to show that y1, y2 ∈ Yc implies y1 + y2 ∈ Yc. This follows by
(C2, iv).
If y ∈ Y is a cone, then {y} ∈ P (Y ) is a cone, too.
The concept of a conlinear space is sufficient to define convexity. In fact, it seems
to be the natural framework for convexity rather than linear spaces. Here, we only give
the definition of convex elements and convex subsets of a conlinear space as well as some
elementary facts.

Definition 7 Let (Y, +) be a conlinear space. An element y ∈ Y is said to be convex iff
∀t1, t2 0 : (t1 + t2) y = t1y + t2y. (2.3)
The set of all convex elements of Y is denoted by Yco.
A subset M ⊂ Y is called a convex subset of Y iff
∀t ∈ (0, 1) : tM ⊕ (1 − t) M ⊆ M. (2.4)
The set of all nonempty convex subsets of Y is denoted by Co (Y ).
Of course, θ ∈ Y is always a convex element.
Proposition 12 Let (Y, +) be a conlinear space. Then (Yco, +), (Co (Y ) , ⊕) and

c
Co (Y ) , ⊕

are conlinear spaces as well. Thereby, c
Co (Y ) = Co (Y ) ∪ {∅}.
Proof. Concerning Yco, it suffices to show that ty, y1 + y2 ∈ Yco whenever y, y1, y2 ∈ Yco
and t ≥ 0. This is straightforward using (C2, i, iv) and (2.3).
Concerning Co (Y ), we have to show that M, M1, M2 ∈ Co (Y ) implies tM ∈ Co (Y )
whenever t ≥ 0 and M1 ⊕ M2 ∈ Co (Y ). This is straightforward as well as to check the
axioms (C1), (C2).
The extension to c
Co (Y ) is obvious.
Note that Yco is a conlinear subspace of (Y, +), while Co (Y ) and

c
Co (Y ) , ⊕

are conlinear
subspaces of (P (Y ) , ⊕).
Putting Proposition 11 and 12 together, the following result is obtained.
Proposition 13 Let (Y, +) be a conlinear space. Then (Yc ∩ Yco, +) is a conlinear space
as well.
Proof. Immediately by Propositions 11 and 12.
The following two propositions answer the question for the relationships between convex
subsets of (Y, +) and convex elements of (P (Y ) , ⊕). In our general framework, the situ-
ation is a bit more complicated than in the linear case, i.e. Y = V is a linear space (see
[105], Theorem 3.2.). This is due to the fact that a convex subset of a conlinear space may
contain nonconvex elements.
Proposition 14 Let (Y, +) be a conlinear space. Then, every convex element of (P (Y ) , ⊕)
is a convex subset of (Y, +).
Proof. Let M ⊆ Y be a convex element of (P (Y ) , ⊕), i.e., for all t1, t2 0,
(t1 + t2) M = t1M ⊕ t2M. (2.5)

We have to show (2.4). Take t ∈ (0, 1). Set t1 = t, t2 = 1 − t. By (2.5), we have
tM ⊕ (1 − t) M ⊆ M.
The most simple condition for a convex subset M ⊆ Y to be a convex element of (P (Y ) , ⊕)
is of course M ⊆ tM ⊕ (1 − t) M whenever t ∈ [0, 1], or equivalently, (t1 + t2) M =
t1M +t2M whenever t1, t2 0. An important special case gives the following proposition.
Proposition 15 Let (Y, +) be a conlinear space. Then, a convex subset M ⊆ Y contain-
ing only convex elements is a convex element of (P (Y ) , ⊕).
Proof. Let M ⊆ Y be a convex subset consisting of convex elements only. We have to
show that (2.3) holds true. By (2.4), we have tM ⊕ (1 − t) M ⊆ M for t ∈ (0, 1). Since
t1 + t2 0, we can replace t by t1
t1+t2
and multiply by t1 + t2. This gives t1M ⊕ t2M ⊆
(t1 + t2) M. Conversely, take y ∈ M. Then (t1 + t2) y = t1y + t2y, since M consists of
convex elements only. Hence (t1 + t2) M ⊆ t1M ⊕ t2M completing the proof.
Remark 2 There are subsets of conlinear spaces satisfying (2.4) but do not consist of
convex elements only. Moreover, a convex subset of a conlinear space Y is not necessarily
a convex element of P (Y ).
For example, take Y = P (IR) and M = P ([0, 1]). M is a convex subset of Y , but
neither it consists only of convex elements nor is it a convex element of P (Y ). Observe
that for y := {0, 1} ∈ M we do not have y ∈ 1
2 M ⊕ 1
2 M, hence M 6= 1
2 M ⊕ 1
2 M. To see
this, assume y = 1
2 y1 + 1
2 y2, y1, y2 ∈ M. Then 1 ∈ y1, y2 as well as 0 ∈ y1, y2. This implies
1
2 ∈ y, a contradiction.
Some important facts about convex subsets of conlinear spaces carry over from the linear
theory. Compare [105], §2.
Theorem 1 The intersection of an arbitrary collection of convex subsets of a conlinear
space is a convex subset.
Proof. Elementary.
Again, there is an additional assumption necessary for convex elements of P (Y ).
Corollary 1 Let (Y, +) be a conlinear space. Let Mα ⊆ Y , α ∈ A be a family of convex
subsets of (P (Y ) , ⊕). If the intersection
M :=

α∈A
Mα
contains only convex elements of Y , then M is a convex element of (P (Y ) , ⊕).

Proof. By Theorem 1, M is a convex subset. Since M contains only convex elements by
assumption, Proposition 15 gives the result.
Let n be a positive integer. We call a sum
t1y1 + t2y2 + . . . + tnyn
a convex combination of the elements yi ∈ Y , i = 1, . . . , n, whenever ti ≥ 0, i = 1, . . . , n
and
Pn
i=1 ti = 1.
Theorem 2 Let (Y, +) be a conlinear space. A subset M ⊆ Y is a convex subset if and
only if it contains all the convex combinations of its elements.
Proof. The if-part is obvious, the only-if-part by induction.
Definition 8 Let (Y, +) be a conlinear space and M ⊆ Y a subset. The convex hull
co M of M is the intersection of all convex subsets of Y containing M.
By Theorem 1, co M is always a convex subset of Y .
Theorem 3 Let (Y, +) be a conlinear space and M ⊆ Y a subset. Then co M coincides
with the set of all convex combinations of elements of M.
Proof. By Theorem 2, the set of all convex combination of elements of M is contained
in co M. Conversely, let
u =
n
X
i=1
tiui, v =
m
X
j=1
sjvj
convex combinations of elements ui, vj ∈ M. Take t ∈ (0, 1). Then
y := tu + (1 − t) v =
n
X
i=1
(tti) ui +
m
X
j=1
((1 − t) sj) vj
is a convex combination of elements of M, too. Hence the set of all convex combinations
of elements of M is a convex subset and contains M. Hence it coincides with co M.
Corollary 2 Let (Y, +) be a conlinear space. Then y ∈ Y is a convex element if and only
if
co {y} = {y} .
Proof. If y is a convex element, then every convex combination of y with itself coincides
with y. Conversely, we have for t ∈ (0, 1)
y = ty + (1 − t) y.
Let t1, t2 0. Substituting t = t1
t1+t2
and multiplying by t1 + t2 we obtain
(t1 + t2) y = t1y + t2y
as desired.

Remark 3 The convex hull of {y} may happen to contain more than one element. In
general,
co {y} =
( n
X
i=1
tiy : ti ≥ 0,
n
X
i=1
ti = 1, n ∈ IN, n ≥ 1
)
.
A convex element of the conlinear space Y being a cone at the same time is called a
convex cone in Y .
Proposition 16 Let (Y, +) be a conlinear space. A cone y ∈ Y is a convex element if
and only if y + y = y, i.e. y is an idempotent element of the monoid constituting Y .
Proof. (1) Let y ∈ Y be a cone and a convex element. Then for t1 = t2 = 1 we obtain
from (2.3) y = 2y = y + y.
(2) Let y ∈ Y be a convex cone. For t1, t2 0 equality (2.3) reduces to y = y.
It turns out that a convex cone of P (Y ) is itself a conlinear space if it contains θ ∈ Y .
Hence, the terms conlinear subspace of Y and convex cone of P (Y ) containing θ ∈ Y are
synonyms in the framework of conlinear spaces.
Moreover, a cone in P (Y ) being a convex subset of Y is almost a convex element.
Proposition 17 Let C ∈ P (Y ) be a cone containing θ ∈ Y . Then C is a convex element
of P (Y ) if and only if it is a convex subset of Y .
Proof. Since every convex element of P (Y ) is a convex subset of Y (Proposition 14),
it remains to show the converse. The cone property and (2.4) for t ∈ (0, 1) imply tC ⊕
(1 − t) C = C ⊕ C ⊆ C. Since θ ∈ C, we have C ⊆ C ⊕ C, hence C = C ⊕ C. Proposition
16 gives the result.
Example 6 Set Y = P (IR), C = P (IR+) {0}. Then C is a convex subset of Y , a cone,
but not a convex element of P (Y ). To see this, take c = {0, 1} ∈ C and assume c = c1+c2,
c1, c2 ∈ C. Then 0 ∈ c1, c2 and γ1 ∈ c1, γ2 ∈ c2 such that γ1, γ2 ≥ 0, γ1 + γ2 = 1. Hence
c = {0, 1} = c1 +c2 ⊇ {0, γ1, γ2, γ1 + γ2}. Without loss of generality, we must have γ1 = 0,
γ2 = 1. This implies c1 = {0} which is not possible.
Let (Y, +) be conlinear space. According to Definition 2, we denote the set of invertible
elements of Y with respect to + by Yin. We finish this section by defining a linear space.
Definition 9 A conlinear space (Y, +) is said to be a (real) linear space iff it consists
only of elements being convex and invertible at the same time.
We shall show that this definition is consistent with the usual definition of a linear (vector)
space. A definition of linear spaces can be found e.g. in [75] vol. I, §. We state the fact
in a more convenient form.

Theorem 4 Let (Y, +) be a conlinear space and Yl := Yin ∩ Yco ⊆ Y . Then (Yl, +) is a
linear space, and it is the largest one contained in Y .
Proof. For y ∈ Yl, we define a multiplication with negative reals by
(−α) y := αy0
where α 0 and y+y0 = θ. It remains to show the following properties for all y, y1, y2 ∈ Yl,
α, β ∈ IR:
(1) (Yl, +) is a commutative group.
(2) α (y1 + y2) = αy1 + αy2.
(3) (α + β) y = αy + βy.
(4) α (βy) = (αβ) y.
Let’s start with (1). We have to show that y1, y2 ∈ Yl implies y1 + y2 ∈ Yl. Since the set
of all invertible elements of a monoid forms a group, y1 + y2 is invertible. Since inverse
elements in groups are unique, we have (y1 + y2) + (y0
1 + y0
2) = (y1 + y0
1) + (y2 + y0
2) = θ,
hence (y1 + y2)0
= y0
1 + y0
2. Applying (C2, iv) the convexity of y1, y2 implies
(α + β) (y1 + y2)
(C2, iv)
= (α + β) y1 + (α + β) y2
y1,y2 convex
= αy1 + βy1 + αy2 + βy2
(C1),(C2, iv)
= α (y1 + y2) + β (y1 + y2) ,
hence y1 + y2 is convex. Hence (Yl, +) is a commutative group.
(2) has to be proven for α 0. This follows from (1) and the convexity of the y0
1, y0
2.
(3) is obvious for α, β 0 and α, β 0. Without loss of generality, consider the case
α 0, β 0 and α + β 0. Then
(α + β) y = βy + βy0
+ (−1) (α + β) y0
β,−(α+β)0
= βy + (β + (−1) (α + β)) y0
= (−1) αy0
+ βy
= αy + βy.
(4) can be proven by a case study with respect to α, β. Exemplary, we check the case
α 0, β 0. Then
α (βy) = α (− |β| y) = α |β| y0

(C2, i)
= (α |β|) y0
= (αβ) y.
The set Yl := Yin ∩ Yco is called the lineality space of Y .
Corollary 3 A conlinear space (Y, +) is linear if and only if Y = Yco ∩ Yin.

Every element of a linear space V is a convex element, hence every subset of V consists
of convex elements only. Hence a subset M ⊆ V is convex if and only if M is a convex
element of (P (V ) , ⊕). For cones, something more can be said.
Corollary 4 Let (V, +) be a linear space and C ⊆ V a cone of (P (V ) , ⊕). Then the
following facts are equivalent:
(i) C is a convex element of (P (V ) , ⊕).
(ii) C is a convex subset of (V, +).
(iii) C ⊕ C ⊆ C.
Proof. The equivalence of (i) and (ii) is clear from the remark above. We have C ⊆ C⊕C,
since c ∈ C implies 1
2 c ∈ C and consequently c = 1
2 c + 1
2 c ∈ C ⊕ C. Hence C ⊕ C = C.
The equivalence of (i) and (iii) follows from Proposition 16.
¿From the results above, one may see that every convex subset of a linear space V contain-
ing θ ∈ V and being a cone in (P (V ) , ⊕) is a conlinear space. However, it is not possible
to reduce the investigation of conlinear spaces to convex cones as subsets of linear spaces.
Theorem 5 A conlinear space with a nontrivial cone can not be embedded into a linear
space.
Proof. Let (Y, +) be a conlinear space and (V, +) a linear space such that Y ⊆ V and
+ coincides on Y . Let y ∈ Y , y 6= θ be a nontrivial cone. Then there is v ∈ V , v 6= θ such
that y + v = θ. Since V is linear, we have 2y = y + y and therefore θ = v + y = v + 2y =
v + y + y = y, a contradiction.
Example 7 Let (V, +) be a linear space. (i) Since (V, +) is especially conlinear, by Propo-
sition 10 (P (V ) , ⊕) and

b
P (V ) , ⊕

are conlinear spaces as well.
(ii) The set of all convex cones of (P (V ) , ⊕) containing θ ∈ V form a conlinear space
consisting only of idempotent elements. This follows from Proposition 13.
2.1.3 Semilinear spaces
In the last paragraph, we have seen that a linear space can be understood as the subset of a
conlinear space contaning those elements which are invertible and convex at the same time.
In this case, the definition of negative multiples was possible. Conversely, considering e.g.
the power set of a linear space, it seems to be a natural idea to have a multiplication with
negative real numbers, even though inverse elements with respect to the addition do not
exists.
Note that not all conlinear spaces admit such an operation. For example, a pointed
convex cone of a linear space is a conlinear space, but does not contain the negative of
any of its elements beside zero.
We call a conlinear space with a (−1)-multiplication a semilinear space. This concept
is very close to that of an almost linear space introduced by G. Godini [42] around 1985.

Definition 10 A set Y , together with an addition +, is said to be a (real) semilinear
space (Y, +) iff the following axioms are satisfied:
(S1) (Y, +) is a commutative monoid with neutral element θ;
(S2) For any two elements y ∈ Y and t ∈ IR there exists the product ty := t · y ∈ Y such
that the following conditions are satisfied:
(i) ∀y ∈ Y, ∀s, t ∈ IR : s (ty) = (st) y;
(ii) ∀y ∈ Y : 1 · y = y;
(iii) ∀y ∈ Y : 0 · y = θ;
(iv) ∀t ∈ IR, ∀y1, y2 ∈ Y : t (y1 + y2) = (ty1) + (ty2).
Again, the second distributive law (s + t) y = (sy)+(ty) does not hold in general, not even
for nonnegative numbers. This is a difference to Godini’s almost linear spaces [42]. The
second distributive law is not valid for the power set of a linear space being a semilinear
but not an almost linear space. The following properties can be proven in the same way
as Proposition 8.
Proposition 18 Let (Y, +) be a semilinear space. Then:
(i) If t ∈ IR, then tθ = θ.
(ii) If t ∈ IR {0}, y1, y2 ∈ Y and ty1 = ty2, then y1 = y2.
(iii) If y ∈ Y {θ} and t ∈ IR {0}, then ty 6= θ.
Starting from a semilinear space we are able to generate new semilinear spaces by passing
to power sets.
Proposition 19 Let (Y, +) be a semilinear space. Defining the product of t ∈ IR and
M ∈ P (Y ) by tM := {ty : y ∈ M} and agreeing on t · ∅ = ∅ for t 6= 0 and 0 · ∅ = {θ}, the
spaces (P (Y ) , ⊕) and

b
P (Y ) , ⊕

are semilinear spaces as well.
Proof. We know from Proposition 1 that (P (Y ) , ⊕) and

b
P (Y ) , ⊕

are commutative
monoids with neutral element Θ = {θ}, hence axiom (S1) is satisfied. The properties (S2,
(i)) to (S2, (iv)) are easy to check.
Let (Y, +) be a semilinear space. If Y 0 ⊆ Y and (Y 0, +) is itself a semilinear space with
the same multiplication with real numbers as Y , then it is called a semilinear subspace
of Y . A subset Y 0 ⊆ Y is a semilinear subspace if and only if t ∈ IR, y, y1, y2 ∈ Y 0 imply
ty ∈ Y 0 and y1 + y2 ∈ Y 0.
Let (Y, +) be a semilinear space. We define the set of invertible, convex and symmetric
elements and the set of cones of Y , respectivly, by
Yin :=

y ∈ Y : ∃y0
∈ Y : y + y0
= θ ,
Yco := {y ∈ Y : ∀t1, t2 ≥ 0 : (t1 + t2) y = t1y + t2y} ,
Ysy := {y ∈ Y : y = (−1) y} ,
Yc := {y ∈ Y : ∀t 0 : ty = y} .
As in the case of conlinear spaces, we denote Yl := Yin ∩ Yco.

Proposition 20 Let (Y, +) be a semilinear space. Then (Yc, +), (Yco, +) and (Ysy, +) are
semilinear spaces as well.
Proof. Take y ∈ Yc. Then (−1) y ∈ Yc by (S2, (i)): For t 0, we obtain t (−1) y =
(−1) (ty) = (−1) y. Taking y1, y2 ∈ Yc, by (S2, (iv)) it follows t (y1 + y2) = ty1 + ty2 =
y1 + y2, hence y1 + y2 ∈ Yc. Therefore, (Yc, +) is a semilinear space.
By similar considerations, one can show that (Yco, +) and (Ysy, +) are semilinear spaces,
too.
Proposition 21 Let (Y, +) be a semilinear space. Then, (Yl, +) is a linear subspace of
(Y, +), and it is the largest one contained in Y .
Proof. Every semilinear space is all the more conlinear, hence the result follows by
Theorem 4.
Of course, every linear space is almost linear, every almost linear space is semilinear and
every semilinear space is conlinear. There exist examples showing that these classes do
not coincide. Several examples are listed below.
Example 8 (i) Let (V, +) be a real linear space. Then it is a semilinear space. We only
have to prove that (S1), (S2) imply (C2, (iii)). We omit the proof noting that either the
group property or (S2, (iii)) has to be involved.
(ii) Let (V, +) be a real linear space. Then (P (V ) , ⊕) is a semilinear space as well as

b
P (V ) , ⊕

, (Co (V ) , ⊕) and

c
Co (V ) , ⊕

.
(iii) The spaces IR ∪ {+∞} and R+ (X) from Example 5, (i) are conlinear, but not semi-
linear.
(iv) The space RM (X) of all functions f : X → IRM is a semilinear space as well as
RO (X).
With the aid of topological properties more examples of semilinear (and conlinear) space
may be obtained. Compare Section 2.3.5.
2.2 Order structures
2.2.1 Basic notation
We recall basic order theoretic notation necessary for the following considerations. We
refer to [32], [36] and [130].
Let W be a nonempty set. A binary relation on W is understood to be a subset
R ⊆ W × W. We say that w1 ∈ W is related to w2 ∈ W iff (w1, w2) ∈ R. In this case,
we shortly write w1Rw2. If wRw for all w ∈ W, the relation R is called reflexive. If
w1, w2, w3 ∈ W, w1Rw2 and w2Rw3 implies w1Rw3, the relation R is called transitive. If
w1Rw2, w2Rw1 for w1, w2 ∈ W implies w1 = w2, the relation R is called antisymmetric.

2.2. Order structures 27
Definition 11 Let W be a nonempty set and R a relation on W. R is called a quasiorder
iff it is reflexive and transitive. R is called a partial order iff it is reflexive, transitive
and antisymmetric.
If R is a quasiorder on W, we write w1 w2 instead of w1Rw2 (or (w1, w2) ∈ R) and
speak about the quasiorder . The couple (W, ) is called a quasiordered set.
Definition 12 Let (W, ) be a quasiordered set. The lower (upper) section Sl (w)
(Su (w)) of w ∈ W are defined by
Sl (w) :=

w0
∈ W : w0
w , Su (w) :=

w0
∈ W : w w0
.
The set of minimal (maximal) elements min (W) (max (W)) is defined by
min (W) := {w ∈ W : Sl (w) ⊆ Su (w)} ,
max (W) := {w ∈ W : Su (m) ⊆ Sl (w)} .
Of course, an element w̄ ∈ W is minimal with respect to iff
w ∈ W, w w̄ =⇒ w̄ w.
If is additionally antisymmetric and w̄ ∈ min (W), then w ∈ W, w w̄ even implies
w = w̄. Analogous conditions hold true for maximal elements.
Having a quasiordered set (W, ), by a standard procedure an equivalence relation ∼
can be defined by
w ∼ w0
⇐⇒ w w0
, w0
w.
Denoting
[w] :=

w0
∈ W : w0
∼ w
and
[w]

w0

⇐⇒ ∀w ∈ [w] , w0
∈

w0

: w w0
,
the set of all equivalence classes [W] together with is a partially ordered set. Compare
[36], p. 13 or [32], Satz 3.19 for more details.
A subset M ⊆ W is called bounded from above (below) in W iff there exist an
w ∈ W such that m w (w m) for all m ∈ M. In this case, w is called upper (lower)
bound of M. A supremum (infimum) of M in W is an upper (lower) bound w ∈ W
such that w w0 (w0 w) for any other upper (lower) bound w0 of M in W. We use
sup M and inf M, respectively, to denote a supremum and infimum of M. If (W, ) is
partially ordered, then sup M and inf M, if they exist, are unique.
If for every pair of elements m1, m2 ∈ M ⊆ W there exists an upper (lower) bound in
M, then M is said to be directed upwards (resp. downwards).
The quasiordered set (W, ) is called Dedekind complete iff every nonempty subset
having an upper bound (lower bound) has a supremum (infimum) in W. Note that the

two conditions are not independent: (W, ) is Dedekind complete if and only if every
nonempty subset having an upper bound has a supremum ([130], Theorem 1.4).
The quasiordered set (W, ) is called order complete iff every nonempty subset has
an infimum and a supremum in W.
The quasiordered set (W, ) is called a lattice iff every subset consisting of two points
has an infimum and a supremum in W.
An element w̄ ∈ W is said to be the largest element iff w w̄ for all w ∈ W. The
smallest element is defined analogously. If (W, ) is partially ordered, then the largest
and smallest element, if they exist, are unique.
Remark 4 Let (W, ) be quasiordered. If W has a largest as well as a smallest element,
then it is Dedekind complete if and only if it is order complete. If W is order complete,
then W has a largest as well as a smallest element. Compare [130], p. 3.
2.2.2 Ordered product sets
The following definition deals with subsets of a product set supplied with a quasiorder.
Definition 13 Let X, Y be two nonempty sets and W = X × Y the set of all ordered
pairs (x, y), x ∈ X, y ∈ Y . The quasiorder on W is called partially antisymmetric
(with respect to X) iff for all (x, y) , (x0, y0) ∈ W
(x, y) x0
, y0

, x0
, y0

(x, y) =⇒ x = x0
.
It is clear that if is a partially antisymmetric quasiorder on W, then a point w̄ = (x̄, ȳ) ∈
W is a minimal point with respect to if and only if
(x, y) ∈ W, (x, y) (x̄, ȳ) =⇒ x = x̄ and (x̄, ȳ) (x, y) .
In some cases, the y-component is not of interest. Therefore, we give the following defini-
tion.
Definition 14 Let X, Y be two nonempty sets and a partially antisymmetric quasiorder
on W = X × Y . A point w̄ = (x̄, ȳ) ∈ W is called a partial minimal point of W iff
(x, y) ∈ W, (x, y) (x̄, ȳ) =⇒ x = x̄.
Analogously, partial maximal points are defined.
Of course, if is a partially antisymmetric quasiorder on W = X ×Y , then every minimal
point of W is also a partial minimal point while the converse is not true in general.

2.2.3 Power sets of ordered sets
Let (W, ) be quasiordered. We extend the ordering to the set b
P (W), the set of all
subsets of W including the empty set, by defining
M1 4 M2 :⇐⇒ ∀m2 ∈ M2 ∃m1 ∈ M1 : m1 m2 (2.6)
M1 2 M2 :⇐⇒ ∀m1 ∈ M1 ∃m2 ∈ M2 : m1 m2 (2.7)
for M1, M2 ∈ P (W). If M2 ⊆ M1, then M1 4 M2 and M2 2 M1 by reflexivity of .
Observe that W 4 M and M 2 W for each M ∈ P (W), i.e. W is the smallest element
for 4 and the largest for 2. Setting M1 = M, M2 = ∅ in (2.6) and M1 = ∅, M2 = M in
(2.7) we may find
∀M ∈ b
P (W) : M 4 ∅, ∅ 2 M. (2.8)
This means, ∅ is the largest element for 4 and the smallest for 2. Note that for Mi = {wi},
wi ∈ W for i = 1, 2, we have
M1 4 M2 ⇔ M1 2 M2 ⇔ w1 w2,
i.e., the ordering relations 4 and 2 can be understood to be extensions of to b
P (W). In
fact, they are quasiorders.
Proposition 22 Let (W, ) be a quasiordered set. Then (P (W) , 4), (P (W) , 2),

b
P (W) , 4

and

b
P (W) , 2

are quasiordered as well.
Proof. Reflexivity and transitivity of 4 and 2 on b
P (W) follow immediately from (2.6),
(2.7) and (2.8).
Note that neither 4 nor 2 are partial orders in general, not even if is antisymmetric.
One can easy construct counterexamples for W = IR1, ≤

. However, if we start with
(W, =), we obtain (P (W) , ⊇) and (P (W) , ⊆) being partial orders.
The next result contains formulas for infima and suprema in b
P (W) with respect to 4
and 2.
Theorem 6 Let (W, ) be quasiordered. Then:
(i) (P (W) , 4) is Dedekind complete. If M ⊆ P (W) is nonempty, then it is bounded
below and
I∗
:=
[
M∈M
[
m∈M
{w ∈ W : m w} (2.9)
is an infimum of M with respect to 4. If M ⊆ P (W) is nonempty and bounded above,
then the set
S∗
:=

M∈M
[
m∈M
{w ∈ W : m w} (2.10)

is a supremum of M.
(ii) (P (W) , 2) is Dedekind complete. If M ⊆ P (W) is nonempty and bounded below,
then the set
I
:=

M∈M
[
m∈M
{w ∈ W : w m} (2.11)
is an infimum of M. If M ⊆ P (W) is nonempty, then M is bounded above and the set
S
:=
[
M∈M
[
m∈M
{w ∈ W : w m} (2.12)
is a supremum of M.
(iii)

b
P (W) , 4

is order complete. If M ⊆ b
P (W) is nonempty, then I∗ from (2.9) is an
infimum of M and S∗ from (2.10) is a supremum of M. If ∅ is the only upper bound of
M, then S∗ = ∅.
(iv)

b
P (W) , 2

is order complete. If M ⊆ b
P (W) is nonempty, then I from (2.11) is
an infimum of M and S from (2.12) is a supremum of M. If ∅ is the only lower bound
of M, then I = ∅.
Proof. (i) Let M ⊆ P (W) be nonempty. Then M is bounded below by ∅ 6= M :=
S
M∈M M since for each M ∈ M we have M ⊆ M and this implies M 4 M. Moreover,
we have M ⊆ I∗ implying that I∗ is a lower bound of M. It remains to show that N 4 I∗
for any other lower bound N of M. To see this, take w ∈ I∗. By definition of I∗, there
is m ∈ M such that m w. Since N is a lower bound of M, there is n ∈ N such that
n m w. Hence, for each w ∈ I∗ there is n ∈ N such that n w, i.e. N 4 I∗.
Now, let M ⊆ P (W) be nonempty and bounded above with respect to 4 by N ∈
P (W). Since M 4 N for all M ∈ M we have
∀M ∈ M : ∀n ∈ N ∃m ∈ M : m n,
hence N ⊆ S∗. Hence S∗ is nonempty and S∗ 4 N. On the other hand, for M ∈ M the
definition of S∗ implies
∀w ∈ S∗
∃m ∈ M : m w,
hence M 4 S∗ for all M ∈ M. This proves that S∗ is a supremum of M.
(ii) By similar arguments as used for the proof of (i).
(iii) According to Remark 4,

b
P (W) , 4

is order complete if and only if it contains a
largest as well as a smallest element. This is true since
∀M ∈ b
P (W) : W 4 M 4 ∅.
Formulas (2.9) and (2.10) remain true: (2.9) yields I∗ = ∅ if ∅ is the only member of M
and (2.10) yields S∗ = ∅ if ∅ ∈ M.
Finally, let ∅ be the only upper bound of M. Assume w ∈ S∗ for some w ∈ W. The
definition of S∗ gives that N = {w} is an upper bound of M with respect to 4. This is a
contradiction, hence S∗ must be empty.
(iv) The proof runs analogous to that of (iii).

Remark 5 Let M ⊆ b
P (W) be given and define
M :=
[
M∈M
M.
(i) The set M is an infimum of M with respect to 4, i.e. I∗ 4 M 4 I∗ holds true.
Since I∗ is an infimum and M a lower bound of M, certainly M 4 I∗ holds. On the other
hand, M ⊆ I∗, hence I∗ 4 M.
(ii) The set M is a supremum of M with respect to 2, i.e. S 2 M 2 S holds true.
Since S is a supremum and M an upper bound of M, certainly S 2 M holds. On the
other hand, M ⊆ S, hence M 2 S.
Proposition 23 Let (W, ) be quasiordered and M ⊆ b
P (W) be given.
If I ∈ b
P (W) is an infimum of M with respect to 4 (with respect to 2), then I ⊆ I∗
(I ⊆ I) holds true.
If S ∈ b
P (W) is a supremum of M with respect to 4 (with respect to 2), then S ⊆ S∗
(S ⊆ S) holds true.
Proof. Let I be an infimum of M with respect to 4. Take w0 ∈ I. Since I∗ 4 I, there
is w ∈ I∗ such that w w0. The definition of I∗ implies
∃M ∈ M ∃m ∈ M : m w.
The transitivity of implies m w0 for all these m’s, hence w0 ∈ I∗.
Let I be an infimum of M with respect to 2. Take w0 ∈ I. Since I 2 I, there is
w ∈ I such that w0 w. The definition of I implies
∀M ∈ M ∃m ∈ M : w m.
The transitivity of implies w0 m for all these m’s, hence w0 ∈ I.
The proofs for the suprema run analogously.
The preceding result shows that the infima and suprema from Theorem 6 are the largest
ones in the sense of set inclusion. The question arises how one can shrink these sets as
much as possible. It turns out that the sets of minimal and maximal points, respectively,
of the largest infima and suprema are good candidates.
In the following two theorems some relationships are established between the infimum
(supremum) of a subset M ⊆ b
P (W) with respect to 4 and 2 on one hand and the set of
minimal points of I∗ (S∗) and maximal points of I (S) with respect to on the other
hand, respectively.
To state the result, we recall the so called domination condition. This concept plays
an important role in vector optimization. Compare the book of Luc [85], [44] and the
references therein.

Definition 15 Let (W, ) be quasiordered. A subset M ⊆ W is said to satisfy the lower
domination condition iff
∀m ∈ M ∃n ∈ min (M) : n m.
A subset M ⊆ W is said to satisfy the upper domination condition iff
∀m ∈ M ∃n ∈ max (M) : m n.
For the sake of simplicity, we state the result for partial orders.
Theorem 7 Let (W, ) be partially ordered and M ⊆ b
P (W) be given.
(i) Let I∗ ⊆ b
P (W) be the set of all infima of M with respect to 4. Then
min (I∗
) =

I∈I∗
I.
The set I∗ satisfies the lower domination condition if and only if min (I∗) ∈ I∗. In this
case, min (I∗) is the smallest set being an infimum of M with respect to 4.
(ii) Let S∗ ⊆ b
P (W) be the set of all suprema of M with respect to 4. Then
min (S∗
) =

S∈S∗
S.
The set S∗ satisfies the lower domination condition if and only if min (S∗) ∈ S∗. In this
case, min (S∗) is the smallest set being a supremum of M with respect to 4.
Proof. (i) Recall that I∗ =
S
M∈M
S
m∈M {w ∈ W : m w}, compare (2.9).
First, we show that min (I∗) ⊆ I for each I ∈ I∗. Take m ∈ min (I∗) ⊆ I∗. Since
I 4 I∗, there is w ∈ I such that w m. Since I∗ 4 I, there is m0 ∈ I∗ such that m0 w.
Since is transitive, we get m0 m and since m is minimal in I∗ and is antisymmetric,
this implies m0 = w = m. Hence m ∈ I. Thus, we have proved that min (I∗) ⊆
T
I∈I∗ I.
To show the converse inclusion, take w ∈
T
I∈I∗ I and assume w 6∈ min (I∗). Then
there must exist a w̄ ∈ I∗ such that w̄ w and w̄ 6= w. For I ∈ I∗ consider the set
I0
:= I {w} ∪ {w̄} .
Then
∀w ∈ I ∃w0
∈ I0
: w0
w,
hence I0 4 I 4 I∗. On the other hand, since I0 ⊆ I∗, we have I∗ 4 I0. Hence I∗ 4 I0 4 I∗,
i.e. I0 ∈ I∗. But w 6∈ I0, a contradiction.
Since min (I∗) ⊆ I∗, we have I∗ 4 min (I∗). The lower domination conditions is
equivalent to min (I∗) 4 I∗, hence min (I∗) is an infimum of M with respect to 4.
(ii) By similar arguments.
Note that min (I∗) = min (I) for every I ∈ I∗. Since M :=
S
M∈M M ∈ I∗, it might be

a good idea to look for minimal points of the union M. This is the underlying idea of
set valued optimization in the sense of Corley [20], Jahn [63] and many others since the
middle of the 80ies. Theorem 7 tells us, among other things, that looking for minimal
points of M yields a subset of an infimum with respect to 4.
On the other hand, the set min (S∗) is not contained in the union M in general. There
are easy to construct examples in IR2 with even min (S∗) ∩ M = ∅.
The following corollary pays special attention to the case when the set M consists only
of singletons. We obtain relationships between the set of minimal elements of a subset
M ⊆ W and the infimum with respect to 4 on one hand and the supremum with respect
to and the supremum with respect to 4 on the other hand.
We denote by sup (M) the set of suprema of M in W with respect to whereas
min (M) ⊆ M is the set of minimal points of M, compare Section 2.2.1. Assuming (W, )
to be a partially ordered set, sup (M) is empty or consists of a single point.
Corollary 5 Let (W, ) be partially ordered and ∅ 6= M ⊆ W. Considering
M := {{m} : m ∈ M} ⊆ P (W)
the following assertions hold true:
(i) The set
I∗
:=
[
m∈M
{w ∈ W : m w}
is an infimum of M with respect to 4. The set min (I∗) = min (M) is contained in every
infimum of M with respect to 4 and is itself an infimum if and only if M satisfies the
lower domination condition.
(ii) The set
S∗
:=

m∈M
{w ∈ W : m w}
is a supremum of M with respect to 4. If sup (M) ∈ W exists, then S∗ = {w ∈ W : sup (M) w}
and min S∗ = {sup (M)}.
Proof. (i) I∗ is an infimum of M with respect to 4 by Theorem 6, (i). The remaining
part follows from Theorem 7, (i).
(ii) S∗ is a supremum of M with respect to 4 by Theorem 6, (i). Moreover, S∗ =
{w ∈ W : sup (M) w}, since S∗ contains by definition all upper bounds of M with
respect to and sup (M) is the smallest upper bound by definition.
Theorem 8 Let (W, ) be partially ordered and M ⊆ b
P (W) be given.
(i) Let I ⊆ b
P (W) be the set of all infima of M with respect to 2. Then
max (I
) =

I∈I
I.

The set I satisfies the upper domination condition if and only if max (I) ∈ I. In this
case, it is the smallest set being an infimum of M with respect to 2.
(ii) Let S ⊆ b
P (W) be the set of all suprema of M with respect to 2. Then
max (S
) =

S∈S
S.
The set S satisfies the upper domination condition if and only if max (S) ∈ S. In this
case, it is the smallest set being a supremum of M with respect to 2.
Proof. (i) Recall that I =
T
M∈M
S
m∈M {w ∈ W : w m}, compare (2.11).
First, we show that max (I) ⊆ I for each I ∈ I. Take m ∈ max (I) ⊆ I. Since
I 2 I, there is w ∈ I such that m w. Since I 2 I, there is m0 ∈ I such that w m0.
Hence m m0 and therefore m = w = m0 since m is maximal in I and is transitive
and antisymmetric. Hence m ∈ I as desired.
To show the converse inclusion, take w ∈
T
I∈I I and assume w 6∈ max (I). Then
there must exist a w̄ ∈ I such that w w̄ and w 6= w̄. For I ∈ I consider the set
I0
:= I {w} ∪ {w̄} .
Then I 2 I 2 I0 by construction of I0. On the other hand, I0 2 I since I0 ⊆ I. Hence
I0 ∈ I and w 6∈ I0, a contradiction.
Since max (I) ⊆ I, we have max (I) 2 I. The upper domination conditions requires
I 2 max (I), hence max (I) is an infimum of M with respect to 2.
(ii) By similar arguments.
The notes after the proof of Theorem 7 apply analogously with reversed roles of infimum
and supremum.
Parallel to Corollary 5 we have the following result. Here, we denote by inf (M) the set
of infima of M in W with respect to whereas max (M) ⊆ M is the set of maximal points
of M, compare Section 2.2.1. Assuming (W, ) to be a partially ordered set, inf (M) is
empty or consists of a single point.
Corollary 6 Let (W, ) be partially ordered and ∅ 6= M ⊆ W. Considering
M := {{m} : m ∈ M} ⊆ P (W)
the following assertions hold true:
(i) The set
I
:=

m∈M
{w ∈ W : w m}
is an infimum of M with respect to 2. If inf (M) ∈ W exists, then I = {w ∈ W : w inf (M)}
and max I = {inf (M)}.
(ii) The set
S
:=
[
m∈M
{w ∈ W : w m}

is a supremum of M with respect to 2. The set max (S) = max (M) is contained in every
supemum of M with respect to 2 and is itself an supremum if and only if M satisfies the
upper domination condition.
Proof. Similar to the proof of Corollary 5.
Example 9 Consider M = {M1, M2} ⊂ P (IR) with M1 = [0, 1] and M2 = [−1, 1000].
Then I∗ = [−1, +∞) and I = (−∞, 1]. If we interpret the numbers contained in M1,
M2 as the possible financial loss we have to expect choosing M1 and M2, respectively, one
might see that it is sometimes better to prefer M2 to M1, i.e. to deal with 2 instead of 4
or instead of simply to look for minimal points in the union.
Since 4 and 2 are only quasiorders in general, one might ask for the structure of the
corresponding partially ordered sets of equivalence classes on b
P (W). They are introduced
as follows (compare Section 2.2.1):
M1
∗
∼ M2 :⇐⇒ M1 4 M2 4 M1; (2.13)
M1

∼ M2 :⇐⇒ M1 2 M2 2 M1 (2.14)
for M1, M2 ∈ b
P (W). The empty set ∅ is equivalent only to itself. For M ∈ b
P (W) we set
[M]∗
:=
n
M0
∈ b
P (W) : M0 ∗
∼ M
o
, M
∗
:=
[
M0∈[M]∗
M0
, (2.15)
[M]
:=
n
M0
∈ b
P (W) : M0
∼ M
o
, M

:=
[
M0∈[M]
M0
. (2.16)
The order relations on the set of equivalence classes have to be defined by
[M1]∗
4 [M2]∗
:⇐⇒ ∀M0
1 ∈ [M1]∗
, M0
2 ∈ [M2]∗
: M0
1 4 M0
2; (2.17)
[M1]
2 [M2]
:⇐⇒ ∀M0
1 ∈ [M1]
, M0
2 ∈ [M2]
: M0
1 2 M0
2. (2.18)
This leads to the following relationships containing ∅:
∀M ∈ b
P (W) : [M] 4 [∅] , [∅] 2 [M] .
The following theorem tells us that the resulting partial order on the set of equivalence
classes can be identified in a sense with ⊇ and ⊆, respectively.
(i) M
∗
= {w ∈ W : ∃m ∈ M : m w}. Moreover, M0 ∗
∼ M
∗
for each M0 ∈ [M]∗
and
[M1]∗
4 [M2]∗
⇐⇒ M1
∗
⊇ M2
∗
. (2.19)
(ii) M

= {w ∈ W : ∃m ∈ M : w m}. Moreover, M0
∼ M

for each M0 ∈ [M]
and
[M1]
2 [M2]
⇐⇒ M1

⊆ M2

. (2.20)

Proof. (i) Denote for the moment M̃∗ = {w ∈ W : ∃m ∈ M : m w}. First, we show
M̃∗ ⊆ M
∗
. Take w̃ ∈ M̃∗ and set M0 := M ∪ {w̃}. Then M0 ∗
∼ M, hence w̃ ∈ M
∗
.
Conversely, we have M
∗
⊆ M̃∗, since w̄ ∈ M
∗
implies the existence of M0 ∈ [M]∗
such
that w̄ ∈ M0. Since M 4 M0, there exists m ∈ M such that m w̄. Hence w̄ ∈ M̃∗.
For the second assertion, it suffices to show M
∗
∼ M
∗
. This is true since M ⊆ M
∗
and
each m ∈ M
∗
belongs to some M0 ∗
∼ M.
It remains to show (2.19). Let [M1]∗
4 [M2]∗
and take m2 ∈ M2
∗
. Since M1
∗
4 M2
∗
,
there is m1 ∈ M1
∗
such that m1 m2. This implies M1
∗
4 M1
∗
∪ {m2}. Since M1
∗
⊆
M1
∗
∪ {m2}, we also have M1
∗
∪ {m2} 4 M1
∗
. Hence M1
∗
∪ {m2} ∈ [M1]∗
implying
m2 ∈ M1
∗
.
Conversely, take M0
2 ∈ [M2]∗
. Then M0
2 ⊆ M2
∗
⊆ M1
∗
. This implies M1
∗
4 M0
2, hence
we have M1 4 M0
2 for each M1 ∈ [M1]∗
.
(ii) The proof for 2 goes analogously.
Remark 6 Let M ∈ b
P (W) and consider M∗ := [M]∗
. Then (M∗, ⊇) is partially ordered
and M
∗
from above is the unique infimum of M∗ that belongs itself to M∗. Hence it is
the smallest element of M∗ with respect to ⊇. If the intersection
T
M0∈[M]∗ M0 belongs to
M∗, then it is the largest element of M∗. However, this is not the case in general.
Of course, every M0 ∈ M∗ is an infimum of M∗ with respect to 4. Applying Theorem
7 to M∗, we get I∗ = M
∗
and the lower domination condition for M
∗
is necessary and
sufficient for min

M
∗

=
T
M0∈[M]∗ M0 being the largest element in (M∗, ⊇).
Analogously, M

is the largest element of M := [M]
in (M, ⊆) being partially
ordered as well. The smallest element is max

M

=
T
M0∈[M] M0 if and only if M

satisfies the upper domination property according to Theorem 8.
Let (W, ) be quasiordered. Set
h
c
W
i∗
:=
n
[M]∗
: M ∈ b
P (W)
o
,
h
c
W
i
:=
n
[M]
: M ∈ b
P (W)
o
.
Proposition 24 Let (W, ) a quasiordered. Then
h
c
W
i∗
, 4

and
h
c
W
i
, 2

are par-
tially ordered.
Proof. This follows from using the standard procedure as described in Section 2.2.1.
The next theorem gives formulas for the supremum and infimum in the set of equivalence
classes
h
c
W
i∗
and
h
c
W
i
.
(i)
h
c
W
i∗
, 4

is order complete. Let [M]∗
⊆
h
c
W
i∗
be nonempty and define M∗ :=
n
M
∗
: [M]∗
∈ [M∗]
o
⊆ b
P (W). Then
I∗
=
[
M
∗
∈M∗
[
m∈M
∗
{w ∈ W : m w}

is an infimum of M∗ and [I∗]∗
is an infimum of [M]∗
. The set
S∗
=

M
∗
∈M∗
[
m∈M
∗
{w ∈ W : m w}
is a supremum of M∗ and [S∗]∗
is a supremum of [M]∗
.
(ii)
h
c
W
i
, 2

is order complete. Let [M]
⊆
h
c
W
i
be nonempty and define M :=
n
M

: [M]
∈ [M]
o
⊆ b
P (W). Then
I
=

M

∈M
[
m∈M

{w ∈ W : w m}
is an infimum of M and [I]
is an infimum of [M]
. The set
S
=
[
M

∈M
[
m∈M

{w ∈ W : w m}
is a supremum of M and [S]
is a supremum of [M]
.
Proof. (i) This part follows from Theorem 6, (iii) and Theorem 9, (i).
(ii) This part follows from Theorem 6, (iii) and Theorem 9, (ii).
2.2.4 Ordered monoids
In this section, we investigate monoids supplied with a reflexive and transitive relation.
The main application will be the monoid being the power set of an ordered group together
with one of the relations 4 and 2 defined in the last subsection.
Definition 16 Let (Y, ◦) be a commutative monoid and be a quasiordering on Y satis-
fying the following condition:
(Q) y1, y2, y3 ∈ Y , y1 y2 implies y1 ◦ y3 y2 ◦ y3.
Then (Y, ◦, ) is called a quasiordered monoid. If is a partial order satisfying (Q),
then (Y, ◦, ) is called an ordered monoid.
Let us note that (Q) is equivalent to
y1, y2, y3, y4 ∈ Y, y1 y2, y3 y4 =⇒ y1 ◦ y3 y2 ◦ y4. (2.21)
since is reflexive and transitive. It is a crucial observation that a quasiordered monoid
can be made into an order complete quasiordered monoid by adding a smallest and a
largest elements, if necessary.
Proposition 25 Every Dedekind complete quasiordered (ordered) monoid can be extended
to an order complete quasiordered (ordered) monoid by adding at most two elements.

Proof. Let (Y, ◦, ) be a Dedekind complete quasiordered monoid. Add an element l (u,
respectively) such that
∀y ∈ Y : l y (y u) ,
∀y ∈ Y : l ◦ y = y ◦ l = l (u ◦ y = y ◦ u = u) ,
i.e., l (u, respectively) is the smallest (largest) element and dominant in the ordered monoid
(Yl := Y ∪ {l} , ◦, ) and (Yu := Y ∪ {u} , ◦, ), respectively.
Add an element u (l, respectively) to Yl (Yu) being the largest (smallest) and again
dominant. The result is the order complete commutative monoid (Y M := Yl ∪ {u} , ◦, )
and (Y O := Yu ∪ {l} , ◦, ), respectively. (Q) is easy to check.
Remark 7 1. Proposition 25 tells us that order completion does not destroy the monoidal
structure. Order completion of a group in this way yields an order complete monoid.
2. Of course, Proposition 25 is a generalization of the order completion of the reals,
widely used in optimization theory, compare Example 4. Especially, IRM is a fundamental
structure in Convex Analysis.
Proposition 26 Let (Y, ◦, ) be an ordered monoid with a largest (smallest) element.
Then it is an idempotent element.
Proof. Let ŷ ∈ Y be the largest element, i.e.
∀y ∈ Y : y ŷ.
Especially, ŷ ◦ ŷ ŷ and θ ŷ. From the latter inequality we obtain by (Q) ŷ ŷ ◦ ŷ.
The antisymmetry of implies ŷ ◦ ŷ = ŷ . The proof for the smallest element is similar.
In view of Proposition 6 we see that an order complete monoid can not be embedded into
a group.
The set of positive elements for an ordered monoid is defined to be
P :=

y ∈ Y : ∀y0
∈ Y : y0
y ◦ y0
= {y ∈ Y : θ y} .
Moreover, (P, ◦, ) is an ordered submonoid of Y .
Next, we discuss ordering relations in the power set of an ordered monoid using the
4– and 2–relation introduced in the last section.
Theorem 11 Let (Y, ◦, ) be a quasiordered monoid. Then:
(i) (P (Y ) , , 4) and (P (Y ) , , 2) are quasiordered, Dedekind complete monoids.
(ii)

b
P (Y ) , , 4

and

b
P (Y ) , , 2

are quasiordered, order complete monoids.

Proof. (i) By Proposition 22 and Theorem 6, (P (Y ) , 4) and (P (Y ) , 2) are qua-
siordered and Dedekind complete. It remains to show that (Q) holds true for 4, 2.
Take M1, M2, M3 ∈ P (Y ) such that M1 4 M2. Then by definition of 4,
∀y2 ∈ M2 ∃y1 ∈ M1 : y1 y2.
(Q) implies
∀y3 ∈ M3 ∀y2 ∈ M2 ∃y1 ∈ M1 : y1 ◦ y3 y2 ◦ y3.
Hence
∀y ∈ M2 M3 ∃y0
∈ M1 M3 : y0
y
which is M1 M3 4 M2 M3. The proof for 2 is similar. Hence (P (Y ) , , 4) and
(P (Y ) , , 2) are quasiordered, Dedekind complete monoids.
(ii) It is easy to check that (Q) is true for 4, 2 if one or more of M1, M2, M3 are the
empty set.
Considering equivalence classes one can obtain ordered monoids. We start with a qua-
siordered monoid (Y, ◦, ). Denote by
h
b
Y
i∗
:=
n
[M]∗
: M ∈ b
P (Y )
o
and
h
b
Y
i
:=
n
[M]
: M ∈ b
P (Y )
o
the set of equivalence classes over b
P (Y ) with respect to 4 and 2, respectively. For
M1, M2 ∈ b
P (Y ) define operations by
[M1]∗
[M2]∗
:= [M1 M2]∗
, [M1]
[M2]
:= [M1 M2]
. (2.22)
Recall the definitions of the equivalence classes in (2.13), (2.14) and the order relations 4,
2 for equivalence classes in (2.17), (2.18), respectively.
Proposition 27 Let (Y, ◦, ) be a quasiordered monoid. Then
h
b
Y
i∗
, , 4

and
h
b
Y
i
, , 2

are order complete ordered monoids.
Proof. First, let us note that the operations in (2.22) are well-defined: Take M1, M0
1, M2, M0
2 ∈
b
P (Y ) such that M1
∗
∼ M0
1 and M2
∗
∼ M0
2. From M1 4 M0
1, M2 4 M0
2 and (2.21) with a
view to Theorem 11 we get M1 M2 4 M0
1 M0
2. Similar, from M0
1 4 M1, M0
2 4 M2
follows M0
1 M0
2 4 M1 M2. Hence M1 M2
∗
∼ M0
1 M0
2. The same procedure applies
for 2. The remaining part of the theorem follows from Theorems 10, 11 and Proposition
24.
Theorem 9 admits the consideration of elements of b
P (Y ) instead of equivalence classes.
Define two subsets of b
P (Y ) by
b
Y∗
:=
n
M
∗
: M ∈ b
P (Y )
o
and b
Y
:=
n
M

: M ∈ b
P (Y )
o
we obtain the following proposition by applying Theorem 9.

Proposition 28 Let (Y, ◦, ) be a quasiordered monoid. Then

b
Y∗, , ⊇

and

b
Y, , ⊆

are order complete ordered monoids.
Proof. Invoke Theorem 9 and Proposition 27.
In the following, we consider the case of a quasiordered group (Y, ◦, ). In this case, the
relation on Y as well as 4 and 2 on b
P (Y ) can be expressed equivalently using the sets
P of positive elements and P0 of ”negative” elements which is defined to be
P0
:=

y0
∈ Y : ∃y ∈ P : y ◦ y0
= θ =

y0
∈ Y : y0
θ .
Moreover, there is a close relationship between 4 and 2.
Theorem 12 Let (Y, ◦, ) be a quasiordered group with neutral element θ, Mi ∈ b
P (Y )
and M0
i := {y0
i ∈ Y : ∃yi ∈ Mi : yi ◦ y0
i = θ} for i = 1, 2. Then it holds
M1 4 M2 ⇐⇒ M2 ⊆ M1 P,
M1 2 M2 ⇐⇒ M1 ⊆ M2 P0
,
M1
∗
∼ M2 ⇐⇒ M1 P = M2 P,
M1

∼ M2 ⇐⇒ M1 P0
= M2 P0
,
M1 4 M2 ⇐⇒ M0
2 2 M0
1.
Proof. The proof of the first four equivalences relies on the fact that in quasiordered
groups we have
y1 y2 ⇐⇒ y2 ∈ {y1} P ⇐⇒ y1 ∈ {y2} P0
.
The relations
M1 4 M2 ⇐⇒ ∀y2 ∈ M2 ∃y1 ∈ M1 : y1 y2
(Q)
⇐⇒ ∀y0
2 ∈ M0
2 ∃y1 ∈ M1 : y1 ◦ y0
2 θ
(Q)
⇐⇒ ∀y0
2 ∈ M0
2 ∃y0
1 ∈ M0
1 : y0
2 y0
1
⇐⇒ M0
2 4 M0
1.
yield the last assertion.
Example 10 The preceding theorem especially applies if V is a real linear space with a
quasiorder K generated by a convex cone K ∈ P (V ) =: Y . By 4K and 2K we denote
the two canonical extensions of K to Y . Then
M1 4K M2 ⇐⇒ (−1) M2 2K (−1) M1 ⇐⇒ (−1) M2 4(−1)K (−1) M1
where (−1) M := {−v : v ∈ M} for M ∈ Y .

2.2.5 Ordered conlinear spaces
We introduce the concept of an ordered conlinear space close to that of an ordered linear
space.
Definition 17 Let (Y, +) be a conlinear space and a quasiordering on Y satisfying the
following conditions:
(Q1) y1, y2, y3 ∈ Y , y1 y2 implies y1 + y3 y2 + y3;
(Q2) y1, y2 ∈ Y , y1 y2, t ≥ 0 implies ty1 ty2.
Then (Y, +, ) is called a quasiordered conlinear space. If is a partial order satis-
fying (Q1), (Q2), then (Y, +, ) is called an ordered conlinear space.
Note that (Q1) of this definition coincides with (Q) of Definition 16 if we consider (Y, +)
to be a commutative monoid. Again, order completiton does not destroy the conlinear
structure.
Proposition 29 Every Dedekind complete quasiordered (ordered) conlinear space can be
extended to an order complete quasiordered (ordered) conlinear space by adding at most
two elements.
Proof. Let (Y, +, ) be a quasiordered conlinear space. Proposition 25 ensures that
(Y, +, ) as a quasiordered monoid can be supplemented, if necessary, by two elements
l and u being the largest and the smallest. We obtain two possibilities for quasiordered
monoids: Y M (l is dominant) and Y O (u is dominant). Defining for t 0 t · l = l, t · u = u
and 0·l = 0·u = θ in both cases, it is easy to check that (C2), (Q1) and (Q2) are satisfied.
Hence (Y M, +, ) and (Y O, +, ) are order complete quasiordered conlinear spaces. If
(Y, +, ) is ordered, so are the resulting spaces.
By Proposition 26, the largest and the smallest element of an ordered conlinear space, if
they exist, are idempotent elements of the underlying monoid. Moreover, they are cones.
Proposition 30 Let (Y, +, ) be an ordered conlinear space. Then the largest and the
smallest element of an ordered conlinear space, if they exist, are cones.
Proof. Let ŷ ∈ Y be the largest element. Then 1
t y ŷ holds for all t 0 and y ∈ Y .
¿From (Q2) we may conclude y tŷ for all t 0 and y ∈ Y . Since the largest element
is unique by antisymmetry, it must be a cone. The proof for the smallest element is the
same.
Theorem 13 Let (Y, +, ) be a quasiordered conlinear space. Then:
(i) (P (Y ) , ⊕, 4) and (P (Y ) , ⊕, 2) are quasiordered, Dedekind complete conlinear spaces.
(ii)

b
P (Y ) , ⊕, 4

and

b
P (Y ) , ⊕, 2

are quasiordered, order complete conlinear spaces.

Proof. (i) By Proposition 22 and (i), (ii) of Theorem 6, (P (Y ) , 4) and (P (Y ) , 2) are
quasiordered and Dedekind complete. On the other hand, by Proposition 10, (P (Y ) , ⊕)
is a conlinear space. It remains to show that (Q1) and (Q2) hold true. Let us consider
the case 4. Take M1, M2, M3 ∈ P (Y ) such that M1 4 M2. We have to show M1 ⊕ M3 4
M2 ⊕ M3, i.e.
∀m ∈ M2 ⊕ M3, ∃m0
∈ M1 ⊕ M3 : m0
m.
Take m ∈ M2 ⊕ M3, i.e. m = m2 + m3, m2 ∈ M2, m3 ∈ M3. Since M1 4 M2, there
is m1 ∈ M1 such that m1 m2. Applying (Q1) of Definition 17 in (Y, +, ) we obtain
m0 := m1 + m3 m2 + m3 = m with m0 ∈ M1 ⊕ M3, i.e. (Q1) is valid in (P (Y ) , ⊕, 4).
(Q2) is immediate. A similar procedure can be applied for 2.
(ii) The extension to b
P (Y ) is straightforward.
We denote by
h
b
Y
i∗
and
h
b
Y
i
the set of equivalence classes of b
P (Y ) with respect to 4 and
2 as defined in (2.15) and (2.16), respectively.
The relations 4 and 2 for equivalence classes are defined as in (2.17) and (2.18),
respectively. The algebraic operation on
h
b
Y
i∗
and
h
b
Y
i
are defined by
[Y1]∗
⊕ [Y2]∗
:= [Y1 ⊕ Y2]∗
, [Y1]
⊕ [Y2]
:= [Y1 ⊕ Y2]
,
t · [Y1]∗
:= [tY1]∗
, t · [Y1]
:= [tY1]
for Y1, Y2 ∈ b
P (Y ) and t ≥ 0. As in (2.15) and (2.16) we set b
Y∗ =
n
M
∗
: M ∈ b
P (Y )
o
and b
Y =
n
M

: M ∈ b
P (Y )
o
where M
∗
=
S
M0∈[M]∗ M0 and M

=
S
M0∈[M] M0, respec-
tively.
Theorem 14 Let (Y, +, ) be a quasiordered conlinear space. Then:
(i)
h
b
Y
i∗
, ⊕, 4

and
h
b
Y
i
, ⊕, 2

are order complete, ordered conlinear spaces.
(ii)

b
Y∗, ⊕, ⊇

and

b
Y, ⊕, ⊆

are order complete, ordered conlinear spaces.
Proof. (i)
h
b
Y
i∗
, ⊕, 4

and
h
b
Y
i
, ⊕, 2

are order complete, ordered monoids by
Proposition 27. The conditions of (C2) of Definition 5 may be checked straightforward.
(ii) Similar to (i) invoking Proposition 28.
Let (Y, +, ) be a quasiordered conlinear space. We define the set K of positive elements
by
K := {y ∈ Y : θ y} .
It can easily be seen that K is a convex subset of Y by (Q1), (Q2) and transitivity and that
it is a cone in P (Y ) by (Q2) containing θ ∈ Y by reflexivity. Therefore, from Proposition
17, we know that K is a convex element of P (Y ).
Since (Y, +) is not a group, it is not possible to get back the relation from K in
general by defining
y K y0
⇐⇒ y0
∈ {y} ⊕ K,

i.e. the relations K and do not coincide. More precisely, we have y1 K y2 implies
y1 y2, but not conversely in general.
It is beyond the scope of this thesis to develop a theory of (quasi)ordered conlinear
spaces although this seems to be worth doing. We shall give only one more example for a
difference to the linear case.
Let (Y, +, R, ) be an ordered conlinear space and y1, y2 ∈ Y . We call the set [y1, y2] :=
{y ∈ Y : y1 y y2} the order intervall between y1 and y2. In ordered linear spaces,
the convex hull of {y1, y2} is always contained in the [y1, y2]. This is no longer true in
conlinear spaces as the following example shows.
Example 11 Take Y = P IR2

, K = IR+
1
1
!
and set for M1, M2 ∈ Y
M1 K M2 ⇐⇒ M2 ⊆ M1 ⊕ K.
Take M1 =
(
0
0
!
,
1
0
!)
, M1 =
(
2
1
!
,
3
3
!)
. Then M1 K M2 but we have
neither M1 K tM1 + (1 − t) M2 nor tM1 + (1 − t) M2 K M2 for t ∈ (0, 1).
2.2.6 Ordered semilinear spaces
The order relations 4 and 2 for the power set of a linear space are the main motivation for
the considerations of this section. Therefore, we extend the definitions to the case where
a multipication with negative real numbers is available but the group property still does
not hold. Recall Definition 10.
Definition 18 Let (Y, +) be a semilinear space and be a quasiorder on Y satisfying
(Q1) and (Q2) of Definition 17. Then (Y, +, ) is called a quasiordered semilinear
space. If is additionally antisymmetric, i.e. a partial order, then (Y, +, ) is called an
ordered semilinear space.
Since a semilinear space is especially conlinear, the results concerning order completion
and the extension of the order to the power set remain true for semilinear spaces. We
shall present the results indicating in the proofs only the main differences to the conlinear
case.
Proposition 31 Every Dedekind complete quasiordered (ordered) semilinear space can be
extended to an order complete quasiordered (ordered) semilinear space by adding at most
two elements.
Proof. Proceed as in Proposition 29: To maintain the semilinear structure of the exten-
sions, one has to define t · l = l and t · u = u for the largest element l and the smallest
u and all t ∈ IR {0} as well as 0 · l = 0 · u = θ. Depending on the dominance property,
two cases are possible: l + u = l or l + u = u. The conditions of Definition 10 are easy to
check.

In the following theorm, we use the definitions of
h
b
Y
i∗
,
h
b
Y
i
, b
Y∗, b
Y and the algebraic
operations and the order relations as in the preceding subsection on conlinear spaces.
Naturally, the multiplication with real numbers is defined as follows:
t · M := {t · m : m ∈ M} , t · [M] := [t · M]
for t ∈ IR, M ∈ b
P (Y ), [M] ∈ {[M]∗
, [M]
}.
Theorem 15 Let (Y, +, ) be a quasiordered semilinear space. Then:
(i)

b
P (Y ) , ⊕, 4

and

b
P (Y ) , ⊕, 2

are quasiordered, order complete semilinear spaces.
(ii)
h
b
Y
i∗
, ⊕, 4

and
h
b
Y
i
, ⊕, 2

are order complete, ordered semilinear spaces.
(iii)

b
Y∗, ⊕, ⊇

and

b
Y, ⊕, ⊆

are order complete, ordered semilinear spaces.
Proof. (i) This is true since

b
P (Y ) , ⊕, 4

and

b
P (Y ) , ⊕, 2

are quasiordered, or-
der complete conlinear spaces by Theorem 13 and

b
P (Y ) , ⊕

is a semilinear space by
Proposition 19.
(ii) The extension of the semilinear structure follows essentially from
t · ([M1] ⊕ [M2]) = t · ([M1 ⊕ M2]) = [t · (M1 ⊕ M2)] = [t · M1] ⊕ [t · M2]
for M1, M2 ∈ b
P (Y ), [·] ∈ {[·]∗
, [·]
}.
(iii) This is a consequence of (ii) and Theorem 14.
2.2.7 Historical comments
In the preceding section, basic order theoretic notation has been presented with special
emphasis on the two canonical extensions of an order relation from a set W to its power
set b
P (W). We refer the reader to the comprehensive 1993 survey [7] of a more algebraic
motivated approach to the topic of power structures. In this paper, the relations 2 and
4 are denoted by R+
0 and R+
1 , respectively. These and similar structures mostly defined
on finite or countable sets are widely used in theoretical infomation sciences, compare for
example the reference list of [7].
However, the question how algebraic and order structures have to be extended from a
given set to its power set has been investigated from several, quite different viewpoints.
Without intending to give a complete list we mention a few authors being of influence for
the present work.
The paper [129] by R. C. Young already contains the definitions of 4 and 2 implicitly
and presents applications to the analysis of upper and lower limits of sequences of numbers.
Nishianidze [95] also used the relations 4 and 2. Construction mainly motivated by
applications in economical and social choice theory can be found e.g. in [88]. Compare
also the references therein, especially [68].

2.3. Topological and uniform structures 45
In [78] one can find a systematic investigation of six extensions of a quasiorder ≤K
on a topological linear space with convex ordering cone K with nonempty interior to its
power set; the relations 4K and 2K are proven to be the only relations being reflexive
and transitive and definitions for in some sense convex setvalued maps are given. Several
subsequent papers of the three authors of [78] contain applications, see for example [77],
[76], [117] within the field of optimization with a setvalued objective function. For this
topic, compare also the book [63], especially Chapter V.
Finally, in [116] an algebraic approach to vector optimization has been presented in-
cluding some results on hull structures being in some sense related to power structures as
used in this section.
The formulas for infimum and supremum with repect to 4 and 2 in this section and
the relationships between extrema for these relations on one hand and infimal/minimal
points for on the other hand seems to be new.
2.3 Topological and uniform structures
For the convenience of the reader, we recall in this section definitions, facts and refer-
ences concerning basic uniform and topological structures that are used in the subsequent
chapters. Moreover, some results are collected not being very much standard such as the
equivalent characterization of a uniformity by means of quasimetrics or an order metric.
Our standard references for this section are [75] and [72], for uniform structures also
[64] and [16].
2.3.1 Topological spaces
There are several possibilities to introduce the concept of a topology. In the following
definition, the neighborhoods of a point are used as the starting point since this is the
most convenient method for the proofs of the next subsection on uniform topologies.
Definition 19 Let Z be a nonempty set and for each z ∈ Z let there be a nonempty set
N (z) ⊆ P (Z) satisfying
(T1) If N ∈ N (z), then z ∈ N;
(T2) If N ∈ N (z) and N ⊆ N0 ∈ P (Z), then N0 ∈ N (z);
(T3) If N1, N2 ∈ N (z), then N1 ∩ N2 ∈ N (z);
(T4) If N ∈ N (z), then there is N0 ∈ N (z) such that N ∈ N (z0) for each z0 ∈ N0.
An element N ∈ N (z) is called a neighborhood of z ∈ Z. The entity N (z) is called a
system of neighborhoods of z ∈ Z. A subset T ⊆ Z is called open set iff T ∈ N (z)
whenever z ∈ T. The set T of all open sets is called a topology on Z and the pair (Z, T )
is called a topological space.
The axioms (T2) and (T3) imply that N (z) is a filter (see Definiton 0.1, 0.3 on p. 5f
of [64]). A filter base for N (z) is called a neighborhood base for z ∈ Z. A subset
B (z) ⊆ P (Z) is a neighborhood base for z ∈ Z if it satisfies (T1), (T4) with N replaced

by B and
(T3’) If B1, B2 ∈ B (z), then there is B ∈ B (z) such that B ⊆ B1 ∩ B2.
In this case, the subset of P (Z) that contains the supersets of members of B (z) satisfies
(T1) – (T4), i.e., it is a neighborhood system N (z) for z ∈ Z. If a neighborhood base for
each z ∈ Z is given, the corresponding topology is uniquely defined.
The complement ZT for T ∈ T is said to be a closed set. Let M ⊆ Z be a subset of Z.
An element z ∈ M is called interior point of M iff there is N ∈ N (z) such that N ⊆ M.
The set of interior points of M is denoted by int M. The set cl M := Zint (ZM) is
called the closure of M.
A topological space is called separated or Hausdorff iff for any two distinct points
z1, z2 ∈ Z there are disjoint open sets T1, T2 ∈ T such that z1 ∈ T1 and z2 ∈ T2. This
is equivalent to the property that the intersection of the closed neighborhoods of a point
z ∈ Z contains only z itself. In a Hausdorff topological space, each set {z} for z ∈ Z is
closed.
Prominent examples of topological spaces are topological (Abelian) groups. We give
the definition, compare for example [72], [54] or [64]. In order to do this one needs the
concepts of a continuous function and of a product topology.
Let (Z1, T1), (Z2, T2) be two topological spaces. A function f : Z1 → Z2 is said to be
continuous iff the inverse image of a member of T2 is a member of T1. The collection of
all Cartesian products T1 ×T2 for T1 ∈ T1, T2 ∈ T2 form the base for a uniquely determined
topology on the Cartesian product Z1 × Z2 called the product topology.
Definition 20 Let (Y, ◦) be a group supplied with a topology T . Suppose further that (i)
the mapping (y1, y2) → y1 ◦ y2 is a continuous function of the Cartesian product Y × Y
onto Y and (ii) the mapping y → y−1 is a continuous function of Y onto Y (y−1 being
the inverse element of y with respect to ◦). Then (Y, ◦, T ) is called a topological group.
If the group is additionally commutative, it is called topological Abelian group.
A neighborhood N of θ in a topological group (Y, ◦, T ) is called symmetric if y ∈ N
implies y−1 ∈ N. The symmetric neighborhoods of the neutral element θ ∈ Y form a
neighborhood base of θ ∈ Y . The topology of a topological Abelian group is uniquely
defined by a base of symmetric neighborhoods of the neutral element since {y} N is
a neighborhood of y ∈ Y if and only if N is a neighborhood of θ ∈ Y . If B (θ) is
a neighborhood base of θ ∈ Y and T the topology generated by B (θ), then (Y, T ) is
separated if and only if
T
B∈B(θ) B = {θ}.
Let (Y, ◦, T ) be a topological Abelian group with neutral element θ ∈ Y . Further,
let ≤ be a quasiorder on Y such that (Y, ◦, ≤) is a quasiordered monoid in the sense of
Definition 16. Then (Y, ◦, ≤, T ) is called a quasiordered topological Abelian group.
We study two properties linking the order structure with the topological structure.
(A) There is a neighborhood base B (θ) of θ ∈ Y such that
∀B ∈ B (θ) : θ ≤ y ≤ y0
, y0
∈ B

=⇒ y ∈ B.

(B) There exists a neighborhood base N (θ) of θ ∈ Y such that
∀N ∈ N (θ) : (y1 ≤ y ≤ y2, y1, y2 ∈ N) =⇒ y ∈ N.
Lemma 1 Let (Y, ◦, ≤, T ) be a quasiordered topological Abelian group with neutral element
θ ∈ Y . Then, the properties (A) and (B) are equivalent.
Proof. First, we show that (A) implies (B). Let M ∈ N (θ). We shall show that there
is N ∈ N (θ), N ⊆ M satisfying N = {y ∈ Y : y1 ≤ y ≤ y2, y1, y2 ∈ N}. Take B ∈ B (θ)
such that B B ⊆ M and a symmetric N0 ∈ N (θ) such that N0 N0 ⊆ B. Let y ∈ Y
such that y1 ≤ y ≤ y2 for y1, y2 ∈ N0. Then
θ ≤ y ◦ y−1
1 ≤ y2 ◦ y−1
1 ∈ N0
N0
⊆ B.
Hence y ◦ y−1
1 ∈ B implying y ∈ {y1} B ⊆ N0 B ⊆ B B ⊆ M. Hence N :=
{y ∈ Y : y1 ≤ y ≤ y2, y1, y2 ∈ N0} as desired.
Conversely, if N ∈ N (θ), N (θ) satisfies (B) and θ ≤ y ≤ y0 ∈ N, then y ∈ N since
θ ∈ N.
Lemma 2 A quasiorder on a separated topological Abelian group satisfying (A) (and (B))
is antisymmetric, i.e., a partial order.
Proof. If θ ≤ y ≤ θ for some y ∈ Y , then y ∈ B for each B ∈ B (θ) satisfying (A). Since
B is separated, the result follows.
Definition 21 A partial order ≤ on a topological Abelian group (Y, ◦, ≤, T ) satisfying (A)
(and (B), too) is called normal. In this case, (Y, ◦, ≤, T ) is called normally ordered.
A set M ⊆ Y satisfying M = {y ∈ Y : y1 ≤ y ≤ y2, y1, y2 ∈ M} is called full or satu-
rated. Thereby, condition (B) can be rewritten as: There is a neighborhood base of the
neutral element consisting of full (saturated) sets.
Let (Y, ◦, ≤, T ) be a quasiordered topological Abelian group. Defining
P := {y ∈ Y : θ ≤ y} , P−1
:=

y0
∈ Y : y0
≤ θ
we may see that
y1 ≤ y2 ⇐⇒ y2 ◦ y−1
1 ∈ P ⇐⇒ y1 ◦ y−1
2 ∈ P−1
.
Hence, a set M ⊆ Y is full (saturated) if and only if M = (M P)
T
M P−1

.
If a quasiorder with the set P of ”positive” elements is given, another quasiorder is
generated by cl P (note that (cl P)−1
= cl P−1):
y1 ≤cl P y2 :⇐⇒ y2 ◦ y−1
1 ∈ cl P ⇐⇒ y1 ◦ y−1
2 ∈ cl P−1
.
The following relationship holds true concerning normality.

Lemma 3 Let (Y, ◦, T ) be a topological Abelian group and ≤P a partial order with P =
{y ∈ Y : θ ≤ y}. Then, ≤P is normal if and only if ≤cl P is normal.
Proof. Since y1 ≤P y2 implies y1 ≤cl P y2, it is clear that the normality of ≤cl P implies
the normality of ≤P .
Conversely, assume the normality of ≤P , i.e. there is a neighborhood base B (θ) of
θ ∈ Y such that B = (P B)
T
P−1 B

for each B ∈ B (θ). Take B ∈ B (θ) and
y1, y2 ∈ Y such that θ ≤cl P y1 ≤cl P y2 ∈ B. Since Y is a topological Abelian group, there
is B0 ∈ B (θ) such that B0 B0 ⊆ B (see e.g. [64], p. 37f). From θ ≤cl P y1 we obtain
y1 ∈ cl P ⊆ P B0 ⊆ P B. From θ ≤cl P y2 we get y−1
2 ∈ cl P−1 ⊆ P−1 B0 and using
this from y1 ≤cl P y2 it follows
y1 ∈

y−1
2 cl P−1
⊆ P−1
B0

P−1
B0

⊆ P−1
B.
Therefore, y1 ∈ (P B)
T
P−1 B0

= B as desired.
The notion of a normal partial order and the corresponding normal cone (in the classical
sense) is a central concept in the theory of ordered topological linear spaces. Details,
especially some more equivalent characterizations can be found e.g. in [99], Chapter 2, or
[107], Chapter V, §3. Compare also [44], p. 24ff, and [58], Chapter 1.2. An interesting
result using the normality of orders on topological Abelian groups is Proposition 12 of
[87], p. 76. Note that condition (0.5) of [108], p. 1 is also a normality condition that is
used for the definition of quasimetric spaces.
2.3.2 Uniform spaces
Let X be a nonempty set. We consider a collection U of subsets E of X × X :=
{(x1, x2) : x1, x2 ∈ X}. The set ∆ := {(x, x) ∈ X × X : x ∈ X} is called the diag-
onal. For E ⊆ X × X we denote E−1 := {(x2, x1) : (x1, x2) ∈ E} and E ◦ E :=
{(x1, x2) ∈ X × X : ∃x ∈ X : (x1, x) , (x, x2) ∈ E}.
Definition 22 Let X be a nonempty set. A set U ⊆ P (X × X) is said to be a uniformity
on X iff
(U1) E ∈ U, E ⊆ E0 implies E0 ∈ U and E1, E2 ∈ U implies E1 ∩ E2 ∈ U;
(U2) If E ∈ U, then ∆ ⊆ E;
(U3) If E ∈ U, then there is E0 ∈ U such that E0 ⊆ E−1;
(U4) For all E ∈ U there is E0 ∈ U such that E0 ◦ E0 ⊆ E.
The pair (X, U) is called a uniform space. The elements of U are called entourages or
surroundings. The uniformity U is called separated iff
(U5)
T
E∈U E = ∆.
A uniformity U on X is a filter by (U1), (U4). A filter base UB for the filter U is called a
base of the uniformity U.

A set UB ⊆ P (X × X) is a base of some uniformity U if it satisfies (U2), (U3), (U4)
with U replaced by UB and, additionally,
(U1’) If E1, E2 ∈ UB, then there is E ∈ UB with E ⊆ E1 ∩ E2.
In this case, a uniformity is obtained by taking supersets in P (X × X) of the members of
UB.
Let (X, U) be a uniform space. The family of sets U (x) := {UE(x) : E ∈ U} where
UE (x) := {x0 ∈ X : (x, x0) ∈ E} is a neighborhood system for x ∈ X, i.e., it satisfies (T1)
– (T4) of Definiton 19. Similarly, a base of the uniformity generates a neighborhood base
for each x ∈ X. In this way, a uniquely defined topology, called the uniform topology
on X can be generated. If the uniformity is separated, then so is the uniform topology.
The class of separated uniform spaces coincides with the class of completely regular
(Tychonoff) spaces. This result is well–known, [72], Corollary 17, p. 188 or [75], p. 48-50.
Metric spaces, topological groups and hence topological linear spaces can be supplied
with a uniform structure such that the given topology is the corresponding uniform topol-
ogy. For example, let (Y, ◦, T ) be a topological Abelian group and B (θ) a neighborhood
base of the neutral element θ ∈ Y . Take B ∈ B (θ) and define a subset of Y × Y by
EB :=

(y1, y2) ∈ Y × Y : y−1
1 ◦ y2 ∈ B .
The set {EB : B ∈ B (θ)} is a base for a uniform structure on Y . Of course, if Y is not
commutative one has to distinguish between a ”left” and a ”right” uniformity. See the
book of James [64] for this and further related results.
There are other possibilities to generate a uniform structure: via a family of realvalued
pseudometrics or quasimetrics and via an order metric. We shall give the definitions since
they will admit a greater degree of freedom in defining an order relation on X that is an
essential ingredient of minimal element theorems.
The first equivalent description of a uniformity is via a family of pseudometrics. This
notion is standard in textbooks on uniform spaces, compare [72], [75], [21], [64], [16].
Definition 23 Let X be a nonempty set. A function p : X × X → IR is called a pseu-
dometric on X iff for all x, x1, x2 ∈ X the following conditions are satisfied:
(UP1) p (x1, x2) ≥ 0 and p (x, x) = 0;
(UP2) p (x1, x2) = p (x2, x1);
(UP3) p (x1, x2) ≤ p (x1, x) + p (x, x2).
Let (Λ, ≺) be a directed set. A set {pλ}λ∈Λ of pseudometrics pλ : X × X → IR satisfying
(UP4) λ, µ ∈ Λ, λ ≺ µ implies pλ (x1, x2) ≤ pµ (x1, x2) for all x1, x2 ∈ X
is called a family of pseudometrics. If, additionally, the condition
(UP5) If pλ (x1, x2) = 0 for all λ ∈ Λ, then x1 = x2.
is satisfied, then the family of pseudometrics is called separating.
The following proposition gives the relationsship between uniformities and pseudometric
spaces. It is a fundamental result on uniform spaces.

Proposition 32 (i) Let (X, U) be a (separated) uniform space. Then there is a (separat-
ing) family {pλ}λ∈Λ of pseudometrics on X such that the entity of sets
Eλ,r := {(x1, x2) ∈ X × X : pλ (x1, x2) r} , λ ∈ Λ, r 0, (2.23)
is a base for the uniformity U.
(ii) Let X be a nonempty set and {pλ}λ∈Λ a (separating) family of pseudometrics on
X. Then the entity of sets given by (2.23) is a base for a (separated) uniformity U on X.
Proof. The proof is standard in text books on uniform spaces. Compare [72], Metrization
Lemma, Theorems 15, 16. The proof of (i) is essentially based on a metrization procedure,
(ii) can be verified directly checking the properties for bases of uniformities.
A family of pseudometrics generates a uniformity and hence a corresponding uniform
topology. More precisely, we have the following result.
Corollary 7 A topological space (X, T ) is a (separated) uniform space if and only if its
topology T can be generated by a (separating) family of pseudometrics.
Proof. If {pλ}λ∈Λ is a family of pseudometrics on X, then the entity of sets
Nλ,r (x) :=

x0
∈ X : pλ x0
, x

r , λ ∈ Λ, r 0, x ∈ X (2.24)
is a neighborhood base for some topology on X that coincides with the uniform topology
generated by {pλ}λ∈Λ. Conversely, if T is a topology on X generated by some uniformity
U on X, then there is a family of pseudometrics yielding a base for U via (2.23) and also
a base for the uniform topology via (2.24).
Fang [33] introduced so called F–type topological spaces using families of quasimetrics.
It has been observed in [51] that this is just another way to generate the topology via a
uniformity.
Definition 24 Let X be a nonempty set and (Λ, ≺) be a directed set. A set {qλ}λ∈Λ of
functions qλ : X × X → IR is called a family of quasimetrics on X iff the following
conditions are satisfied:
(UQ1) qλ (x1, x2) ≥ 0 and qλ (x, x) = 0 for all λ ∈ Λ and all x, x1, x2 ∈ X;
(UQ2) qλ (x1, x2) = qλ (x2, x1) for all λ ∈ Λ and all x1, x2 ∈ X;
(UQ3) For all λ ∈ Λ there is µ ∈ Λ such that λ ≺ µ and qλ (x1, x2) ≤ qµ (x1, x)+qµ (x, x2)
for all x, x1, x2 ∈ X;
(UQ4) λ, µ ∈ Λ, λ ≺ µ implies qλ (x1, x2) ≤ qµ (x1, x2) for all x1, x2 ∈ X.
If, additionally, the condition
(UQ5) if qλ (x1, x2) = 0 for all λ ∈ Λ, then x1 = x2;
is satisfied, then the family of quasimetrics is called separating.

Proposition 33 (i) Let (X, U) be a (separated) uniform space. Then there is a (separat-
ing) family {qλ}λ∈Λ of quasimetrics on X such that the entity of sets
Eλ,r := {(x1, x2) ∈ X × X : qλ (x1, x2) r} , λ ∈ Λ, r 0, (2.25)
(ii) Let X be a nonempty set and {qλ}λ∈Λ a (separating) family of quasimetrics on X.
Then the entity of sets given by (2.25) is a base for a (separated) uniformity U on X.
Proof. (i) Apply Proposition 32, (i) to get a family of pseudometric generating the
topology. Since every pseudometric is all the more a quasimetric, the result follows. (ii)
Check the properties for a base of a uniformity for the entity of sets Eλ,r in (2.25).
Corollary 8 A topological space (X, T ) is a (separated) uniform space if and only if its
topology T can be generated by a (separating) family of quasimetrics.
Proof. This can been seen by the same arguments as in the proof of Corollary 7. See
also [51], p. 579, Theorem 2.4.
The next possibility to introduce a uniformity uses functions on X × X to some ordered
topological group Y with properties very close to the properties of a metric in the usual
sense.
Definition 25 Let X be a nonempty set, (Y, ◦, ≤, T ) a normally ordered topological Abelian
group with neutral element θ ∈ Y . A function D : X × X → Y is called an order pseu-
dometric iff for all x, x1, x2 ∈ X the following conditions are satisfied:
(UM1) θ ≤ D (x1, x2) and D (x, x) = θ;
(UM2) D (x1, x2) = D (x2, x1);
(UM3) D (x1, x2) ≤ D (x1, x) ◦ D (x, x2).
(UM4) D (x1, x2) = θ implies x1 = x2;
is satisfied, then D is called an order metric1. The pair (X, D) is called an order
(pseudo)metric space.
If Y = IR, i.e. the set of real numbers together with the usual addition, order relation and
topology, the widely used definition of a metric space is obtained.
The above definition can be generalized in different directions. For example, Y can
be assumed to be not a group but an ordered monoid. Compare e. g. [108] for details
and the practical importance of more general structures. Also, some of the axioms can
be relaxed. In the following chapter, a generalization is given in order to obtain as much
freedom as possible to define order relations on uniform space.
1
We would prefer simply to speak of (pseudo)metrics. For historical reasons, we keep on using the term
”(pseudo)metric” only in case Y = IR.

Proposition 34 Let X be a nonempty set, (Y, ◦, ≤, T ) a normally ordered topological
Abelian group with neutral element θ ∈ Y and D : X × X → Y an order pseudometric.
Let B (θ) be a neighborhood base of θ ∈ Y consisting of symmetric neighborhoods. Then,
the entity of sets
EB := D−1
(B) = {(x1, x2) ∈ X × X : D (x1, x2) ∈ B} , B ∈ B (θ) (2.26)
is a base of a uniform structure on X. The corresponding uniform topology coincides with
the topology generated by D. If D is an order metric, the uniformity is separated.
Proof. Denote UB = {EB : B ∈ B (θ)}. First, take E1, E2 ∈ UB. Then there are
B1, B2 ∈ B (θ) such that E1 = D−1 (B1), E2 = D−1 (B2). Since B (θ) is a neighborhood
base of θ ∈ Y , there is B ∈ B (θ) such that B ⊆ B1 ∩ B2. It is easy to see EB ∈ UB and
EB ⊆ E1 ∩ E2.
It remains to show that UB satisfies (U2), (U3) and (U4). (U2) follows from (UM1)
whereas (U3) is a consequence of (UM2) since the elements B ∈ B (θ) are symmetric.
To show (U4), take EB ∈ UB. Since (Y, ◦, ≤, T ) is a topological Abelian group, there
is B0 ∈ B (θ) such that B0 B0 ⊆ B. Set E0 = D−1 (B0) and take (x1, x2) ∈ E0 ◦ E0. Then
there is x ∈ X such that (x1, x) , (x, x2) ∈ E0. From (UM3) we obtain
D (x1, x2) ≤ D (x1, x) ◦ D (x, x2) ∈ B0
B0
⊆ B.
Therefore D (x1, x2) ∈ B by normality of ≤. Hence (x1, x2) ∈ EB and E0 ◦ E0 ⊆ EB as
desired.
The last assertions are obvious.
Proposition 35 Let (X, U) be a (separated) uniform space. Then, there is a topological
Abelian group and an order pseudometric (metric) such that D generates the uniform
structure U on X via (2.26).
Proof. Since (X, U) is uniform, there exists a family of pseudometrics {pλ}λ∈Λ such that
the entity of sets
Eλ,r := {(x1, x2) ∈ X × X : pλ (x1, x2) ≤ r} , λ ∈ Λ, r 0
Define y := (rλ)λ∈Λ ∈ IRΛ =: Y , rλ ∈ IR. With the usual componentwise addition and
y0 ≤ y iff rλ − r0
λ ≥ 0 for all λ ∈ Λ, we obtain a partially ordered Abelian group (Y, +, ≤)
with neutral element θ := (0λ)λ∈Λ. It is not hard to see that the sets
Bt,λ(n) := {y ∈ Y : |rλ1 | t, . . . , |rλn | t}
for t 0, n ∈ IN {0}, λ (n) = {λ1, . . . , λn} ⊆ Λ form a neighborhood base of a topology
T such that (Y, +, ≤, T ) is a normally ordered, topological Abelian group. Defining D :
X × X → Y by
D (x1, x2) := (pλ (x1, x2))λ∈Λ

we may easily verify the axioms (UM1)–(UM4) from (UP1)–(UP4). Moreover, the defini-
tion of D ensures that the uniformity generated by D coincides with U. This completes
the proof.
Remark 8 Propositions 34 and 35 show that a space admits a uniformity if and only if
it admits a metric in the sense of Definition 25. This observation is due to [66], but it is
much less standard than the formulation via families of pseudometrics.
Proposition 36 Let X be a nonempty set, (Y, ◦, ≤, T ) a normally ordered topological
Abelian group with neutral element θ ∈ Y and D : X × X → Y an order pseudometric.
The sets
NB (x) :=

x0
∈ X : D x, x0

∈ B , B ∈ B (θ)
form a neighborhood base of x ∈ X. The collection
N := {{NB (x) : B ∈ B (θ)} : x ∈ X}
is a neighborhood base on X generating a topology T . If D is an order metric, the (X, T )
is a separated topological space.
Proof. It suffices to show that N (x) := {NB (x) : B ∈ B (θ)} satisfies (T1), (T3) and
(T4) of Definition 19. (T1) is clear from (UM1).
To prove (T3) take B1, B2 ∈ B (θ). Since B (θ) is a neighborhood base of θ ∈ Y , there is
B ∈ B (θ) such that B ⊆ B1
T
B2. Since for x0 ∈ NB (x) we have D (x, x0) ∈ B ⊆ B1
T
B2
we may conclude NB (x) ⊆ NB1 (x)
T
NB2 (x), hence N (x) satisfies (T3).
To show (T4) take B ∈ B (θ) and x0 ∈ NB (x). Since B (θ) is a neighborhood base of
θ ∈ Y and y := D (x, x0) ∈ B there is a neighborhood N (y) ⊆ Y such that N (y) ⊆ B.
Since

y−1 N (y) is a neighborhood of θ ∈ Y , there is B0 ∈ B (θ) such that B0 ⊆

y−1 N (y). This implies {y} B0 ⊆ N (y) ⊆ B. Take u ∈ NB0 (x0). From (UM3) we
get
D (x, u) ≤ D x, x0

◦ D x0
, u

∈ {y} B0
⊆ B.
The normality of ≤ implies D (x, u) ∈ B, hence u ∈ NB (x) and therefore NB0 (x0) ⊆
NB (x). This completes the proof.
Note that the group Y in Proposition 34 – 36 can be replaced by a locally convex space
with a normal cone, see [91]. A definition of locally convex spaces is given below. To the
opinion of the author, the choice of a normally ordered topological Abelian group admits
a lucid formulation without to much non–standard technicalities. We refer the reader also
to [27] and the various papers of A. B. Nemeth about so called cone valued metrics.
Thus, if a uniform space is given (say, by a base for the uniformity, i.e. for the system of
entourages), there are at least three further possibilities to generate its uniform structure
as well as the corresponding uniform topology.

2.3.3 Completeness in uniform spaces
Many topological concepts carry over from the metric to the uniform case. The concepts
of a Cauchy sequence (net) and completeness are of importance for the formulation of
variational principles. Therefore, we shall give the definitions.
Let (X, U) be a uniform space. Let (A, ≺) be a directed set. A net {xα}α∈A ⊆ X is
called a Cauchy net iff
∀E ∈ U ∃αE ∈ A : (α1, α2 ∈ A, αE ≺ α1 ≺ α2 =⇒ (xα1 , xα2 ) ∈ E) .
The net {xα}α∈A ⊆ X is called convergent to x ∈ X iff
∀E ∈ U ∃αE ∈ A : (α, ∈ A, αE ≺ α =⇒ (xα, x) ∈ E) .
The uniform space (X, U) is called complete iff every Cauchy net converges to some
element x ∈ X. The uniform space (X, U) is called sequentially complete iff every
Cauchy sequence ((A, ≺) = (IN, ≤)) converges to some element x ∈ X.
In the above definitions, it suffices to involve elements E ∈ UB for a base UB of the
uniformity U. A base may be generated by a family of pseudometric, of quasimetrics
or by an order pseudometric. Therefore, there are several possibilities to characterize
Cauchy (convergent) nets. For example, if an order pseudometric D : X ×X → Y is used,
(Y, ◦, ≤, T ) being a normally ordered topological Abelian group, the property of being a
Cauchy net can be expressed as follows:
∀B ∈ B (θ) ∃αB ∈ A : (α1, α2 ∈ A, αB ≺ α1 ≺ α2 =⇒ D (xα1 , xα2 ) ∈ B) ,
where B (θ) is a neighborhood base of θ ∈ Y .
2.3.4 The linear case
Linear spaces are special cases of conlinear spaces defined in Section 2.1.2.
Definition 26 Let (X, +) be a linear space supplied with a topology T . Suppose further
that (i) the mapping (x1, x2) → x1 + x2 is a continuous mapping of the Cartesian product
X × X onto X and (ii) the mapping (t, x) → tx is a continuous mapping of IR × X onto
X. Then (X, +, T ) is called a (real) topological linear space.
A standard reference is the book of Köthe [75]. Since a topological linear space is especially
a topological Abelian group with respect to addition, it is clear that it can be provided
with a unifom structure in the same way, i.e., the topology of a topological linear space
is uniformizable ([75], §15, (3), p. 150). Hence, Propositions 32 and 7 as well as 34 and
36 remain in force, i.e., the uniform structure as well as the topology of a topological
linear space can be generated by an order pseudometric or by a family of realvalued
pseudometrics.
An important subclass of the class of topological linear spaces is the class of locally
convex spaces. A definition is as follows.

Definition 27 A topological linear space (X, +, T ) is called locally convex topological
linear space (for short: locally convex space) iff there is a neighborhood base of θ ∈ X
consisting of convex sets.
Hyers [56], [57] and LaSalle [79] observed that every topological linear space admits a
system of quasinorms generating the topology. The definition is as follows.
Definition 28 Let (X, +) be a real linear space and (Λ, ≺) be a directed set. A system
{k·kλ}λ∈Λ of functions k·kλ : X → IR is called a family of quasinorms on X iff for all
x, x1, x2 ∈ X the following conditions are satisfied:
(NQ1) kxkλ ≥ 0 for all λ ∈ Λ;
(NQ2) ktxkλ = |t| kxkλ for all t ∈ IR and λ ∈ Λ;
(NQ3) For all λ ∈ Λ there is µ ∈ Λ such that λ ≺ µ and kx1 + x2kλ ≤ kx1kµ + kx2kµ.
(NQ4) λ, µ ∈ Λ, λ ≺ µ implies kxkλ ≤ kxkµ.
(NQ5) If kxkλ = 0 for all λ ∈ Λ, then x = θ;
is satisfied, then the family of quasimetrics is called separating.
Given a family of quasinorms {k·kλ}λ∈Λ on the linear space X, the expressions
qλ (x1, x2) := kx1 − x2kλ , λ ∈ Λ, x1, x2 ∈ X
define a family of quasimetrics on X generating a uniform structure. The corresponding
uniform topology T on X can be generated by the sets
Bλ,r := {x ∈ X : kxkλ r} , λ ∈ Λ, r 0
forming a neighborhood base of θ ∈ X on X. The couple (X, T ) is a topological linear
space. The following result is due to Hyers and LaSalle. A concise proof and some more
details can be found in [81].
Proposition 37 A linear space (X, +) is a (separated) topological linear space with respect
to a topology T on X if and only if the topology can be generated by a (separating) family
of quasinorms on X.
Proof. See [56], [79] or [81], Theorem 1.6.
Remark 9 If the choice µ = λ is always possible in (NQ3) of Definition 28, the function
k·kλ is called a (realvalued) seminorm2. If the topology of the linear space X can be
generated by a family of seminorms, the resulting topological linear space is a locally convex
space. In this case, the sets Bλ,r are convex, hence there is a neighborhood base of θ ∈ X
consisting of convex sets.
2
This term is not consistent with Definition 23. It seems to be preferable to replace the term ”seminorm”
by ”pseudonorm” since every seminorm generates a pseudometric in an obvious way. However, we keep on
using ”seminorm” for historical reasons.

As it is the case for uniform spaces, the family of quasinorms can be replaced by a single
norm with values in a set with less structure than the set of nonnegative real numbers.
Definition 29 Let (X, +) be a linear space and (V, +, ≤K) be a quasiordered linear spaces
where K ⊆ V is the convex ordering cone. A function N : X × X → V is called an order
quasinorm iff there is a linear mapping T : V → V such that T (K) ⊆ K and for all
x, x1, x2 ∈ X the following conditions are satisfied:
(N1) θ ≤K N (x) where θ ∈ V is the neutral element of (V, +);
(N2) N (tx) = |t| N (x) for all t ∈ IR;
(N3) N (x1 + x2) ≤K T (N (x1) + N (x2)).
(N4) N (x) = θ ∈ V implies x = θ ∈ X;
is satisfied, then the order quasinorm N is called separating.
If T is the identity, the function N satisfying (N1) – (N3) is called order seminorm. A
separating order seminorm is called order norm.
The concept of an order norm has been introduced by Kantorovich [69]. It is used in
vector optimization and approximation theory, compare the books of Jahn [62], [63] as
well as [44], for example. Order norms are sometimes called vector-valued or cone-valued
norms.
Order quasinorms with values in topological semifields appeared in [2]. The following
two propositions contain a complete characterization of topological linear spaces using
order quasinorms. For this purpose we use a procedure close to that of [70], Theorem 3
and 4, but avoiding the use of topological semifields explicitly.
Proposition 38 Let (X, +) be a linear space and (V, +, S, ≤K) be a normally ordered,
locally convex topological linear space with convex ordering cone K. Let S (θ) be a neigh-
borhood base of θ ∈ V consisting of convex full sets. If N : X → V is an order quasinorm
on X, then the following assertions hold true:
(i) The entity of sets
BS := {x ∈ X : N (x) ∈ S} , S ∈ S (θ)
form a neighborhood base of θ ∈ X for some topology T on X; it is the coarsest topology
on X such that (X, +, T ) is a topological linear space and N is continuous at θ ∈ X;
(ii) The entity of sets
ES := {(x1, x2) ∈ X × X : N (x1 − x2) ∈ S} , S ∈ S (θ)
form a base of a uniformity U on X such that T is the uniform topology generated by U.
(iii) If N is separating and (V, S) is separated, then the topology T and the uniformity U
are separated as well.

Proof. (i) Since V is normally ordered, there is a neighborhood base S (θ) of θ ∈ V
satisfying (B), i.e.
∀S ∈ S (θ) : v1, v2 ∈ S, v1 ≤K v ≤K v2 ∈ S ⇒ v ∈ S. (2.27)
Define B := {BS : S ∈ S (θ)}. Let us show that B is a neighborhood base of θ ∈ X for
some topology T .
First, let B1, B2 ∈ B, i.e. there are S1, S2 ∈ S (θ) such that Bi = {x ∈ X : N (x) ∈ Si},
i = 1, 2. Since S (θ) is a neighborhood base of θ ∈ V , there is S ∈ S (θ) such that
S ⊆ S1
T
S2. It follows that BS ⊆ B1
T
B2.
Next, take B ∈ B, i.e. there is S ∈ S (θ) such that B = {x ∈ X : N (x) ∈ S}. We are
going to show that there is B00 ∈ B such that B00 ⊕ B00 ⊆ B. Since S (θ) is a neighborhood
base of θ ∈ V , there is S0 ∈ S (θ) such that S0 ⊕ S0 ⊆ S. Since T is continuous, there
is S00 ∈ S (θ) such that T (S00) ⊆ S0, hence T (S00) ⊕ T (S00) ⊆ S. Take x1, x2 ∈ B00 :=
{x ∈ X : N (x) ∈ S00}. By (N3) of Definition 29 we obtain
N (x1 + x2) ≤K T (N (x1) + N (x2)) = T (N (x1)) + T (N (x2)) ∈ T S00

⊕ T S00

⊆ S.
(2.27) implies N (x1 + x2) ∈ S, i.e. x1 + x2 ∈ B. Therefore, B00 ⊕ B00 ⊆ B as desired.
Take B ∈ B, i.e. B = BS for some S ∈ S (θ). We show that there is S0 ∈ S (θ) such
that tBS0 ⊆ B whenever |t| ≤ 1. Indeed, since S (θ) is a neighborhood base of θ ∈ V ,
there is S0 ∈ S (θ) such that tS0 ⊆ S whenever |t| ≤ 1. Take x ∈ BS0 , i.e. N (x) ∈ S0.
Then tx ∈ tBS0 and by (N2) of Definition 29 it follows
N (tx) = |t| N (x) ∈ |t| S0
⊆ S,
therefore tx ∈ B and consequently tBS0 ⊆ B.
Finally, we shall show that the sets BS are absorbing. Take S ∈ S (θ) and x ∈ X.
Since S is absorbing, there is t 0 such that N (x) ∈ tS. (N2) of Definition 29 and the
definition of BS imply x ∈ tBS as desired.
Concludingly, there is a topology T such that the couple (X, T ) is a topological linear
space such that B is a neighborhood base of θ ∈ X. Of course, this topology is the coarsest
one making N continuous at θ ∈ X.
(ii) Is clear since the topology of a topological linear space is uniformizable.
(iii) Is obvious from (N5) of Definition 29, since x̄ ∈ {x ∈ X : N (x) ∈ S} for all
S ∈ S (θ) implies N (x̄) ∈
T
S∈S(θ) S. Hence N (x̄) = θ since (V, S) is separated.
Remark 10 If the order quasinorm is such that T can be chosen to be the identity, the
resulting topology on X is locally convex since the sets BS = {x ∈ X : N (x) ∈ S}, S ∈
S (θ) are convex: Take x1, x2 ∈ BS, t ∈ (0, 1). Then N (x1) , N (x2) ∈ S. (N3) and (N2)
of Definiton 29 and the convexity of S imply
N (tx1 + (1 − t) x2) ≤K tN (x1) + (1 − t) N (x2) ∈ S.

The next proposition shows that every topological linear space can be supplied with an
order quasinorm generating the topology.
Proposition 39 Let (X, +, T ) be a topological linear space. Then there are a normally
ordered, locally convex space (V, +, S, ≤K) with convex ordering cone K, an order quasi-
norm N : X → V and a continuous linear operator T : V → V such that T (K) ⊆ K and
(i) For every neighborhood B of θ ∈ X there is a S ∈ S (θ), S (θ) being a neighborhood
base of θ ∈ V such that
{x ∈ X : N (x) ∈ S} ⊆ B;
(ii) For every neighborhood S of θ ∈ V , there is a neighborhood B of θ ∈ X such that
B ⊆ {x ∈ X : N (x) ∈ S} .
Proof. Proposition 37 tells us that there is a family {k·kλ}λ∈Λ of quasinorms generating
the topology, i.e. the family of sets
{x ∈ X : kxkλ ≤ r} , r 0, λ ∈ Λ
is a neighborhood base of θ ∈ X. Consider the locally convex space IRΛ being normally
ordered by K = IRΛ
+. Define
N (x) := (kxkλ)λ∈Λ .
The space IRΛ can be identified with the set of all functions mapping Λ into IR. In this
sense,
N (x) (λ) = kxkλ , λ ∈ Λ.
By (N3) of Definition 28, there is a mapping φ : Λ → Λ such that for each λ ∈ Λ
kx1 + x2kλ ≤ kx1kφ(λ) + kx2kφ(λ) .
Denoting an element of IRΛ by v = (vλ)λ∈Λ, we define a mapping T : IRΛ → IRΛ by
T (vλ)λ∈Λ

:= vφ(λ)

λ∈Λ
.
We claim that T is linear, positive and continuous at θ ∈ IRΛ. Linearity and positivity are
obvious. To show the continuity, take a neighborhood of θ ∈ IRΛ, i.e. choose r1, . . . , rn 0,
λ1, . . . , λn ∈ Λ and consider
S :=

v ∈ IRΛ
: |vλi
| ri, i = 1, . . . , n .
The set
S0
:=

w ∈ IRΛ
: wφ(λi) ri, i = 1, . . . , n
is also a neighborhood of θ ∈ V . Moreover,

v ∈ IRΛ : v = Tw, w ∈ S0 = T (S0) ⊆ S,
since v ∈ T (S0) means v = Tw for some w ∈ S0, hence |vλi
| = wφ(λi) ri for i = 1, . . . , n.
This proves the claim.

The mapping N defined above is an order quasinorm. The conditions (N1), (N2) and
(N3) of Definition 29 are easy to check.
Let us prove (i). It suffices to show that for each
B := {x ∈ X : kxkλ ≤ r} , r 0, λ ∈ Λ
there is S ∈ S (θ), S ⊆ IRΛ such that BS := {x ∈ X : N (x) ∈ S} ⊆ B. This is obvious
for S =

v ∈ IRΛ : |vλ| r .
Finally, we show (ii). Take S ∈ S (θ), S ⊆ IRΛ, i.e.
S =

v ∈ IRΛ
: |vλi
| ri, i = 1, . . . , n .
Then
{x ∈ X : N (x) ∈ S} =
n

i=1

x ∈ X : kxkλi
≤ ri .
The sets

x ∈ X : kxkλi
≤ ri , i = 1, . . . , n, are neighborhoods of θ ∈ X. Hence there
is a neighborhood B ⊆ {x ∈ X : N (x) ∈ S} of θ ∈ X. This completes the proof of the
proposition.
Remark 11 If (X, +, T ) is a locally convex space, the mapping T can be chosen to be the
identity. This is due to the fact that in the proof of Proposition 39 the family of quasinorms
can be replaced by a family of seminorms.
Taking Remark 10 into account, the class of separated locally convex spaces coincides
with that of order normed spaces.
2.3.5 Conlinear spaces via topological constructions
Starting with a topological linear space (V, +, T ) we may construct conlinear subspaces of

b
P (V ) , ⊕

with the help of topological properties. We refer the reader to the thesis [83]
for a far reaching application for the case V = IRn.
Example 12 Let (V, +, T ) be a topological linear space and denote by F (V ) the set of
all closed subsets of V . The Minkowski sum of two closed sets is not closed in general.
Therefore, we define for W1, W2 ∈ F (V )
W1 b
⊕W2 := cl (W1 ⊕ W2) .
Then F (V ) , b
⊕

is a semilinear space, hence also conlinear.
Using the conventions of Proposition 19 we only have to show the law of assoziativity
for b
⊕, see (S1) of Definition 10. We show that cl (W1 ⊕ W2) ⊕ W3 ⊆ cl (W1 ⊕ W2 ⊕ W3).
Indeed, if w ∈ cl (W1 ⊕ W2)⊕W3, then w ∈ (W1 ⊕ W2 ⊕ B)⊕W3 for each member B of a
neighborhood base of θ ∈ V . Since ⊕ is associative, this implies w ∈ (W1 ⊕ W2 ⊕ W3)⊕B,
hence w ∈ cl (W1 ⊕ W2 ⊕ W3). The opposite inclusion follows by symmetry.
In a similiar way, the set of all closed convex subsets of a topological linear space can
be provided with a semilinear structure.

Chapter 3
Order Premetrics and their
Regularity
A basic ingredients for results in the spirit of Ekeland’s variational principle is a metric
space and an order relation on the space defined in terms of the metric itself. If a function
is involved that maps not into the reals but into a more general set, for example a linear
space, the metric has to be replaced by an expression mapping into the same set. On the
other hand, not all properties of a metric are really essential for a proof of a variational
principle. This has been already realized in [8]. Therefore, we extend the concept of a
(realvalued) metric to functions into ordered monoids maintaining only a few but not all
properties of a metric. We do not focus on topological structures which may be generated
by such extensions of a metric as it has been done in Section 2.3 with the concept of order
(pseudo)metrics. Of course, an order (pseudo)metric is an example of an order premetric
that is introduced in the next definition.
Definition 30 Let X be a nonempty set and (Y, ◦, ≤) a quasiordered monoid with neutral
element θ ∈ Y . A function Φ : X ×X → Y is called an order premetric iff the following
conditions are satisfied:
(P1) ∀x ∈ X: θ = Φ (x, x);
(P2) ∀x1, x2 ∈ X: θ ≤ Φ (x1, x2);
(P3) ∀x1, x2, x3 ∈ X: Φ (x1, x3) ≤ Φ (x1, x2) ◦ Φ (x2, x3).
The condition (P1) is not a true restriction as the following lemma shows.
Lemma 4 Let X, Y be as in Definition 30 and Ψ : X × X → Y be a function satisfying
(P2) and (P3). Then the function Φ : X × X → Y defined by
Φ (x1, x2) :=
(
Ψ (x1, x2) : x1 6= x2
θ : x1 = x2
(3.1)
is an order premetric.
61

62 Chapter 3. Order Premetrics and their Regularity
Proof. For Φ, the conditions (P1), (P2), (P3) may be checked straightforward.
Definition 31 Let (X, U) be a uniform space and (Y, ◦, ≤) a quasiordered monoid with
neutral element θ ∈ Y . A function Φ : X × X → Y satisfying (P2), (P3) of Definition 30
is called (sequentially) regular with respect to y1, y2 ∈ Y iff it satisfies:
(P4) If {xn}n∈IN ⊆ X and
∀n ∈ IN : y1 ◦
n
X
k=0
Φ (xk+1, xk) ≤ y2,
then {xn}n∈IN is asymptotic, i.e.
∀E ∈ U ∃nE ∈ IN ∀n ≥ nE : (xn+1, xn) ∈ E.
Note that, if (X, d) is a metric space, a sequence {xn}n∈IN is asymptotic if and only if
limn→∞ d (xn+1, xn) = 0. The definition above applies also to this case.
Let Y be not only a monoid but also a group. Then y1 ◦
Pn
k=0 Φ (xk+1, xk) ≤ y2 for
all n ∈ IN if and only if θ ≤
Pn
k=0 Φ (xk+1, xk) ≤ y2 ◦ y−1
1 for all n ∈ IN. Hence, it is
enough to assume that the boundedness of above of {
Pn
k=0 Φ (xk+1, xk) : n ∈ IN} implies
that {xn}n∈IN is asymptotic.
Lemma 5 Let (X, U) be a uniform space, (Y, ◦, ≤) a quasiordered monoid and Ψ : X ×
X → Y be a function satisfying (P2) and (P3). Then, the order premetric Φ, defined via
(3.1) is regular if and only if Ψ is regular.
Proof. Clearly, the regularity of Φ implies the regularity of Ψ. To show the converse,
assume the regularity of Ψ and take a sequence {xn}n∈IN ⊆ X such that
∀n ∈ IN : y1 ◦
n
X
k=0
Φ (xk+1, xk) ≤ y2.
If xn+1 = xn for some n ∈ IN, we may delete xn+1 from the sequence since (xn+1, xn) ∈ E
for each E ∈ U in this case. Doing this as long as possible, we either obtain only finitly
many elements of the original sequence or a subsequence {xnl
}l∈IN ⊆ X such that xnl+1 6=
xnl
. In the first case, the original sequence is constant up to finitely many elements and
hence asymptotic. In the second case, we have Φ (xnl+1, xnl
) = Ψ (xnl+1, xnl
) for all l ∈ IN.
This implies
∀l ∈ IN : y1 ◦
l
X
k=0
Ψ (xnk+1, xnk
) ≤ y2.
¿From the regularity of Ψ we may deduce that for E ∈ U and nl ∈ IN sufficiently large,
we have (xnl+1, xnl
) ∈ E. This completes the proof.

63
Example 13 In [65], Kada et al. introduced the concept of a w-distance as follows: Let
(X, d) be a metric space and w : X × X → IR+ be a function satisfying (i) w (x1, x3) ≤
w (x1, x2) + w (x2, x3) for all x1, x2, x3 ∈ X; (ii) For each x0 ∈ X, the function x →
w (x0, x) is lower semicontinuous; (iii) For each ε 0, there is δ 0 such that w (x, x1)
δ, w (x, x2) δ imply d (x1, x2) ε.
We show that a w-distance is a regular premetric with (Y, ◦) = (IR+, +) and the usual
≤-relation for real numbers for y1 = 0 and each y2 = r ∈ IR+. To this purpose, take a
sequence {xn}n∈IN ⊂ X such that
Pn
k=0 w (xk+1, xk) ≤ r is true for all n ∈ IN. Since
0 ≤ w (xn+1, xn) for all n ∈ IN, this implies 0 = limn→∞ w (xn+1, xn). Fix ε 0. Then
there is nε ∈ IN such that
∀n ≥ nε : w (xn, xn+1)
δ
2
δ, w (xn+1, xn+2)
δ
2
with δ 0 from (iii). By (i), we obtain
w (xn, xn+2) ≤ w (xn, xn+1) + w (xn+1, xn+2) δ.
and therefore from (iii) d (xn+1, xn+2) ε for all n ≥ nε. Hence the sequence {xn}n∈IN is
asymptotic.
In [65], a list of examples can be found showing that the set of w-distances contains
the metric d but much more elements. Note that already in Brønstedts paper [8] similar
functions has been used on uniform spaces.
Example 14 A simple example of an ordered monoid is (Y := IR+ ∪ {+∞} , +, ≤). Let
(X, d) be a metric space. Then d is a regular order premetric with respect to y1 = 0,
y2 = r ∈ IR+ $ Y , but not with respect to y2 = +∞ ∈ Y , of course.
Example 15 Let (X, d) be a metric space and let (Y, +, T , ≤K) be a normally ordered
separated locally convex space with ordering cone K ⊆ Y . Take k ∈ K {0}. Then
Φ (x1, x2) := kd (x1, x2) is a regular order premetric in the sense of Definition 30. This
result is presented by Isac, compare Proposition 1 and the proof of Theorem 3 of [59].
Example 16 Let (X, U) be a uniform space and let (Y, ◦, ≤, T ) be a normally ordered
topological Abelian group. Then, every order pseudometric D : X × X → Y in the sense
of Definition 25 is an order premetric.

64 Chapter 3. Order Premetrics and their Regularity

Chapter 4
Variational Principles on Metric
Spaces
In this chapter, we are dealt with ordered metric spaces. We ask for circumstances en-
suring the existence of minimal elements with respect to the given order relation. The
completeness of the space in connection with a certain regularity assumption of the order
turns out to be the crucial point. We state three equivalent formulations of the main
result and draw a series of corollaries including new results as well as almost all known
theorems that are equivalent to or generalizations of Ekeland’s variational principle on
metric spaces.
4.1 The basic theorem on metric spaces
4.1.1 Preliminaries
Let (X, d) be a metric space provided with a quasiorder , i.e. a reflexive and transitive
relation. In the following, we simply denote the lower sections Sl (x) = {x0 ∈ X : x0 x}
by S (x) for x ∈ X, compare Definiton 12.
A sequence {xn}n∈IN ⊆ X is said to be decreasing with respect to iff
∀n ∈ IN : xn+1 xn.
The metric space X is said to be –complete iff every decreasing Cauchy sequence
in X converges to some element of X. Of course, every complete metric space is –
complete for every quasiordering while the converse is not true: Take X = [0, 1) and
the usual ≤–relation. A quasiorder is called lower closed iff for any decreasing sequence
{xn}n∈IN ⊆ X converging to some x ∈ X
∀n ∈ IN : x xn
holds true. A quasiorder is lower closed if and only if the sections S (x) are closed with
respect to decreasing sequences, i.e. if {xn}n∈IN ⊆ S (x) and limn→∞ xn = x̂, then x̂ ∈
65

66 Chapter 4. Variational Principles on Metric Spaces
S (x). A quasiorder on a metric space X is called regular iff every decreasing sequence
{xn}n∈IN ⊆ X is asymptotic, i.e.,
lim
n→∞
d (xn+1, xn) = 0.
Regularity improves the properties of the order as the following proposition shows.
Proposition 40 A regular quasiorder on a metric space X is antisymmetric.
Proof. Take x, x0, ∈ X such that x0 x as well as x x0. Define
{xn} =

x, x0
, x, x0
, x, . . .
being a –decreasing sequence. The regularity of implies x = x0.
If is a regular quasiorder on X, Proposition 40 admits to say that x̄ ∈ X is a minimal
point with respect to iff {x̄} = S (x̄).
4.1.2 The basic theorem
The stage is set for the basic minimal element theorem on metric spaces.
Theorem 16 Let the following assumptions be satisfied:
(M1) (X, d) is a metric space;
(M2) is a reflexive and transitive relation on X such that X is –complete;
(M3) is regular;
(M4) is lower closed.
Then, for each x0 ∈ X, there exists x̄ ∈ X such that
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. Starting with x0 ∈ X, we define a sequence by choosing xn+1 ∈ S (xn) such that
d (xn+1, xn) ≥ sup
x∈S(xn)
d (x, xn) −
1
n
if sup
x∈S(xn)
d (x, xn) +∞
or d (xn+1, xn) ≥ 1 if supx∈S(xn) d (x, xn) = +∞. There is n0 ∈ IN such that the latter
case can not occur for each n ≥ n0. Otherwise, we may obtain a decreasing sequence with
d (xn+1, xn) ≥ 1 for all n ∈ IN contradicting the regularity.
The transitivity of implies xm ∈ S (xn) for m ≥ n. From this, we get
d (xm, xn) ≤ sup
x∈S(xn)
d (x, xn) ≤ d (xn+1, xn) +
1
n
for all n ∈ IN, n ≥ n0. Assumption (M3) implies that {xn}n∈IN is a Cauchy sequence,
hence convergent to some x̄ ∈ X. By (M4), we have x̄ ∈ S (xn) for all n ∈ IN. Especially,
x̄ ∈ S (x0).

4.1. The basic theorem on metric spaces 67
Take x0 ∈ S (x̄). Then by transitivity, x0 x̄ xn, hence x0 ∈ S (xn) for all n ∈ IN.
This implies
d x0
, xn

≤ sup
x∈S(xn)
d (x, xn) ≤ d (xn+1, xn) +
1
n
.
Concludingly, xn → x0, hence x0 = x̄ and therefore, {x̄} = S (x̄). This completes the
proof.
Remark 12 Starting with a relation 0 being only transitive, we can obtain a quasiorder-
ing by defining
x0
x :⇐⇒ x0
0
x or x0
= x.
If (M3) and (M4) are satisfied for 0, then for , too. Therefore, the restriction to
quasiorders is not essential. Hence, taking Proposition 40 into account, we may assume
that in Theorem 16 is a partial order without loss of generality.
4.1.3 Equivalent formulations of the basic theorem
At an early stage, it has been observed that Ekeland’s variational principle has a number
of equivalent formulations. The papers [24] and [98] by Danes̆ and Penot, respectively,
are the first systematic surveys about this topic. Further results in this direction can be
found e.g. in [38], [3], [96] and [97].
In [118] (Theorem 3.2) and in [67] (Corollary) a fixed point result has been established
as a corollary of the the main result.
In the 1976 paper [9], Caristi observed that his fixed point theorem is an equivalent
formulation of Ekeland’s variational principle. We establish a generalized version and show
the equivalence to Theorem 16. The following result is close to Theorem 3.1 in [22].
We consider a set-valued mapping T : X → b
P (X). A point x̄ ∈ X is said to be a
fixed point of T iff x̄ ∈ T (x̄). A point x̄ ∈ X is said to be an invariant point of T iff
{x̄} = T (x̄).
Theorem 17 Let (M1) through (M4) of Theorem 16 be in force and, additionally, T :
X → b
P (X) be a set-valued mapping. If T satisfies
∀x ∈ X, ∃x0
∈ T (x) : x0
x, (WC)
then there is x̄ ∈ X such that x̄ ∈ T (x̄), i.e. x̄ is a fixed point of T. If T satisfies
∀x ∈ X, ∀x0
∈ T (x) : x0
x, (SC)
then there is x̄ ∈ X such that {x̄} = T (x̄), i.e. x̄ is an invariant point of T.
Proof. Each point x̄ satisfying the conclusions of Theorem 16 does the job.
Conversely, Theorem 16 can be proven using Theorem 17. To see this, replace X in

Theorem 17 by S (x0) of Theorem 16 and consider the map T (x) := S (x) that satisfies
(SC).
In their 1993 paper [96], Oettli and Théra proved an equivalent formulation of Eke-
land’s principle. See also [97]. This theorem can be generalized in order to produce a
reformulation of Theorem 16.
Theorem 18 Let (M1) through (M4) of Theorem 16 be in force and, additionally:
(M5) The set M ⊆ X satisfies
∀x ∈ S (x0) M ∃x0
∈ S (x) {x} .
Then, there exists x̄ ∈ S (x0) ∩ M.
Proof. By Theorem 16, there exists x̄ ∈ S (x0) such that {x̄} = S (x̄). By assumption
(M5), x̄ ∈ M, hence x̄ ∈ M ∩ S (x0).
Conversely, Theorem 16 can be proven using Theorem 18. To see this, let (M1) through
(M4) be in force. Define M := {x ∈ X : {x} = S (x)}. If x 6∈ M, then there exists
x0 ∈ X such that x0 6= x, x0 x, hence (M5) is satisfied. By Theorem 18, there exists
x̄ ∈ S (x0) ∩ M, hence {x̄} = S (x̄).
Let us note that the equivalence of the Theorems 16, 17 and 18 is understood in the
sense that each of it can be proven using each of the others without any reference to the
induction process that appears in the proof of Theorem 16.
4.1.4 The regularity assumptions
The next proposition shows that under the regularity assumption decreasing sequences
are even Cauchy.
Proposition 41 Let (X, d) be a metric space, quasiordered by . Then is regular if
and only if every decreasing sequence {xn}n∈IN ⊂ X is a Cauchy sequence, i.e.
∀ε 0, ∃p ∈ IN, ∀n, m ≥ p : d (xm, xn) ≤ ε.
Proof. Of course, every Cauchy sequence is asymptotic. Conversely, let be regular and
{xn}n∈IN ⊂ X a decreasing, hence asymptotic sequence. Assume {xn}n∈IN is not Cauchy.
Then there is ε 0 such that
∀p ∈ IN, ∃m n ≥ p : d (xm, xn) ≥ ε.
Set p = 1. Then there are m1 n1 ≥ p such that d (xm1 , xn1 ) ≥ ε. Set x0
1 = xn1
and x0
2 = xm1 . Set p = m1. Then there are m2 n2 ≥ p such that d (xm2 , xn2 ) ≥ ε.
Set x0
3 = xn2 and x0
4 = xm2 . Continue this procedure to obtain a subsequence {x0
k}k∈IN
of {xn}n∈IN with d x0
2k+2, x0
2k+1

≥ ε for k ∈ IN. This sequence is decreasing, but not
asymptotic contradicting the assumption.
Proposition 41 yields: If is regular, then X is –complete if and only if every decreasing
sequence in X converges to some element of X.

Remark 13 The regularity assumption (M3) has been used by Turinici [118] as well as
Dancs, Hegedüs and Medvegyev [22]. Theorem 3.1 in [118] has been proved using Zorn’s
Lemma whereas Theorem 3.2 in [22] deals with complete metric spaces.
Several attempts have been made to replace the regularity assumption (M3) by a more
general one. We mention the following conditions:
∀ε 0, ∀x ∈ X, ∃u ∈ S (x) : diam S (u) ≤ ε; (M3-1)
∀ε 0, ∀x ∈ X, ∃u ∈ S (x) : sup
x0∈S(u)
d x0
, u

≤ ε; (M3-2)
∀ε 0, ∀x ∈ X, ∃u ∈ S (x) : x00
x0
u =⇒ d x0
, x00

ε

. (M3-3)
The following relationships can be established.
Lemma 6 Let (X, d) be a metric space. Then, the conditions (M3-1), (M3-2) and (M3-3)
are mutually equivalent. Moreover, (M3) implies each of (M3-1), (M3-2) and (M3-3).
Proof. The implications (M3-1) ⇒ (M3-3) ⇒ (M3-2) are immediate. Let (M3-2) be
in force. Fix ε 0 and x ∈ X. Take u ∈ S (x) such that supx0∈S(u) d (x0, u) ≤ ε
2 . For
x0, x00 ∈ S (u) we obtain
d x0
, x00

≤ d x0
, u

+ d u, x00

≤ ε,
hence (M3-1) is satisfied.
Finally, we show that regularity implies (M3-2). Assume the contrary. Then there are
ε 0 and x ∈ X such that
∀u ∈ S (x) : sup
x0∈S(u)
d x0
, u

ε.
Starting with x0 = x one may find a sequence such that xn ∈ S (x) and d (xn+1, xn) ≥ ε
for all n ∈ IN. This contradicts regularity.
By a simple modification of the proof of Theorem 16 one obtains the same conclusions
assuming (MP-i), i = 1, 2, 3, instead of (M3). However, it is not sure if this is a true
generalization.
Theorem 19 Let the assumptions (M1), (M2) and (M4) of Theorem 16 be in force and
additionally either of (M3-1), (M3-2) and (M3-3). Then, for each x0 ∈ X, there exists
x̄ ∈ X such that
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. We use (M3-1) for the proof. Starting with x0 ∈ X, we define a sequence
by choosing xn+1 ∈ S (xn) such that diam S (xn) ≤ 1
n. The transitivity of implies
xm ∈ S (xn) for m ≥ n. From this, we get
d (xm, xn) ≤ diam S (xn) ≤
1
n
for all n ∈ IN. The remaining part of the proof is the same as that for Theorem 16.

4.1.5 Completeness
By several authors, it has been observed that Ekeland’s principle or its reformulations
characterize the completeness of the metric space X. Compare [74], [126], [111], [22] and
[112]. The same is true for Theorem 16. We state the result in the following form.
Theorem 20 Let (X, d) be a metric space. If for all reflexive and transitive relations
being regular and lower sequentially closed the assertions of Theorem 16 hold true, then
X is complete.
Proof. Assume that X is not complete. Then there is a sequence X = M0 ⊇ M1 ⊇ M2 ⊇
. . . of nonempty closed subsets of X such that diam Mn → 0 as n → ∞, but ∩∞
n=1Mn = ∅.
Define an ordering relation by
S (x) := Mn+1 ∪ {x} if x ∈ Mn and x 6∈ Mn+1
and x0 x iff x0 ∈ S (x). One may check that is reflexive, transitive (lower sequentially)
closed and regular by assumption. A minimal point x̄ of would satisfy {x̄} = S (x̄) =
Mn̄ ∪ {x̄} for some n̄ ∈ IN. This would imply Mn̄ = ∅, a contradiction.
In fact, it is enough that the assumption of Theorem 20 holds true for all ordering relations
generated by uniformly continuous functions f : X → IR, being bounded below, in the
following way:
x0
x ⇐⇒ f x0

+ d x0
, x

≤ f (x) .
This shows the proof of the theorem of Weston in [126].
4.1.6 Set relation formulation
The variational principle of Ekeland is bound up with so–called minimal point theorems
in procduct spaces. This idea probably goes back to Phelps (compare [102]) and has been
put explicitely in [3].
At this early stage of our development, we shall establish a minimal point theorem
involving order relations on product sets X × Y . The crucial point of the proof is to
generate a suitable order relation defined on X only in order to apply Theorem 16. This
is possible using the set relations introduced in Section 2.2.1.
Note that the set Y is merely assumed to be nonempty. Neither algebraic nor topo-
logical structure concerning Y appears.
We need some notation to formulate the results. Let (X, d) be a metric space and
Y as well as M ⊆ X × Y be nonempty sets. For x ∈ X, let us define M (x) :=
{(x0, y) ∈ X × Y : x0 = x, (x0, y) ∈ M} ∈ b
P (X × Y ) and MY (x) := {y ∈ Y : (x, y) ∈ M} ∈
b
P (Y ). Let be a quasiorder on M. Then, ({M (x) : x ∈ X} , 4) as well as ({M (x) : x ∈ X} , 2)
is quasiordered. Here, the relations 4 and 2 are the extensions of to subsets of
b
P (X × Y ), compare (2.6), (2.7). Note that M (x0) 4 M (x) if and only if
∀y ∈ MY (x) , ∃y0
∈ MY x0

: x0
, y0

(x, y) (4.1)

and M (x0) 2 M (x) if and only if
∀y0
∈ MY x0

, ∃y ∈ MY (x) : x0
, y0

(x, y) . (4.2)
(M1’) (X, d) is a metric space and Y as well as M ⊆ X × Y are nonempty sets;
(M2’) is a quasiorder, i.e., a reflexive and transitive relation on X × Y ;
(M3’) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence, i.e.
∀n ∈ IN : (xn+1, yn+1) (xn, yn)
and {xn}n∈IN converges to x ∈ X, then there exists y ∈ Y such that (x, y) ∈ M and
∀n ∈ IN : (x, y) (xn, yn) ;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence, then limn→∞ d (xn+1, xn) = 0.
Then, for each x0 ∈ X with MY (x0) 6= ∅, there exists x̄ ∈ X such that MY (x̄) 6= ∅ and
(i) M (x̄) 4 M (x0)
(ii) M (x) 4 M (x̄) =⇒ x = x̄.
Proof. We define a binary relation on X be setting
x0
X x ⇐⇒ M x0

4 M (x) .
in order to apply Theorem 16. Of course, X is reflexive and transitive. Let us check the
regularity assumption (M3) of Theorem 16. Take a sequence {xn}n∈IN decreasing with
respect to X, i. e.
∀yn ∈ MY (xn) ∃yn+1 ∈ MY (xn+1) : (xn+1, yn+1) (xn, yn) . (4.3)
Take y0 ∈ MY (x0). Find y1 ∈ MY (x1) via (4.3) such that (x1, y1) (x0, y0). Find
y2 ∈ MY (x2) via (4.3) such that (x2, y2) (x1, y1). Continuing this procedure, one gets a
sequence {(xn, yn)}n∈IN being decreasing with respect to . Since is regular by (M4’),
we obtain limn→∞ d (xn+1, xn) = 0 as desired.
Finally, we show that X is lower closed. Take a sequence {xn}n∈IN decreasing with
respect to X and converging to x ∈ X. We have to show that x X xn for each n ∈ IN.
Fix n ∈ IN. By (4.3), one can find yn+1 ∈ MY (xn+1) such that (xn+1, yn+1) (xn, yn)
and, as before, gets a sequence {(xn+m, yn+m)}m∈IN being decreasing with respect to .
Of course, we still have limm→∞ xn+m = x. Assumption (M3’) implies the existence of
y ∈ MY (x) such that for each m ∈ IN
(x, y) (xn+m, yn+m) (xn, yn) .
This procedure is applicable for every n ∈ IN (the corresponding y ∈ MY (x) may depend
on n). The lower closedness of X is proven.

The final step of the proof is an application of Theorem 16 to the metric space (X, d)
and the relation X in order to obtain (i) and (ii). This is straightforward.
Analyzing the proof above, one may see that it is not possible to show the regularity and
lower closedness of the order X if simply 4 is replaced by 2. The corresponding result
for 2 reads as follows.
(M1’) (X, d) is a metric space and Y as well as M ⊆ X × Y are nonempty sets;
(M3’) If {(xn, yn)}n∈IN ⊆ M is a increasing sequence, i.e.
∀n ∈ IN : (xn, yn) (xn+1, yn+1)
∀n ∈ IN : (xn, yn) (x, y) ;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a increasing sequence, then limn→∞ d (xn+1, xn) = 0.
(i) M (x0) 2 M (x̄)
(ii) M (x̄) 2 M (x) =⇒ x = x̄.
Proof. There are at least two proofs possible. The first idea is to reformulate Theorem
16 as a maximal element theorem and proceed as in the proof of Theorem 21. Another
plan is to apply Theorem 21 to a suitable order relation. We shall do the latter. Define a
binary relation 0 on X × Y by
x0
, y0

0
(x, y) ⇐⇒ (x, y) x0
, y0

.
Of course, 0 is a quasiorder. Moreover, a sequence {(xn, yn)}n∈IN ⊆ X × Y is decreasing
with respect to 0 if and only if it is increasing with respect to . Hence (M3’) and
(M4’) of Theorem 21 are satisfied for 0 if and only if (M3’) and (M4’) of Theorem 22 are
satisfied for , respectively. Denote by 40 the relation defined by (4.1) replacing by 0.
We can apply Theorem 21 to get an x̄ ∈ X such that MY (x̄) 6= ∅ and
(i0) M (x̄) 40 M (x0)
(ii0) M (x) 40 M (x̄) =⇒ x = x̄.
Observing that M (x0) 40 M (x) if and only if M (x) 2 M (x0) we see that (i0) and (ii0)
are equivalent to (i) and (ii) of Theorem 22, respectively. This completes the proof.
Remark 14 A special case of Theorem 21 (as well as of Theorem 22) is the case if Y
consists of a single element only. In this case, Theorem 21 reduces to Theorem 16 (as
well as Theorem 22 to a maximal element reformulation of Theorem 16). On the other
hand, Theorem 21 (as well as Theorem 22) are proven using Theorem 16 without any
reference to the induction process in the proof of Theorem 16. In this sense, the theorems
are equivalent.

4.2. Results with functions into ordered monoids 73
Remark 15 Another special case is M (x) = {(x, y)}, i.e., M (x) is a singleton. In this
case, the set M ⊆ X × Y defines a function f : X → Y . The relation coincides with 4
and X, and they compare arguments and values of f at the same time:
x0
X x ⇐⇒ x0
, f x0

(x, f (x)) .
Remark 16 Assumption (M3’) of the Theorems 21 and 22 coincides with assumption
(H1) in [47] if Y is assumed to be a topological linear space. Thanks to this assumption,
we can get rid of assumptions concerning topological and algebraic properties of Y or the
concrete form of . On the other hand, it is by no means a trivial task to verify assumption
(M3’) in special cases. Compare the discussion in [47] and [44], Section 3.10. Finally,
note that assumption (2) of the famous Theorem 1 due to Brézis and Browder has a similar
structure, but does not involve product sets.
4.2 Results with functions into ordered monoids
The Theorems 16, 17 and 18 are formulated in such a way that the order relation may
or may not depend on the metric d. The charming character of Ekeland’s variational
principle and related theorems relies on its recursive structure: the order relation is defined
in terms involving the metric itself. Actually, this ensures the topological requirements
such as (M3) and (M4) of Theorem 16. It has already been observed by Brønsted [8] that
not all properties of a metric are necessary for defining order relations in order to produce
Ekeland type theorems.
To the authors opinion, it should be possible to prove (almost) all Ekeland type the-
orems and its equivalent reformulations on metric spaces by defining a suitable order
relation and applying Theorem 16. To carry out this program, is the main goal of the
remaining part of this chapter.
Results in the spirit of Ekeland’s principle usually involve a function f mapping the
metric space X to a set Y provided with some algebraic, order and topological structure.
The next sections are devoted to such results supplying Y with more and more structure
from step to step. Each of the Theorems 16, 17 and 18 (or even Theorems 21 and 22) may
be chosen as starting points for the these developments. We prefer to use Theorem 16.
The first corollary seems to require less algebraic structure of the image space than any
other result in this direction up to now. It involves functions from a metric space (X, d) to
an ordered monoid. Most of the so called setvalued or vectorvalued variants of Ekeland’s
principle on metric spaces are special cases of the theorems of the next subsection.
4.2.1 Ekeland’s variational principle
The following result is parallel to Ekeland’s variational principle from [30], but for functions
with values in ordered monoids.

Corollary 9 Let the following assumptions be satisfied:
(A1) (X, d) is a metric space and (Y, ◦, ≤) a quasiordered monoid;
(A2) Φ : X × X → Y is an order premetric;
(A3) The function f : X → Y and ỹ ∈ Y are such that
(i) ỹ ≤ f (x) for all x ∈ X;
(ii) Φ is regular with respect to ỹ, f (x0) ∈ Y for x0 ∈ X;
(iii) if {xn}n∈IN ⊆ X is a Cauchy sequence with
∀n ∈ IN : f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) (4.4)
then it converges to some x ∈ X;
(A4) If {xn}n∈IN ⊆ X converges to x ∈ X and satisfies (4.4), then f (x)◦Φ (xn, x) ≤ f (xn)
for all n ∈ IN.
Then, there exists x̄ ∈ X such that
(i) f (x̄) ◦ Φ (x̄, x0) ≤ f (x0)
(ii) x ∈ X, f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Proof. The proof is by checking the assumptions of Theorem 16 for the relation
x0
x :⇐⇒ f x0

◦ Φ x0
, x

≤ f (x) .
The relation is reflexive since ≤ is reflexive and Φ satisfies (P1) of Definition 30. It is
transitive by (P2) of Definition 30 and the transitivity of ≤. The –completeness of X
follows from (A3, (iii)). (M4) follows directly from assumption (A4). It remains to check
(M3). Let {xn}n∈IN ⊆ X be such that xn+1 xn for all n ∈ IN , i.e.
f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) .
The transitivity of ≤ implies
f (xn+1) ◦ Φ (xn+1, xn) ◦ Φ (xn, xn−1) ≤ f (xn) ◦ Φ (xn, xn−1) ≤ f (xn−1) .
Continuing this process, we obtain for each n ∈ IN
f (xn+1) ◦
n
X
k=0
Φ (xk+1, xk) ≤ f (x0) ,
where the ”sum” (
P
) is understood with respect to the operation ◦. Since ỹ ≤ f (xm) for
each m ∈ IN by (A3, (i)), it follows
ỹ ◦
n
X
k=0
Φ (xk+1, xk) ≤ f (x0) .
Since by (A3, (ii)) Φ is regular with respect to ỹ, f (x0) ∈ Y (see Definition 31), this
implies d (xn+1, xn) → 0 as n → ∞. An application of Theorem 16 yields the desired
result.

Remark 17 Let Ψ : X × X → Y be a function satisfying (P2) and (P3) of Definition
30. Then Φ according to Lemma 4 is a regular order premetric. Getting x̄ from Corollary
9, we have either x̄ = x0 or f (x̄) ◦ Ψ (x̄, x0) ≤ f (x0). Relationship (ii) of the corollary
remains in force substituting Φ by Ψ. In the following, we do not mention this possibility,
but work with regular order premetrics.
Remark 18 We do not need topological structure in Y . Note further, that it is not
necessary to have Y being a group. Thus, Corollary 9 generalizes the result of [93] with
respect to the image space Y . In fact, this generalization makes it possible for dealing with
setvalued maps: Let (Y, ◦, ≤) be a quasiordered monoid with neutral element θ ∈ Y . Then,

b
P (Y ) , , 4

and

b
P (Y ) , , 2

are quasiordered monoids as well. Hence Corollary 9
can be applied to functions f : X → b
P (Y ). We shall discuss this situation e.g. in Section
4.3.1 obtaining Ekeland type theorems for setvalued maps.
We shall indicate a sufficient condition for (A4) of Corollary 9. A function f : X → Y is
called lower monotone iff for each sequence {xn}n∈IN ⊆ X converging to some x ∈ X
and satisfying f (xn+1) ≤ f (xn) the inequality f (x) ≤ f (xn) holds true for all n ∈ IN.
Compare [93] for this kind of condition.
An order premetric Φ : X × X → Y is called lower monotone with respect to the
first variable iff for each x0 ∈ X and each sequence {xn}n∈IN ⊆ X converging to x ∈ X
and y1, y2 ∈ Y the condition
∀n ∈ IN : y1 ◦ Φ xn, x0

≤ y2
implies y1 ◦ Φ (x, x0) ≤ y2.
Lemma 7 Let (X, d) be a metric space and (Y, ◦, ≤) be an ordered monoid. Let the
function f : X → Y be lower monotone and the order premetric Φ : X × X → Y lower
monotone with respect to the first variable. Then, (A4) of Corollary 9 is satisfied.
Proof. Take a sequence {xn}n∈IN ⊆ X converging to x ∈ X such that
∀n ∈ IN : f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) .
This implies f (xm) ◦ Φ (xm, xn) ≤ f (xn) for m ≥ n on the one hand and, since θ ≤
Φ (xn+1, xn),
f (xn+1) ≤ f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn)
on the other hand. Therefore f (xn+1) ≤ f (xn) for all n ∈ IN since ≤ is transitive. The
lower monotonicity of f implies f (x) ≤ f (xn) for all n ∈ IN. For m ∈ IN, m ≥ n we
obtain
f (x) ◦ Φ (xm, xn) ≤ f (xm) ◦ Φ (xm, xn) ≤ f (xn) .
The lower monotonicity property of Φ yields the result.

4.2.2 Kirk-Caristi fixed point theorem
The next result is a fixed point theorem. The original variant goes back to Caristi and
Kirk, see [9], [74], [127], [26] and [25]. A concise proof, being constructive in some sense,
can be found in [110]. See also [4] for a thorough discussion of the proof as well as several
applications. There are many generalizations and variants, see for example [96] for an
equilibrium version, [73] and [115] for vector valued variants. Most of them are special
cases of the following corollary of Theorem 16.
We consider a set valued mapping T : X → b
P (X). Recall the definitons of a fixed
point and an invariant point of T given in Section 1.1.3.
∀n ∈ IN : f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) (4.5)
for all n ∈ IN.
If, additionally, the mapping T : X → b
P (X) satisfies the weak contraction condition
∀x ∈ X, ∃x0
∈ T (x) : f x0

◦ Φ x0
, x

≤ f (x) , (WC)
then T has a fixed point.
If the mapping T : X → P (X) satisfies the strong contraction condition
∀x ∈ X, ∀x0
∈ T (x) : f x0

◦ Φ x0
, x

≤ f (x) , (SC)
then T has an invariant point.
Proof. By contradiction: Assume there is no fixed point and no stationary point, re-
spectively. By Corollary 9, there is x̄ ∈ X such that
f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Hence, x̄ is the only point that can satisfy (WC) and (SC), respectively. This proves the
corollary.
Conversely, Corollary 9 can be proven using the fixed point result above. Indeed, assume
that (ii) of Corollary 9 does not hold, i.e.
∀x ∈ X, ∃x0
6= x : f x0

◦ Φ x0
, x

≤ f (x) .

Then, the mapping T : X → b
P (X) satisfies (SC) and has no invariant point, i.e., the
assertions of Corollary 10 can not hold. In this sense, the two corollaries are equivalent.
Of course, Corollary 10 is also a direct consequence of Theorem 17.
4.2.3 Takahashi’s existence principle
The following existence principle is, for the real valued case, due to Takahashi [113]. It’s
equivalence to Ekeland’s principle has been observed in [96] and [48]. Compare e.g. [97]
and [128] for similiar results.
∀n ∈ IN : f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) (4.6)
for all n ∈ IN.
Assume, additionally, that
x1, x2 ∈ X, f (x1) ≤ f (x2) , f (x2) 6≤ f (x1)
implies
∃x3 ∈ X : x3 6= x1, f (x3) ◦ Φ (x3, x1) ≤ f (x1) .
Then, there exists x̄ ∈ X such that f (x̄) ∈ min f (X) where f (X) =
S
x∈X {f (x)}.
Proof. By contradiction: By Corollary 9, there is x̄ ∈ X such that
f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄. (4.7)
Let f (x̄) 6∈ min f (X), i.e.
∃u ∈ X : f (u) ≤ f (x̄) , f (x̄) 6≤ f (u) .
By assumption, there is x0 6= x̄ such that
f x0

◦ Φ x0
, x̄

≤ f (x̄) ,
contradicting (4.7).

Assume additionally that Φ is symmetric, i.e.,
∀x0
, x ∈ X : Φ x0
, x

= Φ x, x0

.
Then, Corollary 9 can be proven using Corollary 11. Indeed, assume that (ii) of Corollary
9 does not hold, i.e.
∀x ∈ X, ∃x0
6= x : f x0

◦ Φ x0
, x

≤ f (x) .
Take x̄ ∈ X such that f (x̄) ∈ min f (X), i.e. x ∈ X, f (x) ≤ f (x̄) imply f (x̄) ≤ f (x).
Such a point does exist by Corollary 11. By assumption, there is also x̄0 ∈ X such that
f x̄0

◦ Φ x̄0
, x̄

≤ f (x̄) .
¿From θ ≤ Φ (x̄0, x̄) we obtain
f x̄0

≤ f x̄0

◦ Φ x̄0
, x̄

.
The transitivity of ≤ implies f (x̄0) ≤ f (x̄) and therefore the minimality of f (x̄) gives
f (x̄) ≤ f (x̄0). Using this and the symmetry of Φ, one may conclude
f (x̄) ◦ Φ x̄0
, x̄

≤ f x̄0

◦ Φ x̄0
, x̄

≤ f (x̄) ≤ f x̄0

.
Since is antisymmetric (this is due to the regularity of Φ, compare Remark 40), x̄0 = x̄,
a contradiction.
4.2.4 The flower petal theorem
In [98], Penot proved a geometric theorem being equivalent to Ekeland’s principle for
extended realvalued functions. It is called the flower petal theorem because of the shape
of certain sets in case X = IR2. We need the following constructions to formulate a similar
theorem in the present general framework.
Let X, Y be as in the last corollaries. Let Φ : X × X → Y be an order premetric and
Ψ : X × X → Y a function satisfying (P1) and (P2) of Definition 30. We call the set
PΦ (u, v) := {x ∈ X : Ψ (x, v) ◦ Φ (x, u) ≤ Ψ (u, v)}
the flower petal generated by u, v ∈ X. It is always nonempty since u ∈ PΦ (u, v) for
each u ∈ X because Φ (u, u) = θ and ≤ is reflexive. The metric space (X, d) is called
Φ–complete iff every Cauchy sequence {xn}n∈IN satisfying xn+1 ∈ PΦ (xn, v) converges
to some x ∈ X.
(A1) (X, d) is a metric space and (Y, ◦, ≤) a quasiordered monoid; M ⊆ X, x0 ∈ M and
v ∈ XM;
(A2) The function Ψ : X × X → Y satisfies (P1) and (P2) of Definition 30;

(A3) Φ : X × X → Y is a regular order premetric with respect to θ, Ψ (x0, v) ∈ Y for
x0 ∈ X such that X is Φ–complete;
(A4) If {xn}n∈IN ⊆ X converges to x ∈ X and
∀n ∈ IN : xn+1 ∈ PΦ (xn, v) ∩ M,
then x ∈ PΦ (xn, v) ∩ M for all n ∈ IN.
Then, there exists x̄ ∈ M such that
x̄ ∈ PΦ (x0, v) ∩ M and {x̄} = PΦ (x̄, v) ∩ M.
Proof. We check the assumptions of Theorem 16 for the relation
x0
x :⇐⇒ x0
∈ PΦ (x, v) ∩ M.
on X0 := PΦ (x0, v)∩M. Of course, (X0, d) is a metric space. The relation is reflexive and
transitive since Φ is an order premetric. X0 is –complete since X is Φ–complete. (M4)
of Theorem 16 follows directly from (A4). It remains to show the regularity assumption
(M3). Since xn+1 xn if and only if Ψ (xn+1, v) ◦ Φ (xn+1, xn) ≤ Ψ (xn, v) we obtain for
each n ∈ IN
Ψ (xn, v) ◦
n−1
X
k=0
Φ (xk+1, xk) ≤ Ψ (x0, v) .
The regularity of Φ implies that the sequence {xn}n∈IN is asymptotic, i.e., (M3) is satisfied.
Hence there exists x̄ ∈ S (x0) = PΨ (x0, v) ∩ M such that {x̄} = S (x̄) = PΨ (x̄, v).
This completes the proof.
The flower petal theorem is also a consequence of Corollary 9. To see this, simply take
f (x) := Ψ (x, v).
4.2.5 An equilibrium formulation of Ekeland’s principle
The next result deals with a function F : X × X → Y instead of f : X → Y . As far as the
author is aware, for realvalued functions this idea is due to Oettli and Théra [96] causing
several subsequent similiar considerations, see for example [59], [97].
(A2) F : X × X → Y is a function and x0 ∈ X, ỹ ∈ Y such that
(i) F (x1, x3) ≤ F (x1, x2) ◦ F (x2, x3) for all x1, x2, x3 ∈ X;
(ii) ỹ ≤ F (x0, x) for all x ∈ X;
(A3) Φ : X × X → Y is a regular order premetric with respect to ỹ, θ ∈ Y ;
(A4) If {xn}n∈IN ⊆ X is a Cauchy sequence such that
F (xn, xn+1) ◦ Φ (xn+1, xn) ≤ θ whenever xn+1 6= xn, (4.8)

(A5) If {xn}n∈IN ⊆ X is a sequence satisfying (4.8) and converging to x ∈ X, then
F (xn, x) ◦ Φ (x, xn) ≤ θ for all n ∈ IN with x 6= xn.
(i) F (x0, x̄) ◦ Φ (x̄, x0) ≤ θ
(ii) x ∈ X, F (x̄, x) ◦ Φ (x, x̄) ≤ θ =⇒ x = x̄.
Proof. We check the assumptions of Theorem 16 for the relation
x0
x :⇐⇒ x0
= x or F x, x0

◦ Φ x0
, x

≤ θ.
being reflexive and transitive by the properties of Φ, F and ≤. (M4) of Theorem 16 follows
directly from (A5). It remains to check (M3). Let {xn}n∈IN ⊆ X be such that xn+1 xn
for all n ∈ IN , i.e.,
xn+1 = xn or F (xn, xn+1) ◦ Φ (xn+1, xn) ≤ θ
for all n ∈ IN. Deleting xn+1 from the sequence if xn+1 = xn we obtain a finite number of
xn’s or a subsequence again denoted by {xn}n∈IN. In the first case, the original sequence
is constant up to finitely many elements, hence asymptotic. In the second case, using (A2,
(i)), the properties of Φ and the transitivity of ≤ we obtain for n ∈ IN:
F (x0, xn) ◦
n−1
X
k=0
Φ (xk+1, xk) ≤ θ.
With the help of assumption (A2, (ii)) we may conclude
ỹ ◦
n−1
X
k=0
Φ (xk+1, xk) ≤ θ.
Therefore, the regularity of Φ implies d (xn+1, xn) → 0. We can apply Theorem 16 in
order to obtain the desired result.
Remark 19 We are moving within the setting of Corollary 13. Define a function f :
X → Y by
f (x) := F (x0, x) , x ∈ X
and an order relation on X by
x0
f x :⇐⇒ x0
= x or f x0

◦ Φ x0
, x

≤ f (x) .
being reflexive and transitive. Observe that x0 x implies x0 f x since by (A2, i)
f x0

◦ Φ x0
, x

= F x0, x0

◦ Φ x0
, x

≤ F (x0, x) ◦ F x, x0

◦ Φ x0
, x

≤ F (x0, x)
whenever F (x, x0) ◦ Φ (x0, x) ≤ θ.

Let the assumptions of Corollary 9 be satisfied for f as defined above. Then we get
x̄ ∈ X such that
(i0) f (x̄) ◦ Φ (x̄, x0) ≤ f (x0)
(ii0) x ∈ X, f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Assuming that F (x0, x0) ≤ θ we obtain (i) of Corollary 13 from (i’). Moreover, if x̄
satisfies (ii’), then it satisfies (ii) of Corollary 13 since x0 x implies x0 f x.
These considerations show that Corollary 13 is a consequence of Corollary 9 in case if
F (x0, x0) ≤ θ and the function f (x) = F (x0, x) satisfies (A5) of Corollary 9.
4.2.6 Ekeland’s variational principle on groups
We shall consider the case Y being a group separately since it is interesting from a theo-
retical point of view. Especially, Corollary 13 will turn out to be an equivalent formulation
to Corollary 9 in this situation. Besides, in many applications Y is even a linear space.
Nemeth [93] first investigated the case of an ordered topological Abelian group (G, ◦).
In the following corollary, G is an ordered group not order complete in general. We can
adjoin a largest element yl (as well as a smallest one if necessary) obtaining an ordered
monoid. Note that no topological requirements concerning G do appear in contrast to
Nemeth’s results [93], [94].
(A1) (X, d) is a metric space and (G, ◦, ≤) a quasiordered Abelian group;
(A2) f : X → G ∪ {yl} is a function and ỹ ∈ G such that and ỹ ≤ f (x) for all x ∈ X;
(A3) Φ : X × X → Y is a regular order premetric with respect to ỹ ◦ [f (x0)]−1
for x0 ∈ G;
(A4) If {xn}n∈IN ⊆ X converges to x ∈ X and
∀n ∈ IN : f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn)
then f (x) ◦ Φ (xn, x) ≤ f (xn) for all n ∈ IN.
Then, there is x̄ ∈ X such that
(i) f (x̄) ◦ Φ (x̄, x0) ≤ f (x0)
(ii) x ∈ X, f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Proof. Set Y := G∪{yl} and extend ◦ to Y according to Proposition 25. Apply Corollary
9 with the ordered monoid (Y, ◦, ≤).
Remark 20 In the present case, one can define
F x, x0

:= f x0

◦ [f (x)]−1
,
where y−1 ∈ G denotes the inverse element of y ∈ G with respect to ◦: y ◦ y−1 = y−1 ◦ y =
θ ∈ G. Then, F (x, x0) ◦ Φ (x0, x) ≤ θ if and only if f (x0) ◦ Φ (x0, x) ≤ f (x). This means,
the assumptions of Corollary 13 are satisfied if f satisfies the assumptions of Corollary
14. In this sense, Corollary 14 is a special case of Corollary 13.
On the other hand, it seem to be not possible to obtain Corollary 9 from Corollary 13
since Y is not a group.

4.3 Ekeland’s principle for set valued maps
The first Ekeland type theorems for set valued maps seems to be the results of Chen and
Huang in [11]. More can be found in [12], [50], [52], [117], [55]. In fact, the appearance
of set valued variants of Ekeland’s principle has been the main motivation to allow the
image space a more general algebraic structure than a linear space or a group. Note that
the results of Nemeth [92], [93], [94] do not cover Ekeland type theorems with set valued
mappings since he assumed the image space of f to be a topological Abelian group.
4.3.1 Power set of ordered monoids
The following two corollaries involve the power set of an ordered monoid (Y, ◦, ≤) supplied
with the relation 4 and 2, respectively.
(A1) (X, d) is a metric space, (Y, ◦, ≤) an ordered monoid and (Y, , 4) the quasiordered
monoid generated by Y := b
P (Y );
(A3) The function f : X → Y and M ∈ Y are such that
(i) M 4 f (x) for all x ∈ X;
(ii) Φ is regular with respect to M, f (x0) ∈ Y for x0 ∈ X;
∀n ∈ IN : f (xn+1) Φ (xn+1, xn) 4 f (xn) (4.9)
(A4) If {xn}n∈IN ⊆ X is a sequence converging to x ∈ X and satisfying (4.9), then
∀n ∈ IN : f (x) Φ (x, xn) 4 f (xn) .
(i) f (x̄) Φ (x̄, x0) 4 f (x0)
(ii) x ∈ X, f (x) Φ (x, x̄) 4 f (x̄) =⇒ x = x̄.
Proof. By Theorem 11, (Y, , 4) is an order complete quasiordered monoid. Defining
the relation
x0
x :⇐⇒ f x0

Φ x0
, x

4 f (x)
on X, the assumptions of Corollary 9 are easy to check. Its conclusions yield the desired
result.
(A1) (X, d) is a metric space, (Y, ◦, ≤) an ordered monoid and (Y, , 2) the quasiordered
monoid generated by Y := b
P (Y );

4.3. Ekeland’s principle for set valued maps 83
(A3) The function f : X → Y and M ∈ Y are such that
(i) M 2 f (x) for all x ∈ X;
(ii) Φ is regular with respect to M, f (x0) ∈ Y for x0 ∈ X;
∀n ∈ IN : f (xn+1) Φ (xn+1, xn) 2 f (xn) (4.10)
(A4) If {xn}n∈IN ⊆ X is a sequence converging to x ∈ X and satisfying (4.10), then
∀n ∈ IN : f (x) Φ (x, xn) 2 f (xn) .
(i) f (x̄) Φ (x̄, x0) 2 f (x0)
(ii) x ∈ X, f (x) Φ (x, x̄) 2 f (x̄) =⇒ x = x̄.
Proof. Replace 4 by 2 in the proof of Corollary 15.
The next results deals with an image space being the power set of a topological linear
space (V, +, T ). Let K ∈ P (V ) be a cone and a convex element at the same time, i.e., K
is a convex cone in the classical sense containing θ ∈ V . It generates the quasiorder ≤K
by v0 ≤K v iff v ∈ {v0} ⊕ K. We denote by 4K and 2K the two quasiorders generated in
V := b
P (V ) by ≤K.
(A1) (X, d) is a metric space and (V, +, T ) a topological linear space;
(A2) K ∈ V is a cone and a convex element in (V, ⊕), K0 ⊆ K (−cl K) is a nonempty
convex and sequentially compact set;
(A3) The function f : X → V and the topological bounded set M ⊆ V are such that
M 4K f (x) for all x ∈ X;
(A4) ϕ : X × X → IR+ is a regular premetric;
(A5) If {xn}n∈IN ⊆ X is a Cauchy sequence with
∀n ∈ IN : f (xn+1) ⊕ ϕ (xn+1, xn) K0
4K f (xn) (4.11)
(A6) If {xn}n∈IN ⊆ X is a sequence satisfying (4.11) such that {xn}n∈IN converges to
x ∈ X, then
∀n ∈ IN : f (x) ⊕ ϕ (x, xn) K0
4K f (xn) .
Then, for each x0 ∈ X with f (x0) 6= ∅, there exists x̄ ∈ X such that
(i) f (x̄) ⊕ ϕ (x̄, x0) K0 4K f (x0)
(ii) x ∈ X, f (x) ⊕ ϕ (x, x̄) K0 4K f (x̄) =⇒ x = x̄.

Proof. Consider (V, ⊕, 4K) being a quasiordered monoid. We check the assumptions of
Corollary 15 replacing Y by V , Y by V and setting Φ := ϕK0. Most of the assumptions
can be checked straightforward, only the regularity of Φ requires some extra words. In
fact, Φ is regular with respect to M, f (x0) ∈ V. To see this, let
∀n ∈ IN : M ⊕
n
X
k=0
Φ (xk+1, xk)
#
= M ⊕
n
X
k=0
ϕ (xk+1, xk) K0
#
4K f (x0)
be satisfied. Since K0 is a convex subset of a linear space, (t + s) K0 = sK0 ⊕ tK0 holds
for all s, t ≥ 0. Therefore,
∀n ∈ IN : M ⊕
n
X
k=0
ϕ (xk+1, xk) K0
#
= M ⊕
n
X
k=0
ϕ (xk+1, xk)
#
K0
4K f (x0) .
Hence, for all v ∈ f (x0) and n ∈ IN there exist ṽn ∈ M, k0
n ∈ K0 such that
ṽn +
n
X
k=0
ϕ (xk+1, xk)
#
k0
n ≤K v.
Assume αn :=
Pn
k=0 ϕ (xk+1, xk) → ∞. Then
1
αn
ṽn + k0
n ≤K
1
αn
v ⇔
1
αn
v −
1
αn
ṽn − k0
n ∈ K.
Since K0 is sequentially compact, there exists a subsequence of {n}n∈IN such that k0
n →
k0 ∈ K0 along this subsequence. Since {v} and M are bounded subsets of V , the above
relationship implies −k0 ∈ cl K, a contradiction to assumption (A2). Therefore, the αn’s
remain bounded, hence ϕ (xn+1, xn) → 0. The regularity of ϕ gives limn→∞ d (xn+1, xn) =
0.
The conclusions of Corollary 15 yield (i) and (ii) of Corollary 17.
Corollary 17 may be considered as a minimal element theorem in the product space X ×
b
P (V ) with respect to the order relation
x0
, M0

4K0 (x, M) ⇐⇒ M0
⊕ ϕ x0
, x

K0
4K M.
Indeed, let a subset A ⊆ X × V be given and define a function (a set valued map)
f : X → b
P (V ) by
f (x) := {v ∈ V : (x, v) ∈ A} .
If f (x0) 6= ∅ and assumptions (A3), (A5) of Corollary 17 are satisfied for this f, we obtain
a minimal point (x̄, f (x̄)) with respect to 4K0 of the set
{(x, f (x)) : x ∈ X} ⊆ X × P (V ) .
Hence, the question ”authentic” or ”not authentic” ([47], [44]) depends on the order rela-
tion. A theorem of the type of Corollary 17 has been called a Minimal Set Theorem in [50].

4.3. Ekeland’s principle for set valued maps 85
Note that more assumptions are necessary to ensure the existence of minimal elements of
a subset A ⊆ X × V with respect to order relations in X × V such as
x0
, v0

≤k0 (x, v) ⇐⇒ v0
+ ϕ x0
, x

k0
≤K v
where k0 ∈ K (−cl K). We refer to Sections 4.7 and 7.2.
It follows the counterpart of the last corollary for the relation 2K. One may notice
the difference concerning the sets M and K0: They have to be only nonempty. On the
other hand, the set f (x0) must be topological bounded.
(A2) K ∈ V is a cone and a convex element in (V, ⊕), K0 ⊆ K (−cl K) is a nonempty
convex set;
(A3) The function f : X → V and the nonempty set M ⊆ V are such that M 2K f (x)
for all x ∈ X;
∀n ∈ IN : f (xn+1) ⊕ ϕ (xn+1, xn) K0
2K f (xn) (4.12)
(A6) If {xn}n∈IN ⊆ X is a sequence satisfying (4.12) and {xn}n∈IN converges to x ∈ X,
then
∀n ∈ IN : f (x) ⊕ ϕ (x, xn) K0
2K f (xn) .
Then, for each x0 with f (x0) topological bounded, there exists x̄ ∈ X such that
(i) f (x̄) ⊕ ϕ (x̄, x0) K0 2K f (x0)
(ii) x ∈ X, f (x) ⊕ ϕ (x, x̄) K0 2K f (x̄) =⇒ x = x̄.
Proof. We check the assumptions of Corollary 16 replacing

b
P (Y ) , , 2

by (V, ⊕, 2K)
and setting Φ := ϕK0. Again, the regularity of Φ is the crucial point to check. Let
∀n ∈ IN : M ⊕
n
X
k=0
Φ (xk+1, xk)
#
= M ⊕
n
X
k=0
ϕ (xk+1, xk) K0
#
2K f (x0)
be satisfied. As in the proof of Corollary 17, we may conclude
∀n ∈ IN : M ⊕
n
X
k=0
ϕ (xk+1, xk)
#
K0
2K f (x0) .
According to the definition of 2K, for all v ∈ M and all k0 ∈ K0 there is vn ∈ f (x0) such
that
v + k0
αn ≤K vn,

where again αn :=
Pn
k=0 ϕ (xk+1, xk) Assuming αn → ∞, we arrive at the contradiction
k0 ∈ −cl K since f (x0) is bounded. This means, Φ = ϕK0 is a regular order premetric
with respect to M, f (x0). Applying Corollary 16 we obtain the desired result.
Of course, Corollary 18 can be considered as a minimal element theorem in the product
space X × b
P (V ) in a similar way as Corollary 17.
4.4 Ekeland’s principle for single valued Functions
It seems to the author that Nemeth [89], [91], [90] proved the first vector valued versions of
Ekeland’s theorem, even for functions on a space X more general than a complete metric
space. Compare also [92]. Related results including a fixed point theorem of Kirk–Caristi
type have been obtained by Khanh [73]
In [84], Proposition 4.2., a variant has been given for functions mapping a real Banach
space into IRp, p ∈ IN, p 1: The proof is an elementary application of Ekeland’s original
result [30] to a scalarized problem.
Using nonlinear scalarization technique, Tammer [114] established an Ekeland type
theorem for functions mapping a complete metric space into an topological linear space
with an order relation not necessarily generated by a cone.
Related results can be found in [59], [58], [10], [14], [80], [132], [34], [35].
Another approach has been developed by Göpfert, Tammer and Zălinescu by proving
a so–called minimal point theorem in the product space X × Y and deriving from this
Ekeland type theorems. Compare [43], [46], [47] and the book [44].
The next result is a special case of Corollary 17 as well as of Corollary 18. In [47], [44]
it is called a non–authentic minimal point theorem.
We consider an quasiordered linear space (V, +, ≤K) where the order relation ≤K is
generated by a convex cone K. Again, a largest element vl can be added to V obtaining
a quasiordered monoid (V ∪ {vl} , +).
(A2) K ∈ V is a cone and a convex element in (V, ⊕) and k0 ∈ K (−cl K);
(A3) The function f : X → V ∪ {vl} and the topological bounded set M ⊆ V are such that
∀x ∈ X : f (x) ∈ (M ⊕ K) ∪ {vl} ;
∀n ∈ IN : f (xn+1) + ϕ (xn+1, xn) k0
≤K f (xn) (4.13)
(A6) If {xn}n∈IN ⊆ X is a sequence satisfying (4.13) and converging to x ∈ X, then
∀n ∈ IN : f (x) + ϕ (x, xn) k0
≤K f (xn) .

4.5. Ekeland’s principle for real valued functions 87
Then, for each x0 with f (x0), there exists x̄ ∈ X such that
(i) f (x̄) + ϕ (x̄, x0) k0 ≤K f (x0)
(ii) x ∈ X, f (x) + ϕ (x, x̄) k0 ≤K f (x̄) =⇒ x = x̄.
First Proof. Specialize Corollary 17.
Second Proof. Set (Y, ◦, ≤) = (V ∪ {vl} , +, ≤K) and apply Corollary 9.
4.5 Ekeland’s principle for real valued functions
In this section, we prove a series of corollaries of Theorem 16 all being equivalent to
Ekeland’s principle from 1972 for extended real valued functions. This includes results
with a more geometric nature such as the drop theorem, the flower petal theorem and
Phelps’ lemma. We do not focus on the equivalence proofs. They are well–known and can
be found e.g. in [98], [96], [97].
In contrast, our proofs rely on Theorem 16. As in the last section, in each case, we
shall construct an order relation and check the assumptions of Theorem 16. In order
to simplify the exposition we assume (X, d) to be a complete metric space and only use
the metric instead of a real valued premetric. The corresponding generalizations can be
obtained easily parallel to the results in Section 4.2.
The first corollary is due to Dancs, Hegedüs, Medvegyev and can be found in [22].
Compare also [4], chapter 6. Therein, the set valued mapping f occuring in the corollary
below is called a dynamical system.
(A1) (X, d) is a complete metric space and f : X → P (X) a set valued map;
(A2) x ∈ f (x) for all x ∈ X;
(A3) x0 ∈ f (x) implies f (x0) ⊆ f (x);
(A4) If {xn}n∈IN ⊂ X is a sequence such that xn+1 ∈ f (xn) for all n ∈ IN, then
limn→∞ d (xn+1, xn) = 0;
(A5) f (x) is closed for each x ∈ X.
Then, for each x0 ∈ X, there is x̄ ∈ X such that
x̄ ∈ f (x0) and {x̄} = f (x̄) .
Proof. Define a relation on X by
x0
x :⇐⇒ x0
∈ f (x)
being reflexive and transitive by (A2) and (A3), respectively. Then S (x) = f (x). As-
sumptions (A4) and (A5) imply (M3) and (M4) of Theorem 16. Its conclusion yields the
result.
Next, we prove a slightly generalized version of Ekeland’s original theorem [30], [31], [98].

(A1) (X, d) is a complete metric space;
(A2) f : X → IR ∪ {+∞} is bounded from below;
(A3) For each x ∈ X, the set {x0 ∈ X : f (x0) + d (x0, x) ≤ f (x)} is closed.
Then, for each x0 ∈ X with f (x0) ∈ IR, there is x̄ ∈ X such that
(i) f (x̄) + d (x̄, x0) ≤ f (x0)
(ii) ∀x ∈ X {x̄} : f (x̄) f (x) + d (x, x̄)
Proof. Define a partial order by
x0
x :⇐⇒ f x0

+ d x0
, x

≤ f (x) .
By (A3), the set S (x) is closed for each x ∈ X. We check (M4) of Theorem 16. Take a
sequence {xn}n∈IN ⊂ X such that xn+1 xn, i.e.
f (xn+1) + d (xn+1, xn) ≤ f (xn)
for each n ∈ IN. Since f (x0) ∈ IR we have {f (xn)}n∈IN ⊂ IR. The sequence is nonincreas-
ing and bounded from below by (A2), hence convergent. This implies limn→∞ d (xn+1, xn) =
0. Applying Theorem 16, we obtain x̄ ∈ X with properties (i) and (ii).
Remark 21 1. Assumption (A3) is weaker than lower semiconituity of f. Consider the
function f : IR → IR defined by f (x) = Exp (− |x|) if x 6= 0 and f (0) = 2 not being lower
semicontinuous at x = 0. This example is taken from [47]. The attempt to weaken the
classical assumptions to f such as lower semicontinuity is due to [37].
2. The above formulation of the conclusions (i), (ii) is probably due to Penot [98].
3. The original formulation of Ekeland is as follows. Start with x0 ∈ X and ε 0, λ 0
such that f (x0) ≤ infx∈X f (x) + ε. Replace d in Corollary 21 by ε
λd. Then (i) implies
that f (x̄) ≤ f (x0) and
1
λ
d (x̄, x0) ≤
1
ε
[f (x0) − f (x̄)] ≤
1
ε

inf
x∈X
f (x) + ε − f (x̄)

≤ 1,
hence d (x̄, x0) ≤ λ. Choosing λ =
√
ε one can ensure that the difference f ( ¯
x0) − f (x̄) is
small as well as the distance d (x̄, x0).
4. Conclusion (ii) can be interpreted as follows: The point x̄ is the unique global minimizer
of the function x → f (x) + d (x, x̄). This observation, probably due to Clarke, is the
starting point of many applications, e.g. the proof of Clarke’s multiplier rule for nonsmooth
optimization problems as well as the maximum principle for optimal control problems in
[17], [18].
Of course, choosing a special function f yields a special case of Corollary 21. One of
them is the nice geometric result due to Penot [98] called flower petal theorem. Compare
Corollary 12 for a general version. We chose f (x) = d (x, v) for some v ∈ X.

4.5. Ekeland’s principle for real valued functions 89
Let 0 γ 1 and u, v ∈ X be given. The set
Pγ (u, v) := {x ∈ X : d (x, v) + γd (x, u) ≤ d (u, v)}
is called the flower petal belonging to u and v. Note that Pγ (x, v) is a closed set.
Corollary 22 Let (X, d) be a complete metric space, M ⊂ X, v ∈ XM and 0 γ 1.
Then, for each x0 ∈ M, there exists x̄ ∈ X such that
x̄ ∈ Pγ (x0, v) and {x̄} = Pγ (x̄, v) .
Proof. Set X0 = M ∩ Pγ (x0, v) and replace in Corollary 21 (X, d) by (X0, γd) as well as
f (x) by d (x, v).
Remark 22 1. From the proofs of Corollary 21 and Corollary 22 it is clear that the
relation
x0
x :⇐⇒ x0
∈ Pγ (x, v)
is a partial order.
2. Remarkably, Corollary 22 is not only a special case, but also equivalent to Corollary
21. Compare [98], [38] for a proof.
The next corollary is the original version of Kirk-Caristi’s fixed point theorem, see [9],
[26], [25]. As in the general version (Corollary 10), we give two variants involving fixed
points and stationary points of a setvalued map.
Let T : X → P (X) be a set valued map. Recall that a point x̄ ∈ X is a fixed point
of T iff x̄ ∈ T (x̄) and an invariant point of T iff {x̄} = T (x̄).
(A2) There exists a function f : X → IR ∪ {+∞} not being identical +∞ and satisfying
(A2), (A3) of Corollary 21.
If T : X → P (X) satisfies the weak contraction condition
∀x ∈ X, ∃x0
∈ T (x) : f x0

+ d x0
, x

≤ f (x) , (WC)
then it has a fixed point.
If T : X → P (X) satisfies the strong contraction condition
∀x ∈ X, ∀x0
∈ T (x) : f x0

+ d x0
, x

≤ f (x) , (SC)
then it has an invariant point.

Proof. Define a partial order by
x0
x :⇐⇒ f x0

+ d x0
, x

≤ f (x) .
Theorem 16 (or Corollary 21) implies the existence of x̄ ∈ X such that S (x̄) = {x̄}. Then
(WC) implies x̄ ∈ T (x̄). If (SC) is satisfied we even have {x̄} = T (x̄).
Ekeland’s principle admits a reformulation as an existence principle for minimizers. This
observation is due to Takahashi [113]. Compare also [96] and [48] as well as [128] for
further applications of this idea.
(A2) f : X → IR ∪ {+∞} is bounded from below;
(A3) For each x ∈ X, the set {x0 ∈ X : f (x0) + d (x0, x) ≤ f (x)} is closed;
(A4) For each x ∈ X with infu∈X f (u) f (x) there exists x0 ∈ X such that
x 6= x0
and f x0

+ d x0
, x

≤ f (x) .
Then, there exists x̄ ∈ X such that f (x̄) = infu∈X f (u).
Proof. By assumption (A1), (A2), (A3) and Corollary 21, there exists x̄ ∈ X such that
∀x 6= x̄ : f (x̄) f (x) + d (x, x̄) .
This contradicts (A4) if x̄ is not a minimizer of f.
In the spirit of Corollary 13, we state an equilibrium version of Ekeland’s principle. This
idea originates from [96].
(A2) F : X × X → IR ∪ {+∞} is a function and x0 ∈ X, r ∈ IR such that
(i) F (x1, x3) ≤ F (x1, x2) + F (x2, x3) for all x1, x2, x3 ∈ X and F (x, x) = 0 for all
x ∈ X;
(ii) r ≤ F (x0, x) for all x ∈ X;
(iii) The function x → F (x0, x) is lower semicontinuous.
(i) F (x0, x̄) + d (x̄, x0) ≤ 0
(ii) x ∈ X, x 6= x̄ =⇒ F (x̄, x) + d (x, x̄) 0.
Proof. We check the assumptions of Corollary 21 for the function f (x) := F (x0, x).
(A1) and (A2) are obvious. (A3) follows from the lower semicontinuity of F (x0, ·). By

4.6. Geometric variational principles in Banach spaces 91
Corollary 21 we get some point x̄ ∈ X such that f (x̄) + d (x0, x̄) = F (x0, x̄) + d (x0, x̄) ≤
F (x0, x0) = 0 satisfying
x ∈ X, x 6= x̄ =⇒ f (x̄) f (x) + d (x, x̄) . (4.14)
Assume that F (x̄, x) + d (x, x̄) ≤ 0 for some x ∈ X, x 6= x̄. Then we obtain from (4.14)
and (A2, (i))
f (x̄) = F (x0, x̄) F (x0, x̄) + F (x̄, x) + d (x, x̄) ≤ F (x0, x̄) ,
a contradiction. Hence x̄ satisfies (ii).
Another proof is possible using the order relation
x 6= x0
and F x0
, x

+ d x, x0

≤ 0.
and applying Theorem 16. Also, one may specialize Corollary 13 to the case Y = IR ∪
{+∞}, Φ = d.
Note that Corollary 25 is in fact an equivalent reformulation of Corollary 21 in the
following sense. Setting F (x1, x2) := f (x2) − f (x1) we obtain the assertions of Corollary
21 from those of Corollary 25.
4.6 Geometric variational principles in Banach spaces
The ”grandfather” (Ekeland 1979) of all Ekeland type theorems is Lemma 1 in the paper
[5] by Bishop and Phelps. Its proof already contains some essentials for the proof of
Theorem 16 and its corollaries. A more general version in topological linear spaces can be
found in [101]. Therefore, we formulate linear - Banach space - variants of the corollaries
of the last section.
Throughout this section, (V, k·k) is a Banach space with topological dual (V ∗, k·k∗).
The expressions v∗ (v) = (v∗, v) denote the value of the linear continuous functional v∗ ∈
V ∗ at v ∈ V . We start with the Bishop–Phelps lemma. Actually, it is a minimal element
theorem, i.e., an existence theorem for minimal points with respect to an order relation
generated by a cone of P (V ).
4.6.1 Results in Banach spaces
(A1) (V, k·k) is a Banach space and M ⊆ V is a nonempty and closed subset;
(A2) B ⊆ V is nonempty, closed, bounded, convex such that 0 /
∈ B;
(A3) K := IR+B = {t · b : t ≥ 0, b ∈ B};
(A4) v0 ∈ M such that M ∩ ({v0} ⊕ K) is bounded.
Then, there exists v̄ ∈ V such that
v̄ ∈ M ∩ ({v0} ⊕ K) and {v̄} = M ∩ ({v̄} ⊕ K) .

Proof. Define V 0 = M ∩ ({v0} ⊕ K). Then, by (A1), (A2), (A3), (V 0, k·k) is a complete
metric space. We introduce an order relation on V 0 by
v0
v :⇐⇒ v0
∈ M ∩ ({v} ⊕ K) .
Since the cone K contains 0 ∈ V and is convex by construction, is reflexive and tran-
sitive. The sets S (x) = {v0 ∈ V 0 : v0 v} = V 0 ∩ ({v} ⊕ K) are closed, i.e., assumption
(M3) of Theorem 16 is satisfied. To check (M4), take {vn}n∈IN ⊆ V 0 such that vn+1 vn
for all n ∈ IN. This especially means vn − vn+1 ∈ K, hence there exist tn ≥ 0 and bn ∈ B
such that vn − vn+1 = tnbn.
Applying a separation theorem to B and {0}, we can find v∗ ∈ V ∗, r ∈ IR such that
0 r ≤ inf {v∗
(b) : b ∈ B} .
Hence
rtn ≤ tnv∗
(bn) = v∗
(vn − vn+1)
for all n ∈ IN. Adding up these inequalities from n = 0 to n = m − 1 we obtain
v∗
(vm) + r
m−1
X
n=0
tn ≤ v∗
(v0) .
The set of numbers {v∗ (vm)}m∈IN is bounded since vm ∈ V 0 and V 0 is a bounded subset
of V . This implies tn → 0 for n → ∞. Therefore,
kvn − vn+1k = ktn · bnk = tn kbnk
tends to zero for n → ∞ since B is a bounded subset of V . Hence assumption (M4)
of Theorem 16 is satisfied. Applying this theorem, we arrive at the conclusions of the
corollary.
The equivalence of Corollary 26 and Ekeland’s principle (Corollary 21, V a Banach
space) has been established in [38] and [3]. Especially, it is easy to prove that Corol-
lary 26 implies a Banach space version of Corollary 21 replacing V by V × IR and setting
M := epi f, K = {(x, r) ∈ V × IR : r + kxk ≤ 0}. The set B can be identified with
{(v, r) ∈ V × IR : r + kvk = −1}.
This observation gave rise to ask if this procedure can be generalized to product spaces
X × Y , X being a complete metric space, Y a topological linear (locally convex) space.
Results in this direction have been obtained by Göpfert, Tammer and Zălinescu, compare
[43], [46], [47] and also the book [44]. Even Theorem 16 implies results of this type
involving the set relations 4 and 2 in P (Y ). Compare the remarks after Corollary 17
and 18, respectively.
The so called drop theorem, established by J. Danes̆ in 1972 [23], is another important
result being an equivalent formulation of Ekeland’s principle. Moreover, the drop theorem
itself is a reformulation of a renorming theorem due to Zabreiko and Krasnosel’skii from
1971, compare [131]. This observation can also be found in [23].

Let B ⊆ V be closed convex set. The drop D (v, B) generated by v ∈ V B and B is
defined to be the set
D (v, B) := co {{v} , B} = {tv + (1 − t) b : b ∈ B, t ∈ [0, 1]} .
We give a proof of the drop theorem using Theorem 16.
(A1) (V, k·k) is a Banach space and M ⊆ V is a nonempty and closed subset;
(A2) B ⊆ V is a nonempty closed convex and bounded subset of V ;
(A3) It holds
0 r := dist (B, M) := inf {kb − xk : b ∈ B, x ∈ M} .
Then, for each v0 ∈ M there exists v̄ ∈ V such that
v̄ ∈ M ∩ D (v0, B) and {v̄} = M ∩ D (v̄, B) .
Proof. Define V 0 = M ∩ D (v0, B). Since B is closed, D (v0, B) is closed by construction
and (V 0, k·k) is a complete metric space. We define a relation on V 0 by
v0
v :⇐⇒ v0
∈ M ∩ D (v, B) .
Of course, is reflexive. It is also transitive. To see this, take v3 ∈ D (v2, B), v2 ∈
D (v1, B). Then there are t1, t2 ∈ [0, 1] and b1, b2 ∈ B such that v3 = t2v2 +(1 − t2) b2 and
v2 = t1v1 + (1 − t1) b1. This gives
v3 = t2 [t1v1 + (1 − t1) b1] + (1 − t2) b2
= t1t2v1 + (1 − t1t2)

t2 (1 − t1)
1 − t1t2
b1 +
(1 − t2)
1 − t1t2
b2

.
Since t2(1−t1)
1−t1t2
+ (1−t2)
1−t1t2
= 1, this implies that v3 is a convex combination of v1 and an
element of b, hence v3 ∈ D (v1, B) which proves the transitivity of .
Next, we show the regularity of . Let {vn}n∈IN ⊆ V 0 be a decreasing sequence, i.e.,
∀n ∈ IN : vn+1 ∈ M ∩ D (vn, B) .
This means, for all n ∈ IN there are tn ∈ [0, 1], bn ∈ B such that
vn+1 = tnvn + (1 − tn) bn. (4.15)
Then, for all b ∈ B and n ∈ IN, we have
kvn+1 − vnk = (1 − tn) kbn − vnk ≤ (1 − tn) (kb − vnk + kbn − bk)
≤ (1 − tn) (kb − vnk + diam B) .

Define dB (x) := infb∈B kb − vk. Then, dB is convex since B is a convex set. From the
latter inequality chain, we obtain
kvn+1 − vnk ≤ (1 − tn) (dB (vn) + diam B) . (4.16)
On the other hand, the convexity of dB implies
0 dB (vn+1) = dB (tnvn + (1 − tn) bn) ≤ tndB (vn) .
Therefore, tn ≥ dB(vn+1)
dB(vn) . Invoking (4.16) we obtain
kvn+1 − vnk ≤

1 −
dB (vn+1)
dB (vn)

(dB (vn) + diam B)
= (dB (vn) − dB (vn+1)) +
dB (vn) − dB (vn+1)
dB (vn)
diam B
= (dB (vn) − dB (vn+1))

1 +
diam B
dB (vn)

.
Since 0 r ≤ dB (v) for all v ∈ M by assumption, this implies
kvn+1 − vnk ≤ (dB (vn) − dB (vn+1))

1 +
diam B
R

.
Setting α−1 := 1 + diam B
R and adding the above inequalities from n = 0 to m − 1, we
obtain for m = 1, 2, . . .
dB (vm) + α
m−1
X
n=0
kvn+1 − vnk ≤ dB (v0)
implying limn→∞ kvn+1 − vnk = 0 which proves the regularity of .
The conclusions of the corollary are obtained by applying Theorem 16.
Corollary 27 that is an equivalent formulation of the original drop theorem of [23] has
been proven by J. Danes̆ in [24]. Besides the more or less straightforward application of
Theorem 16, the proof above contains the essentials of Lemma 1 and Lemma GKZ of [24].
4.6.2 Results in locally complete locally convex spaces
Several attempts have been made to give a formulation of the drop theorem on locally
convex spaces. The first one seems to be [86], in which Mizoguchi proved variational
principles on complete uniform spaces among them the drop theorem in locally convex
spaces. In [15] and [53] proofs for the drop theorem in sequentially complete, locally convex
spaces are given which seem to be not complete. Results of Qiu [103], [104] show that it
is more appropriated to assume local completeness rather than sequential completeness.
Using this concept, we present the drop theorem as well as Phelps’ lemma in locally
complete, locally convex spaces using an idea of [53] in order to apply the Banach space
versions of the corresponding theorems as the essential tool for the proof.

The following definitions as well as many results on local completeness can be found
in [100]. Let (V, +, T ) be a separated, locally convex space. A bounded and absolutely
convex subset D ⊆ V is called a disc. We denote by pD the Minkowski gauge of D, i.e.,
pD (v) := inf {t 0 : v ∈ tD} , v ∈ V.
The linear subspace of V spanned by D is denoted by VD. (VD, pD) is a normed space. A
disc D ⊆ V is called a Banach disc if (VD, pD) is a Banach space. A sequence {vn}n∈IN
is called a locally Cauchy (locally convergent) sequence iff it is Cauchy (convergent)
in (VD, pD) for some disc D.
The space (V, +, T ) is called locally complete iff every locally Cauchy sequence
converges locally in V . It is well–known that every sequentially complete separated locally
convex space is locally complete, but the converse is not true in general, compare [100],
Corollary 5.1.8 and Example 5.1.12. The crucial result for the following proofs is the fact
that a separated locally convex space is locally complete if and only if every bounded set
is contained in a Banach disc, see [100], Proposition 5.1.6.
A subset M ⊆ V is called locally closed iff for a {vn}n∈IN ⊆ M converging locally to
v ∈ V we have v ∈ M.
(A1) (V, +, T ) is a locally complete, separated locally convex space and M ⊆ V is a
nonempty and locally closed subset;
(A2) B ⊆ V is nonempty, locally closed, bounded, convex such that 0 /
∈ B;
(A3) K := IR+B = {t · b : t ≥ 0, b ∈ B};
(A4) v0 ∈ M such that M ∩ ({v0} ⊕ K) is bounded.
v̄ ∈ M ∩ ({v0} ⊕ K) and {v̄} = M ∩ ({v̄} ⊕ K) .
Proof. Define the set
B0 := B ∪ {v0} .
Since B is bounded, so is B0. Since V is locally complete, there is a Banach disc D ⊆ V
such that B0 ⊆ D and hence v0 ∈ VD, K ⊆ VD, M ∩ ({v0} ⊕ K) ⊆ VD.
In order to apply Corollary 26 in (VD, pD) we have to check its assumptions. First,
we check (A1): The set M ∩ VD is closed in (VD, pD). To see this, take a sequence
{vn}n∈IN ⊆ M ∩ VD converging to v with respect to pD. Especially, this means {vn}
converges locally to v ∈ V . Then v ∈ VD since VD is Banach. Since M is locally closed,
v ∈ M is also true, hence v ∈ M ∩ VD.
Assumption (A2) of Corollary 26 is satisfied since obviously B as a subset of VD has
the desired properties. (A3) is clear by construction.
It remains to check (A4): Denote M0 := M ∩ ({v0} ⊕ K) ⊆ VD. We shall show
M0 ⊆ {v0} ⊕ sB for some fixed s 0. This implies the boundedness of M0 in (VD, pD)
since B ⊆ D.

Assume the contrary, i.e., there are sequences {vn}n∈IN ⊆ M0, {tn}n∈IN ∈ IR+ {0},
{bn}n∈IN ⊆ B such that limn→∞ tn = +∞ and vn = v0 + tnbn. This implies
∀n ∈ IN :
vn
tn
−
v0
tn
= bn ∈ B.
Letting n → ∞ we obtain 0 ∈ B, a contradiction. This follows since M0 is bounded in V ,
hence vn
tn
→ 0 in V as well as v0
tn
→ 0 in V if n → ∞.
Applying Corollary 26 to M0 in (VD, pD) we get a point v̄ ∈ M ∩ ({v0} ⊕ K) such that
{v̄} = M ∩ ({v̄} ⊕ K). Since M ∩ ({v̄} ⊕ K) ⊆ VD, this implies the desired result.
The same idea is used to establish the drop theorem in locally convex spaces. The definition
of a drop as given above applies also in this case.
(A1) (V, +, T ) is a locally complete, separated locally convex space and M ⊆ V is a
nonempty and locally closed subset;
(A2) B ⊆ V is nonempty, locally closed, bounded, convex;
(A3) If N (θ) be a neighborhood base of θ ∈ V for T , then there is N ∈ N (θ) such that
M ∩ (B ⊕ N) = ∅.
v̄ ∈ M ∩ D (v0, B) and {v̄} = M ∩ D (v̄, B) .
Proof. Define the set
B0 := B ∪ {v0} .
Since B is bounded, so is B0. Since V is locally complete, there is a Banach disc D ⊆ V
such that B0 ⊆ D and hence v0 ∈ VD. Define the set MD = M ∩ VD in order to apply
Corollary 27 in VD. The assumptions (A1) and (A2) of Corollary 27 can be verified with
similar arguments as used in the proof of Corollary 28. It remains to check (A3) for B, MD.
We must have pD (b − v) ≥ r 0 for all b ∈ B, v ∈ MD. This is equivalent to b − v 6∈ rD
for all b ∈ B, v ∈ MD. Assume the contrary, i.e., there are sequences {vn}n∈IN ⊆ MD,
{tn}n∈IN ∈ IR+, {bn}n∈IN ⊆ B such that limn→∞ tn = 0 and
∀n ∈ IN : bn − vn ∈ tnD.
Take N ∈ N (θ). Since D is bounded in V , there is t 0 such that tD ⊆ N. Hence there
is nN ∈ IN such that bn − vn ∈ N for all n ≥ nN . Since N is arbitrary in N (θ), this
contradicts (A3) of the present corollary.
We may apply Corollary 27 to obtain the desired result.
Although the space V is not a Banach space (or a complete metric) in the last two
corollaries, their proofs rely essentially on the Banach space versions presented before.
Therefore, the latter results are included in this chapter.

4.7. Minimal elements on product spaces 97
4.7 Minimal elements on product spaces
As mentioned before, Bishop and Phelps [5], [102] as well as Ekeland [30] already observed
that the so called variational principle is nothing else than a minimal element theorem for
orders on X × IR defined by
x0
, r0

(x, r) ⇐⇒ r0
+ d x0
, x

≤ r
for (x0, r0) , (x, r) ∈ X × IR. Applied to epigraphs of functions f : X → IR, this relation
generates an order on X simply by setting
x0
X x ⇐⇒ f x0

+ d x0
, x

≤ f (x) .
The question arises if it is possible to obtain minimal element theorems on product spaces
X × V where the set in question can not necessarily be interpretated as the epigraph of a
function f : X → V . Göpfert, Tammer and Zălinescu established results in this direction
in a series of papers [43], [45], [46], [47]. Compare also the book [44], Section 3.10. A
subsequent paper is [55] and more general results can be found in [50], [51].
In this section, we state a minimal element theorem on a product space X × V where
X is a complete metric space and V is a Banach space. The method of proof is again an
application of Theorem 16. For a more general setting, compare the following chapters.
(A1) (X, d) is a complete metric space and (V, k·k) a Banach space;
(A2) K ⊆ V is a closed convex set and a cone in P (V ) generating the quasiorder ≤K on
V and k0 ∈ K − K;
(A3) The nonempty closed set A ⊆ X × V and the topological bounded set M ⊆ V are
such that
(x, v) ∈ A =⇒ v ∈ M ⊕ K;
(A4) If {(xn, vn)}n∈IN ⊆ A satisfies
∀n ∈ IN : vn+1 + k0
d (xn+1, xn) ≤K vn, (4.17)
and {xn}n∈IN ⊆ X is convergent, then {vn}n∈IN ⊆ V is asymptotic.
Then, for each (v0, v0) ∈ A, there exists (x̄, v̄) ∈ A such that
(i) v̄ + k0d (x̄, v0) ≤K v0
(ii) (x, v) ∈ A, v + k0d (x, x̄) ≤K v̄ =⇒ (x, v) = (x̄, v̄) .
Proof. Define a relation on the complete metric space A ⊆ X × V by
x0
, v0

(x, v) ⇐⇒ v0
+ d x0
, x

k0
≤K v.
Invoking the properties of d, we can see that is reflexive and transitive. To prove
regularity, take a –decreasing sequence {(xn, vn)}n∈IN ⊆ A, i.e., it satisfies (4.17). Take

v∗ ∈ K+ := {v∗ ∈ V ∗ : ∀v ∈ K : v∗ (v) ≥ 0} such that v∗ k0

= 1. Such a v∗ does exist
according to classical separation arguments (see [133], Theorem 1.1.5). We obtain
∀n ∈ IN : v∗
(vn+1) + d (xn+1, xn) ≤ v∗
(vn) .
From (A3) we obtain that v∗ is bounded below on {v ∈ V : ∃x ∈ X : (x, v) ∈ M}. There-
fore, the above relation implies that the sequence {xn}n∈IN is Cauchy, hence convergent to
some x̂ ∈ X. Assumption (A4) ensures that {vn}n∈IN is asymptotic. Therefore, assump-
tion (M3) of Theorem 16 is satisfied for .
To check the lower closedness, i.e., assumption (M3) of Theorem 16, take a -decreasing
sequence {(xn, vn)}n∈IN contained in S (v0, v0) and converging to some (x, v) ∈ X × V .
The triangle inequality for d, (4.17) and the definition of ≤K imply
∀n ∈ IN, ∀m ≥ n : vn − vm − k0
ϕ (xm, xn) ∈ K.
Since K is closed, we obtain via m → ∞
∀n ∈ IN : vn − v − k0
d (x, xn) ∈ K.
The transitivity of implies (x, v) ∈ S (v0, v0) as desired.
We may apply Theorem 16 to obtain the result of the corollary.
Remark 23 1. Corollary 30 is an example of an minimal element theorem on a product
space being itself a complete metric space. More general theorems of this type, denoted
as ”authentic minimal point theorems”, have been proven in [47] and [44] using different
techniques. Compare also Section 7.1.
2. The assumptions used in Corollary 30 are strongly related to the assumptions used
in the cited references. For example, the conclusions of the corollary remain true if the
set M is replaced by {ṽ}, ṽ ∈ V and assumption (A4) by (SP4) of [44], p. 97, namely
(A40) Every ≤K-decreasing sequence being bounded from below is asymptotic.
3. It is possible to weaken the assumptions even in the setting of uniform spaces. This
requires different methods for the proofs, namely a scalarization technique. See Section
7.1..
Using different order relations, further results can be obtained immediately. One example
is the following corollary, a generalization of Theorem 8 of [61].
(A1) (X, d) is a complete metric space and (V, k·k) a Banach space;
(A2) K ⊆ V is a closed convex set and a cone in P (V ) generating the quasiorder ≤K on
V and k0 ∈ K − K;
(A3) The nonempty closed set A ⊆ X × V and the topological bounded set M ⊆ V are
such that
(x, v) ∈ A =⇒ v ∈ M ⊕ K;

4.7. Minimal elements on product spaces 99
Then, for each (v0, v0) ∈ A, there exists (x̄, v̄) ∈ A such that
(i) v̄ + k0 (d (x̄, x0) + kv̄ − v0k) ≤K v0
(ii) (x, v) ∈ M, v + k0 (d (x, x̄) + kv + v̄k) ≤K v̄ =⇒ (x, v) = (x̄, v̄) .
Proof. The assumptions of Theorem 16 are checked for the relation
x0
, v0

(x, v) ⇐⇒ v0
+ k0
d x0
, x

+ v0
− v

≤K v.
defined on the complete metric space (A, d (·, ·) + k· − ·k). Reflexivity is obvious and
transitivity can be checked straightforward.
To show regularity, take a –decreasing sequence {(xn, vn)}n∈IN, i.e.,
∀n ∈ IN : vn+1 + k0
[d (xn+1, xn) + kvn+1 − vnk] ≤K vn.
The transitivity of yields
∀m ∈ IN : v0 − vm − k0
m−1
X
n=0
(d (xn+1, xn) + kvn+1 − vnk) ∈ K.
Assume that αm :=
Pm−1
n=0 (d (xn+1, xn) + kvn+1 − vnk) → +∞. Then, for sufficiently
large m,
1
αm
v0 − k0
∈
1
αm
vm ⊕ K ⊆
1
αm
M ⊕ K.
Letting m → ∞, the contradiction −k0 ∈ K is obtained since M is bounded.
Finally, the lower closedness of follows from the closedness of K and A.
We may apply Theorem 16 to get the desired result.
The preceding corollary generalizes Theorem 8 of [61] in different directions. At first,
X is assumed to be a complete metric space rather than a Banach space. At second,
the set M does not consist necessarily of a single element and K is not assumed to be
pointed. However, one may check that, under our assumptions, if (X, k·k) is assumed to
be a Banach space too, the set

(x, v) ∈ X × V : v + k0
(kxk + kvk) ∈ −K
is a closed convex pointed cone, actually generating the relation in this case.
Under the assumptions of Corollary 30, it is also possible to use order relations involving
functionals v∗ ∈ K+ explicitely. Compare Theorem 3.10.7. of [44] and Theorem 4 of [47].
We establish similar results in Chapter 7 on uniform spaces.

Chapter 5
Partial Minimal Element
Theorems on Metric Spaces
5.1 The basic theorem on metric spaces
Let (X, d) be a complete metric space and Y a nonempty set. The goal is to extend
Theorem 16 to order relations on X × Y . Let us note that the assumptions to X and
Y , respectively, are completely different. This is the main feature of the results of this
chapter in contrast to Corollary 30 and 31. The difference to Theorem 21 and 22 is the
order relation: Here we use a relation on X × Y directly, not one on X × b
P (Y ).
(MP1) (X, d) is a complete metric space, Y a nonempty set and M ⊆ X × Y a nonempty
set;
(MP2) is a reflexive and transitive relation on X × Y .
(MP3) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence with respect to , i.e.
∀n ∈ IN : (xn+1, yn+1) (xn, yn) ,
then limn→∞ d (xn+1, xn) = 0.
(MP4) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence and {xn}n∈IN converges to x ∈ X,
then there exists y ∈ Y such that (x, y) ∈ M and
∀n ∈ IN : (x, y) (xn, yn) .
Then, for each (x0, y0) ∈ M, there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) (x0, y0)
(ii) (x, y) ∈ M, (x, y) (x̄, ȳ) =⇒ x = x̄.
Proof. We may assume that d is bounded. Otherwise, it can be replaced by d0 = d
1+d
observing that (MP3), (MP4) remain in force for d0. For (x, y) ∈ X × Y , we set
SX (x, y) :=

x0
∈ X : ∃y0
∈ Y : x0
, y0

∈ M, x0
, y0

(x, y) .
101

102 Chapter 5. Partial Minimal Element Theorems on Metric Spaces
Note that always x ∈ SX (x, y) whenever (x, y) ∈ M. Starting with (x0, y0) ∈ M, a
sequence {(xn, yn)}n∈IN ⊆ M can be defined as follows: Choose xn+1 ∈ SX (xn, yn) such
that
d (xn+1, xn) ≥ sup
x∈SX (xn,yn)
d (x, xn) −
1
n
and yn+1 ∈ Y such that (xn+1, yn+1) (xn, yn) as well as (xn+1, yn+1) ∈ M.
The transitivity of and the definition of SX (xn, yn) imply xm ∈ SX (xn, yn) whenever
m ≥ n. Moreover,
d (xm, xn) ≤ sup
x∈SX (xn,yn)
d (x, xn) ≤ d (xn+1, xn) +
1
n
.
Therefore, by (MP3), {xn}n∈IN is Cauchy and hence convergent to x̄ ∈ X. By (MP4),
there is ȳ ∈ Y such that (x̄, ȳ) ∈ M and
∀n ∈ IN : (x̄, ȳ) (xn, yn) ,
especially (x̄, ȳ) (x0, y0).
Assume, there is (x0, y0) ∈ M such that (x0, y0) (x̄, ȳ). Then by transitivity, (x0, y0)
(xn, yn) for all n ∈ IN implying x0 ∈ SX (xn, yn). This gives
d x0
, xn

≤ sup
x∈SX (xn,yn)
d (x, xn) ≤ d (xn+1, xn) +
1
n
.
Since the right hand side of the last expression tends to 0, we may conclude xn → x0.
Since the limit in complete metric spaces is unique, we obtain x0 = x̄. This proves the
theorem.
Theorem 23 for a separated locally convex space Y and special order relations is called
”non authentic minimal point theorem” in [47] and [44, Section 3.10] since conclusion (ii)
of the theorem only involves the x-variable and not a true minimal element of M with
respect to .
Theorem 16 happens to be a special case of Theorem 23. To see this, take M = X×{ys}
where ys ∈ Y is a fixed single element. In this case, the quasiorder on X × Y generates
a quasiorder X on X by
x0
X x ⇐⇒ x0
, ys

X (x, ys) .
Therefore, (MP3) for is (M3) of Theorem 16 for X. The sets SX (x, ys) coincide with
the section of X at x. This means, (MP4) passes into (M4). Of course, the conclusions
of Theorem 23 specialize to those of Theorem 16.
Conversely, it seems to be not possible to derive Theorem 23 directly from Theorem
16 without an additional induction argument with respect to the y -variable like in the
above proof of Theorem 23.
Roughly speaking, assumption (MP4) can be understood as partial lower closedness
of the sections of . It is a generalization of condition (H1) of [47] and [44, Section 3.10],
playing in locally convex spaces Y . In [47], [44], also sufficient condition for (H1) are given.

5.2. Results involving ordered monoids 103
Remark 24 Assumption (MP3) of Theorem 23 ensures partial antisymmetry of the re-
lation in the sense of Definition 13. To see this, follow the arguments of Remark 40: If
(x, y) (x0, y0) as well as (x0, y0) (x, y), define a sequence by

(x, y) , x0
, y0

, (x, y) , x0
, y0

, . . .
being decreasing with respect to . (MP3) implies d (x, x0) = 0.
In the following sections we provide Y with more algebraic and topological structure from
step to step in order to obtain new as well as several known existence results for partial
minimal elements of product sets.
Finally, we remark that (X, d) has not to be complete. This assumption can be slightly
relaxed similar to the last chapter. However, the completeness is assumed in this chapter
to make the presentation more clearly.
5.2 Results involving ordered monoids
First, we assume (Y, ◦, ≤) to be a quasiordered monoid.
(A1) (X, d) is a complete metric space and (Y, ◦, ≤) a quasiordered monoid;
(A2) The nonempty set M ⊆ X × Y , (x0, y0) ∈ M and y ∈ Y are such that
(x, y) ∈ M =⇒ y ≤ y;
(A3) Φ : X × X → Y is a regular order premetric with respect to y, y0 ∈ Y ;
(A4) If {(xn, yn)}n∈IN ⊆ M is a sequence such that
∀n ∈ IN : yn+1 ◦ Φ (xn+1, xn) ≤ yn
∀n ∈ IN : y ◦ Φ (x, xn) ≤ yn.
Then, there exists (x̄, ȳ) ∈ M such that
(i) ȳ ◦ Φ (x̄, x0) ≤ y0
(ii) (x, y) ∈ M, y ◦ Φ (x, x̄) ≤ ȳ =⇒ x = x̄.
Proof. Define a relation relation
x0
, y0

(x, y) :⇐⇒ y0
◦ Φ x0
, x

≤ y
on X × Y and check the assumptions of Theorem 23. (MP2) is obvious. (A4) implies
(MP4) directly. To check (MP3), take a decreasing sequence {(xn, yn)}n∈IN ⊆ M, i.e.
yn+1 ◦ Φ (xn+1, xn) ≤ yn

for all n ∈ IN. This implies (compare the proof of Theorem 9)
ỹ ◦
n
X
k=0
Φ (xk+1, xk) ≤ y0.
The regularity of Φ ensures (MP4). The conclusions of Theorem 23 yield (i), (ii).
Remark 25 Corollary 9 is a special case of Corollary 32. To see this, take
M = {(x, f (x)) ∈ X × Y : x ∈ X} .
One can easily check the assumptions of Corollary 32. Its conclusions passes into those of
Corollary 9. Rougly speaking, this shows that a partial minimal element theorem suffices
to derive a corresponding Ekeland type theorem. For special cases, this connection has
been observed in [47].
5.3 Results involving power sets of ordered monoids
The set valued situation deserves special attention. Again, we consider the two order
relations introduced in Section 2.2.1.
(A1) (X, d) is a complete metric space, (Y, ◦, ≤) an ordered monoid and (Y, , 4) the
quasiordered monoid generated by Y := P (Y );
(A2) The nonempty set M ⊆ X × Y, (x0, M0) ∈ M and W ∈ Y are such that
(x, M) ∈ M =⇒ W 4 M;
(A3) Φ : X × X → Y is a regular order premetric with respect to W, M0 ∈ Y;
(A4) If {(xn, Mn)}n∈IN ⊆ M is a sequence satisfying
∀n ∈ IN : Mn+1 Φ (xn+1, xn) 4 Mn
such that {xn}n∈IN converges to x ∈ X, then there exists M ∈ Y such that (x, M) ∈ M
and
∀n ∈ IN : M Φ (x, xn) 4 Mn.
Then, there exists x̄, M

∈ M such that
(i) M Φ (x̄, x0) 4 M0
(ii) (x, M) ∈ M, M Φ (x, x̄) 4 M =⇒ x = x̄.
Proof. Define a reflexive and transitive relation on X × Y by
x0
, M0

(x, M) :⇐⇒ M0
Φ x0
, x

4 M.

5.4. Results involving linear spaces 105
(MP1) and (MP2) of Theorem 23 are obviously satisfied. (MP3) follows from (A4). The
regularity of Φ and (A2) ensure (MP4). The result follows from the conclusions of Theorem
23.
Of course, Corollary 33 is a special case of Corollary 32. The counterpart for the relation
2 reads as follows.
Corollary 34 Let the assumptions (A1) – (A4) of Corollary 33 be satisfied with 4 re-
placed by 2. Then, there exists x̄, M̄

∈ M such that
(i) M Φ (x̄, x0) 2 M0
(ii) (x, M) ∈ M, M Φ (x, x̄) 2 M =⇒ x = x̄.
Proof. Define a reflexive and transitive relation on X × Y by
x0
, M0

(x, M) :⇐⇒ M0
Φ x0
, x

2 M.
(MP1) and (MP2) of Theorem 23 are obviously satisfied. (MP4) follows from (A4). The
regularity of Φ and (A2) ensure (MP3). The result follows from the conclusions of Theorem
23.
5.4 Results involving linear spaces
This section contains special cases of the results of the last one whereas Y is replaced by
a topological linear space V and the power set of such a space, respectivly. The following
two corollaries deal with b
P (V ) involving the relations 4 and 2, respectively.
(A1) (X, d) is a complete metric space and (V, +) a separated topological linear space;
(A2) K ⊆ V is a convex set containing θ ∈ X and a cone in

V := b
P (V ) , ⊕

, K0 ⊆
K (−cl K) is a nonempty convex and sequentially compact set;
(A3) The nonempty set M ⊆ X × V and the topological bounded set W ∈ V are such that
(x, M) ∈ M =⇒ W 4 M;
(A5) If {(xn, Mn)}n∈IN ⊆ M is a sequence satisfying
∀n ∈ IN : Mn+1 ⊕ ϕ (xn+1, xn) K0
4 Mn
such that {xn}n∈IN converges to x ∈ X, then there exists M ∈ V such that (x, M) ∈ M
and
∀n ∈ IN : M ⊕ ϕ (x, xn) K0
4 Mn.
Then, for each (x0, M0) ∈ M with M0 6= ∅, there exists x̄, M

∈ M such that
(i) M ⊕ ϕ (x̄, x0) K0 4 M0
(ii) (x, M) ∈ M, M ⊕ ϕ (x, x̄) K0 4 M =⇒ x = x̄.

Proof. Defining the relation
x0
, M0

(x, M) :⇐⇒ M0
⊕ ϕ x0
, x

K0
4 M,
on can easily see from the properties of 4 and ϕ that is reflexive and transitive on
X × V. Clearly, (A5) is (MP4) for this order relation. We are going to check (MP3). If
for all n ∈ IN
Mn+1 ⊕ ϕ (xn+1, xn) K0
4 Mn,
we can add ϕ (xn, xn−1) K0 to both sides of this equality obtaining
Mn+1 ⊕ ϕ (xn+1, xn) K0
⊕ ϕ (xn, xn−1) K0
4 Mn ⊕ ϕ (xn, xn−1) K0
4 Mn−1.
Since K0 is a convex subset of the linear space V , it is a convex element of V, therefore
sK0 ⊕ tK0 = (s + t) K0 whenever s, t ≥ 0. This fact and the transitivity of 4 imply
Mn+1 ⊕ (ϕ (xn+1, xn) + ϕ (xn, xn−1)) K0
4 Mn−1.
Continuing this process if necessary we arrive at
Mn+1 ⊕
n−1
X
k=0
ϕ (xk+1, xk)
!
K0
4 M0
for all n ∈ IN. Assumption (A2) implies
∀n ∈ IN : W ⊕
n−1
X
k=0
ϕ (xk+1, xk)
!
K0
4 M0.
Define αn :=
Pn−1
k=0 ϕ (xk+1, xk). Take m0 ∈ M0 being nonempty by assumption. The
definition of 4 implies
∀n ∈ IN : ∃wn ∈ W, k0
n ∈ K0
: m0 − wn + k0
nαn

∈ K.
Assume αn → +∞. Then
1
αn
−
wn
αn
− k0
n ∈ K
for all n ∈ IN sufficiently large. Since K0 is sequentially compact, there is a subsequence
of

k0
n n∈IN
converging to some k0 ∈ K0. Since W is bounded, 1
αn
wn → 0 ∈ V as well as
1
αn
m0 → 0 ∈ V . Hence −k0 ∈ cl K contradicting assumption (A2). Therefore, the αn’s
remain bounded and ϕ (xn+1, xn) → 0. Hence d (xn+1, xn) → 0 by regularity of ϕ. This
shows that (MP3) is satisfied. The conclusions of Theorem 23 yield the conclusions the
present theorem.
As before (compare Corollary 17 and 18, respectively), the assumptions involving the set
K0, M0 and W are different if 4 is replaced by 2.

5.4. Results involving linear spaces 107
(A1) (X, d) is a complete metric space and (V, +) a separated topological linear space;
(A2) K ⊆ V is a convex set containing θ ∈ V and a cone in

V := b
P (V ) , ⊕

, K0 ⊆
K (−cl K) is a nonempty convex set;
(A3) The nonempty set M ⊆ X × V and the nonempty set W ∈ V are such that
(x, M) ∈ M =⇒ W 2 M;
(A5) If {(xn, Wn)}n∈IN ⊆ M is a sequence satisfying
∀n ∈ IN : Mn+1 ⊕ ϕ (xn+1, xn) K0
2 Mn
such that {xn}n∈IN converges to x ∈ X, then there exists M ∈ V such that (x, M) ∈ M
and
∀n ∈ IN : M ⊕ ϕ (x, xn) 2 Mn.
Then, for each (x0, M0) ∈ M such that M0 ⊆ V is nonempty and topological bounded,
there exists x̄, M

∈ M such that
(i) M ⊕ ϕ (x̄, x0) K0 2 M0
(ii) (x, M) ∈ M, M ⊕ ϕ (x, x̄) K0 2 M =⇒ x = x̄.
Proof. Theorem 23 should be applied to the order relation
x0
, M0

(x, M) :⇐⇒ M0
⊕ ϕ x0
, x

K0
2 M.
Again, only (MP3) calls for a proof. By similar considerations as in the proof of Corollary
35, we arrive at
W ⊕
n−1
X
k=0
ϕ (xk+1, xk)
!
K0
2 M0
for all n ∈ IN. Define αn :=
Pn−1
k=0 ϕ (xk+1, xk). Choose w ∈ W and k0 ∈ K0 which
is possible by (A2), (A3). According to the definition of 2, for each n ∈ IN, there is
(m0)n ∈ M0 such that
(m0)n − w − αnk0
∈ K.
Assume that αn → +∞. Then we may conclude k0 ∈ cl K, a contradiction. Hence
limn→+∞ ϕ (xn+1, xn) = 0. The regularity of ϕ ensures that (MP3) is satisfied. Applying
Theorem 23 gives the desired result.
The next corollary involves a subset M ⊆ X ×V rather than M ⊆ X ×P (V ). However, it
is still more general than Theorem 1 of [47]. This is due to the sets W, K0 not necessarily
containing just a single point.

(A1) (X, d) is a complete metric space and (V, +) a topological linear space;
(A2) K ⊆ V is a convex set containing θ ∈ V and a cone in

V := b
P (V ) , ⊕

generating
the quasiorder ≤K on V and k0 ∈ K (−cl K);
(A3) The nonempty set M ⊆ X × V and the topological bounded set W ⊆ V are such that
(x, v) ∈ M =⇒ v ∈ W ⊕ K;
(A5) If {(xn, vn)}n∈IN ⊆ M is a sequence satisfying
∀n ∈ IN : vn+1 + k0
ϕ (xn+1, xn) ≤K vn
such that {xn}n∈IN converges to x ∈ X, then there exists v ∈ V such that (x, v) ∈ M and
∀n ∈ IN : v + k0
ϕ (x, xn) ≤K vn.
Then, for each (x0, v0) ∈ M, there exists (x̄, v̄) ∈ M such that
(i) v̄ + k0ϕ (x̄, x0) ≤K v0
(ii) (x, v) ∈ M, v + k0ϕ (x, x̄) ≤K v̄ =⇒ x = x̄.
Proof. Setting M := {(x, {v}) : (x, v) ∈ M}, K0 =

k0 and observing that {v0} 4 {v}
if and only if v0 ≤K v for v ∈ V one can see that Corollary 37 is a special case of Corollary
35 (as well as of Corollary 36).
Of course, using the relation
x0
, v0

(x, v) :⇐⇒ v0
+ k0
ϕ x0
, x

≤K v
one can also prove Corollary 37 by an application of Theorem 23.
The results of this subsection are essentially due to Hamel and Löhne [50], [52]. Let
us note that Corollary 32 can be applied especially to conlinear subspaces of b
P (V ), where
(V, +, T ) is a topological linear space. Possible candidates are for example the set of all
closed sets or the set of all closed convex sets with suitable addition. Compare Example
12.

Chapter 6
Variational Principles on
Complete Uniform Spaces
This chapter is devoted to variational principles on complete uniform spaces. The main
difference to the metric case is the appearance of a transfinite induction argument such as
Zorn’s lemma. Without additional assumptions, i.e., simply transforming Theorem 16 into
the context of uniform spaces we are not able to avoid such an argument. The situation
completely changes if a scalarization functional is present or can be constructed. This is
the theme of the next chapter.
Minimal element theorem on uniform spaces are a common generalization of Phelps’
lemma (Lemma 1 in [101] from 1963) on the one hand and Ekeland’s variational principle
from 1972 ([28], [30]) on the other hand. The former is in toplogical linear spaces, the
latter in metric spaces, both classes of spaces belong to the class of uniform spaces. The
first result in this direction is Theorem 1 in [8] due to Brønstedt. A very general result
has been given by Vályi in the 1985 paper [124]. Besides, he proved also the first so called
vector valued version of Ekeland’s principle on uniform spaces (Theorem 5 of [124]).
6.1 The basic theorem on complete uniform spaces
6.1.1 Preliminaries
Let (X, U) be a uniform space with uniformity U ⊆ P (X × X).
Let be a quasiorder on X, i.e., a reflexive and transitive relation. As before, we
denote the lower sections Sl (x) = {x0 ∈ X : x0 x} by S (x) for x ∈ X, compare Definiton
12.
Let (A, ) be a directed set (compare [72], p. 65). A net {xα}α∈A ⊆ X is said to be
decreasing with respect to iff
∀α, β ∈ A, α β : xα xβ.
In this chapter, (X, U) is assumed to be complete. We note that the results can be modified
in order to replace the completeness by –completeness as in Chapter 4.
109

110 Chapter 6. Variational Principles on Complete Uniform Spaces
A quasiorder is called regular iff every decreasing sequence {xn}n∈IN ⊂ X is asymp-
totic, i.e.,
∀E ∈ U, ∃nE ∈ IN, ∀n ≥ nE : (xn+1, xn) ∈ E.
As in the case of a metric space, regularity forces antisymmetry.
Proposition 42 A regular quasiorder on a separated uniform space X is antisymmet-
ric.
Proof. Take x, x0 ∈ X such that x x0 x. Then, the sequence {x, x0, x, x0, . . .} is
decreasing. Regularity implies
∀D ∈ U : x, x0

, x0
, x

∈ E.
Hence x = x0 since X is separated.
A quasiorder is called lower closed iff for any decreasing net {xα}α∈A ⊆ X converging
to some x ∈ X
∀α ∈ A : x xα
holds true. A quasiorder is lower closed if and only if the sections S (x) are closed with
respect to decreasing nets, i.e. if {xα}α∈A ⊂ S (x) and limα xα = x, then x ∈ S (x).
The following theorem is parallel to Theorem 16. A theorem of this type has been estab-
lished by Vályi in [124].
(M1) (X, U) is a complete uniform space;
(M2) is a reflexive and transitive relation on X;
(M3) is regular;
(M4) is lower closed.
Then, for each x0 ∈ X there exists x̄ ∈ X such that
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. Consider the set S (x0) := {x ∈ X : x x0}. Let S0 ⊆ S (x0) be a totally
ordered subset of S (x0). Consider S0 to be a decreasing net,
S0 = {xα}α∈A , xα xα0 for α α0
for some index set A, directed by . We claim that {xα}α∈A is Cauchy. Assume the
contrary. Then there exist E ∈ U and {xn}n∈IN ⊆ {xα}α∈A ⊆ S0 such that
(xn+1, xn) /
∈ E for n ∈ IN.

6.1. The basic theorem on complete uniform spaces 111
Indeed, if {xα}α∈A is not Cauchy, there is E ∈ U such that
∀α ∈ A ∃α2 α1 α : (xα1 , xα2 ) 6∈ E.
Hence we can find α1, α2 ∈ A such that α2 α1 and (xα1 , xα2 ) 6∈ E. Set x1 := xα1 ,
x2 := xα2 . Similiarly, α3, α4 ∈ A can be found such that α4 α3 α2 and (xα3 , xα4 ) 6∈ E.
Set x3 := xα3 , x4 := xα4 and continue this procedure. A decreasing sequence {xn}n∈IN is
obtained being not asymptotic. This contradicts (M3).
Since X is complete, {xα}α∈A converges to some x̄0 ∈ X. From (M3) we obtain that
x̄0 ∈ S (xα) for each α ∈ A, especially x̄0 ∈ S (x0). Hence x̄0 xα for each α ∈ A, i.e., x̄0
is a lower bound of S0.
By Zorn’s lemma, there exists a minimal element x̄ in S (x0). Moreover, {x̄} = S (x̄)
because if x 6= x̄, x x̄ we obtain by transitivity x x̄ x0 contradicting the minimality
of x̄ in S (x0).
Remark 26 The uniform structure U on X can be equivalently generated by a family of
pseudometrics {pλ}λ∈Λ according to Definition 23. This means, each E ∈ U contains a
set of the form
Eλ,r :=

x, x0

: dλ x, x0

r , λ ∈ Λ, r 0.
The sets Eλ,r, λ ∈ Λ, r 0 form a base of the uniform structure U on X. Hence, a
sequence {xn}n∈IN ⊂ X is asymptotic if and only if
∀r 0, ∀λ ∈ Λ, ∃nr,λ ∈ IN, ∀n ≥ nr,λ : pλ (xn+1, xn) r.
Similarly, the property of being asymptotic can be described by quasimetrics (see Definition
23) or an order metric (see Definition 25).
Without serious difficulties it is possible to transform the equivalent formulations of the
basic minimal element theorem for metric spaces to the case of uniform spaces. We shall
give the statements and refer for the details of the proofs to Section 4.1.3.
X → P (X) be a set-valued mapping. If T satisfies
∀x ∈ X, ∃x0
∈ T (x) : x0
x, (WC)
then there is x̄ ∈ X such that x̄ ∈ T (x̄), i.e. x̄ is a fixed point of T. If T satisfies
∀x ∈ X, ∀x0
∈ T (x) : x0
x, (SC)
then there is x̄ ∈ X such that {x̄} = T (x̄), i.e. x̄ is an invariant point of T.

∀x ∈ S (x0) M ∃x0
∈ S (x) {x} .
(M5), x̄ ∈ M, hence x̄ ∈ M ∩ S (x0).
Theorem 24 can be derived from Theorem 25 and Theorem 26 in the same way as Theorem
16 from Theorem 17 and Theorem 18, respectively.
6.1.4 Set relation formulation
In this section, the analogues to the Theorems 21 and 22 shall be established.
Let (X, U) be a uniform space and Y as well as M ⊆ X × Y be nonempty sets.
For x ∈ X, let us define M (x) := {(x0, y) ∈ X × Y : x0 = x, (x0, y) ∈ M} ∈ b
P (X × Y )
and MY (x) := {y ∈ Y : (x, y) ∈ M} ∈ b
P (Y ). Let be a quasiorder on M. Then,
({M (x) : x ∈ X} , 4) as well as ({M (x) : x ∈ X} , 2) is quasiordered. As in Section
4.1.6 we have M (x0) 4 M (x) if and only if
∀y ∈ MY (x) , ∃y0
∈ MY x0

: x0
, y0

(x, y) (6.1)
and M (x0) 2 M (x) if and only if
∀y0
∈ MY x0

, ∃y ∈ MY (x) : x0
, y0

(x, y) . (6.2)
(M1’) (X, U) is a uniform space and X, Y as well as M ⊆ X × Y are nonempty sets;
(M2’) is a quasiorder, i.e. a reflexive and transitive relation on X × Y ;
(M3’) If {M (xα)}α∈A is a decreasing net with respect to 4, i.e.
α, β ∈ A, α β =⇒ ∀yβ ∈ MY (xβ) , ∃yα ∈ MY (xα) : (xα, yα) (xβ, yβ)
and the net {xα}α∈A converges to x ∈ X, then
∀α ∈ A : M (x) 4 M (xα) ;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence with respect to , then {xn}n∈IN is
asymptotic.
(i) M (x̄) 4 M (x0)
(ii) M (x) 4 M (x̄) =⇒ x = x̄.

Proof. We define a binary relation on X be setting
x0
X x ⇐⇒ M x0

4 M (x) .
in order to apply Theorem 24. With the help of (6.1), one can see that X is reflexive
and transitive. (MP3) gives the lower closedness of X. It remains to show the regularity.
This can be done in the same way as the regularity for 4 in the proof of Theorem 21.
Finally, a straightforward application of Theorem 24 yields (i) and (ii).
Note that the closedness assumption (M3’) can not be formulated merely in terms of the
order relation on X × Y whereas the regularity assumption (M4’) can. This is due to
the fact that closedness in uniform spaces requires nets whereas regularity involves only
sequences. Compare the proof of Theorem 21.
The corresponding result for 2 reads as follows.
(M1’) (X, U) is a uniform space and X, Y as well as M ⊆ X × Y are nonempty sets;
(M3’) If {M (xα)}α∈A ⊆ M is an increasing net with respect to 2, i.e.
α, β ∈ A, α β =⇒ ∀yβ ∈ M (xβ) ∃yα ∈ M (xα) : (xβ, yβ) (xα, yα)
and the net {xα}α∈A converges to x ∈ X, then
∀α ∈ A : M (xα) 2 M (x) ;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a increasing sequence with respect to , then {xn}n∈IN is
asymptotic.
(i) M (x0) 2 M (x̄)
(ii) M (x̄) 2 M (x) =⇒ x = x̄.
Proof. The proof is an application of Theorem 27 using the same arguments as in the
proof of Theorem 22 applying Theorem 21.
Remark 27 As in the metric case, one can consider the special Y = {yS}, a singleton. In
this case, Theorem 27 reduces to Theorem 24 (as well as Theorem 28 to a maximal element
reformulation of Theorem 24). On the other hand, Theorem 27 (as well as Theorem 28)
are proven using Theorem 24 without any reference to the constructions in the proof of
Theorem 24, especially not to Zorn’s lemma. In this sense, the theorems are equivalent.

6.1.5 Special cases of Theorem 24
In this section, we shall show that the fundamental lemma of Phelps (Lemma 1 of [101])
as well as its generalizations of Brønsted (Theorem 1 of [8]) and Mizoguchi (the lemma in
[86]) are special cases of Theorem 24.
To begin with, we reformulate Brønsted’s theorem.
(A1) (X, U) is a complete uniform space;
(A2) is a quasiorder on X with lower closed section S (x) = {x0 ∈ X : x0 x};
(A3) The function f : X → IR ∪ {+∞} is bounded below and monotone with respect to ,
i.e.,
x1 x2 =⇒ f (x1) ≤ f (x2) ;
(A4) For each E ∈ U, there is δ 0 such that x1 x2 and f (x2) − f (x1) δ implies
(x1, x2) ∈ E.
Then, for each x0 ∈ X with f (x0) ∈ IR, there is x̄ ∈ X such that
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. It suffices to verify the regularity of in order to apply Theorem 24. Take a
decreasing sequence {xn}n∈IN, i.e.
∀n ∈ IN : xn+1 xn.
Fix E ∈ U and take δ 0 from assumption (A4). Since f is monotone and bounded
below, the sequence {f (xn)}n∈IN is convergent. Hence, there is nδ ∈ IN such that
∀n ≥ nδ : f (xn) − f (xn+1) δ.
Assumption (A4) implies (xn, xn+1) ∈ E for all n ≥ nδ, hence {xn}n∈IN is asymptotic. This
proves the regularity of . The assertions of the theorem follow from those of Theorem
24.
The original lemma of Phelps (Lemma 1 in [101]) is a consequence of Corollary 38. The
details are not repeated here and can be found in [8].
(A1) (X, U) is a uniform space and {pλ}λ∈Λ a family of pseudometrics generating the
uniformity;
(A2) is a quasiorder on X with lower closed section S (x) = {x0 ∈ X : x0 x};
(A3) {fλ}λ∈Λ is a family of functions fλ : X → IR such that each fλ is bounded below on
X and monotone with respect to , i.e.
∀λ ∈ Λ : (x1 x2 =⇒ fλ (x1) ≤ fλ (x2)) ;

(A4) For each λ ∈ Λ and each ε 0, there is δλ 0 such that x1 x2 and fλ (x2) −
fλ (x1) δλ implies pλ (x1, x2) ε.
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. Again, it suffices to verify the regularity of in order to apply Theorem 24. Let
{xn}n∈IN be a decreasing sequence with respect to . Repeating the arguments from the
proof of Corollary 38 with f replaced by fλ, we obtain
∀λ ∈ Λ : ∃nλ : ∀n ≥ nλ : pλ (xn+1, xn) ε.
Since the sets
Eλ,ε =

x, x0

∈ X × X : pλ x, x0

ε , λ ∈ Λ, ε 0
form a base of the uniformity, the sequence {xn}n∈IN is asymptotic. Hence Theorem 24
can be applied to finish the proof.
In view of Proposition 33, the above corollary can be formulated replacing the family of
pseudometrics by a family of quasimetrics. Finally, we give a formulation with a function
f : X → Y , (Y, ≤, ◦) a normally ordered, topological Abelian group. If X is separated
uniform, such a group exists and additionally an order metric D : X × X → Y generating
the uniform structure and the topology on X, cf. Section 2.2.3.
To formulate the result, an additional condition is needed. Let X, Y as above such
that the following condition is satisfied:
(R) Every sequence {yn}n∈IN, that is decreasing with respect to ≤ and bounded from
below, is asymptotic, i.e.,
∀B ∈ B (θ) , ∃nB ∈ IN, ∀n ≥ nB : yn ◦ (yn+1)−1
∈ B,
where B (θ) is a neighborhood base of θ ∈ Y consisting of full sets.
(A1) (X, U) is a complete uniform space, (Y, ◦, ≤) is a normally ordered, topological
Abelian group satisfying condition (R) above;
(A2) is a quasiorder on X with lower closed sections S (x) = {x0 ∈ X : x0 x};
(A3) The function f : X → Y is bounded below and monotone with respect to , i.e.,
x1 x2 =⇒ f (x1) ≤ f (x2) ;
(A4) For all E ∈ U there is B ∈ B (θ) such that
x1 x2, f (x2) ◦ (f (x1))−1
∈ B =⇒ (x1, x2) ∈ E.
x̄ ∈ S (x0) and {x̄} = S (x̄) .

Proof. Again, the only thing te check is the regularity of . Take a decreasing sequence
{xn}n∈IN. Then, the sequence {f (xn)}n∈IN ⊆ Y is decreasing with respect to 4. Take
E ∈ U and consider B ∈ B (θ) from (A4). There is nB ∈ IN such that
∀n ≥ nB : xn+1 xn, f (xn) ◦ (f (xn+1))−1
∈ B,
since {f (xn)}n∈IN is asymptotic according to (R). (A4) implies (xn, xn+1) ∈ E for all
n ≥ nB. Hence {xn}n∈IN is asymptotic as desired and we may apply Theorem 24 to
obtain the assertions of the corollary.
Of course, Corollary 40 is a generalization of Corollary 38. A related result is Theorem 5
of [124].
6.2 Results with functions into ordered monoids
6.2.1 Ekeland’s principle over quasiordered monoids
It is possible to give a uniform space formulation of all results of Section4.2. We pick out
three main theorems to show the principal procedure at work, namely Ekeland’s principle
and its equilibrium version as well as Caristi’s fixed point theorem.
We start with Ekeland’s principle for functions mapping a uniform space into a qua-
siordered monoid. The first result of this type for extended real valued function on uniform
spaces can be found in [8]. Therein, Brønsted proved a common generalization of Eke-
land’s theorem [30], Theorem 1.1, and Lemma 1 of [101] due to Phelps playing in linear
topological spaces. Mizoguchi [86] gave a slight generalization of Brønsted’s results as well
as a fixed point theorem of Kirk–Caristi type for uniform spaces and a drop theorem in
locally convex spaces. Moreover, she established the equivalence of these results. In sev-
eral papers [89], [91], [90], [92], [93], Nemeth generalized Ekeland’s principle to functions
mapping a uniform space into an ordered topological Abelian group. Also, Khanh [73]
dealt with functions mapping so called L-spaces into ordered linear spaces. Finally, in
[51] set valued variants of Ekeland’s principle and fixed point theorems for uniform spaces
have been proven.
The following results involve order premetrics on uniform spaces in the sense of Defi-
nition 30.
(A1) (X, U) is a complete uniform space and (Y, ◦, ≤) a quasiordered monoid;
(A3) The function f : X → Y and ỹ ∈ Y , x0 ∈ X are such that
(ii) Φ is regular with respect to ỹ, f (x0) ∈ Y ;
(A4) If the net {xα}α∈A ⊆ X converges to x ∈ X and
∀α, β ∈ A, α β : f (xα) ◦ Φ (xα, xβ) ≤ f (xβ) ,

then f (x) ◦ Φ (xα, x) ≤ f (xα) for all α ∈ A.
(i) f (x̄) ◦ Φ (x̄, x0) ≤ f (x0)
(ii) f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Proof. The proof is by checking the assumptions of Theorem 24 for the relation
x0
x :⇐⇒ f x0

◦ Φ x0
, x

≤ f (x) .
The relation is reflexive since ≤ is reflexive and (P1) of Definition 30 holds. It is
transitive by (P2) and the transitivity of ≤. (M4) follows directly from assumption (A4).
It remains to check the regularity of . Let {xn}n∈IN ⊆ X be such that xn+1 xn for all
n ∈ IN, i.e.,
f (xn+1) ◦ Φ (xn+1, xn) ≤ f (xn) .
The transitivity of ≤ implies
f (xn+1) ◦ Φ (xn+1, xn) ◦ Φ (xn, xn−1) ≤ f (xn) ◦ Φ (xn, xn−1) ≤ f (xn−1) .
Continuing this process, we obtain for each n ∈ IN
f (xn+1) ◦
n
X
k=0
Φ (xk+1, xk) ≤ f (x0) .
Since ỹ ≤ f (xm) for each m ∈ IN by (A2), it follows
ỹ ◦
n
X
k=0
Φ (xk+1, xk) ≤ f (x0) .
Since by (A3) Φ is regular with respect to ỹ, f (x0), the sequence {xn}n∈IN is asymptotic.
Applying Theorem 24 yields the desired result.
We consider a set valued mapping T : X → b
P (X) in order to prove a fixed point theorem
of Kirk–Caristi type.
Let (A1) to (A4) of Corollary 41 be in force. If the mapping T : X → b
P (X) satisfies the
weak contraction condition
∀x ∈ X, ∃x0
∈ T (x) : f x0

◦ Φ x0
, x

≤ f (x) , (WC)
then T has a fixed point, i.e., there is x̄ ∈ X such that x̄ ∈ T (x̄).
If the mapping T : X → P (X) satisfies the strong contraction condition
∀x ∈ X, ∀x0
∈ T (x) : f x0

◦ Φ x0
, x

≤ f (x) , (SC)
then T has an invariant point, i.e., there is x̄ ∈ X such that {x̄} = T (x̄).

Proof. Use Corollary 41 instead of Corollary 9 in the proof of Corollary 10.
Conversely, Corollary 41 can be proven using the fixed point result above. To see this,
one has to proceed along the same lines as in the metric case. Compare the remarks after
Corollary 10.
The Theorems 6 and 8 of [124] are also fixed point theorems of Kirk–Caristi type
on uniform spaces. They involve real valued functions and a family of pseudometrics
generating the uniformity, respectively.
The next result deals with a function F : X × X → Y instead of f : X → Y .
(A1) (X, U) is a complete uniform space and (Y, ◦, ≤) a quasiordered monoid;
(A2) The function F : X × X → Y , ỹ ∈ Y and x0 ∈ X are such that
(i) F (x1, x3) ≤ F (x1, x2) ◦ F (x2, x3) for all x1, x2, x3 ∈ X;
(ii) ỹ ≤ F (x0, x) for all x ∈ X;
(A2) Φ : X × X → Y is a regular order premetric with respect to ỹ, θ ∈ Y ;
∀α, β ∈ A, α β : F (xα, xβ) ◦ Φ (xβ, xα) ≤ θ,
then F (xα, x) ◦ Φ (x, xα) ≤ θ for all α ∈ A.
(i) F (x0, x̄) ◦ Φ (x̄, x0) ≤ θ
(ii) F (x̄, x) ◦ Φ (x, x̄) ≤ θ =⇒ x = x̄.
First Proof. We check the assumptions of Theorem 16 for the relation
x0
x :⇐⇒ x0
= x or F x, x0

◦ Φ x0
, x

≤ θ.
being reflexive and transitive by the properties of Φ, F and ≤. (M4) follows directly from
assumption (A4). The regularity of can be seen in the same way as in the proof of
Corollary 13. Therefore, we may apply Theorem 24 obtaining the desired result.
We shall indicate a sufficient condition for (A4) of Corollary 41. A function f : X → Y
is called lower monotone iff for each net {xα}α∈A ⊆ X converging to some x ∈ X and
satisfying xα xβ for α β the inequality f (x) ≤ f (xα) holds true for all α ∈ A.
Compare [93] for this kind of condition. It can be interpreted as a generalization of lower
semicontinuity.
Moreover, an order premetric Φ : X ×X → Y is called lower monotone with respect
to the first variable iff for each net {xα}α∈A ⊆ X converging to x ∈ X and satisfying
xα xβ for α β we have for all x0 ∈ X
y1, y2 ∈ Y, ∀α ∈ A : y1 ◦ Φ xα, x0

≤ y2 =⇒ y1 ◦ Φ x, x0

≤ y2.

Lemma 8 Let X, Y be as in Corollary 41, the function f : X → Y be lower monotone
and the order premetric Φ be lower monotone with respect to the first variable. Then (A4)
of Corollary 41 is satisfied.
Proof. Take a net {xα}α∈A ⊆ X converging to x ∈ X such that
∀α, β ∈ A, α β : f (xα) ◦ Φ (xα, xβ) ≤ f (xβ) .
Then, since θ ≤ Φ (xα, xβ),
f (xα) ≤ f (xα) ◦ Φ (xα, xβ) ≤ f (xβ)
and therefore f (xα) ≤ f (xβ) for all α β since ≤ is transitive. The lower monotonicity
of f implies f (x) ≤ f (xα) for all α ∈ A. For α, β ∈ A, α β we obtain
f (x) ◦ Φ (xα, xβ) ≤ f (xα) ◦ Φ (xα, xβ) ≤ f (xβ) .
Since Φ is lower monotone with respect to the first variable, this implies
f (x) ◦ Φ (x, xβ) ≤ f (xβ)
as desired.
6.2.2 Power sets of quasiordered monoids
This subsection contains results parallel to those of Section 1.3.1.
(A1) (X, U) is a complete uniform space, (Y, ◦, ≤) an ordered monoid and (Y, , 4) the
ordered monoid generated by Y := b
P (Y );
(A2) The function f : X → Y and W ∈ Y are such that
∀x ∈ X : W 4 f (x) ;
(A3) Φ : X × X → Y is a regular order premetric with respect to W, f (x0) ∈ Y;
∀α, β ∈ A, α β : f (xα) ◦ Φ (xα, xβ) 4 f (xβ) ,
then f (x) ◦ Φ (xα, x) 4 f (xα) for all α ∈ A.
(i) f (x̄) Φ (x̄, x0) 4 f (x0)
(ii) x ∈ X, f (x) Φ (x, x̄) 4 f (x̄) =⇒ x = x̄.

Proof. By Theorem 11,

b
P (Y ) , , 4

is a quasiordered monoid. Defining the relation
x0
x :⇐⇒ f x0

Φ x0
, x

4 f (x)
on X, the assumptions of Corollary 41 are easy to check. Its conclusions yield the desired
result.
Corollary 45 Let the assumptions of Corollary 44 be satisfied with 4 replaced by 2.
(i) f (x̄) Φ (x̄, x0) 2 f (x0)
(ii) x ∈ X, f (x) Φ (x, x̄) 2 f (x̄) =⇒ x = x̄.
Proof. Replace 4 by 2 in the proof of Corollary 44.
6.2.3 Single valued functions
We show that Nemeth’s results in [93] follow from Theorem 24. They involve a function
f mapping a uniform space into a toplogical Abelian group. Compare Corollary 46.
In the following corollary, Y is an ordered group not order complete in general. As
usual, we can adjoin a largest element yl obtaining an ordered monoid.
Corollary 46 Let the following assumptions be satisfied: (A1) (X, U) is a complete uni-
form space and (Y, ◦, ≤) an ordered topological Abelian group;
(A2) The function f : X → Y ∪ {yl} and ỹ ∈ Y are such that ỹ ≤ f (x) for all x ∈ X;
(A3) Φ : X × X → Y is a regular order premetric with respect to ỹ, f (x0) for x0 ∈ X;
∀α, β ∈ A, α β : f (xα) ◦ Φ (xα, xβ) ≤ f (xβ) ,
then f (x) ◦ Φ (x, xα) ≤ f (xα) for all α ∈ A.
(i) f (x̄) ◦ Φ (x̄, x0) ≤ f (x0)
(ii) f (x) ◦ Φ (x, x̄) ≤ f (x̄) =⇒ x = x̄.
Proof. Apply Corollary 41 to (Y ∪ {yl} , ◦, ≤).
Taking Y = V a topological linear space and corresponding order premetrics we are led to
further results parallel to those of Section 4.3, 4.4 and 4.5. The results of Section 4.7 also
have counterparts for a uniform space X. Let us mention that Theorem 1 of Brønsted in
[8] is also a special case of Corollary 46.
Similar results with a family of quasimetrics instead of an order premetric can be found
in [49].

6.3. A partial minimal element theorem on complete uniform spaces 121
6.3 A partial minimal element theorem on complete uni-
form spaces
Let (X, U) be a complete uniform space and Y a nonempty set. The goal is to extend
Theorem 24 to order relations on X × Y . The result is parallel to Theorem 23. Let be
a quasiordering on X × Y . A net {(xα, yα)}α∈A ⊆ X × Y is said to be decreasing iff
∀α, β ∈ A, α β : (xα, yα) (xβ, yβ) .
A quasiorder on X × Y is called regular on M ⊆ X × Y iff for each decreasing sequence
{(xn, yn)}n∈IN ⊆ M the sequence {xn}n∈IN is asymptotic, i.e.,
∀E ∈ U, ∃nE ∈ IN, ∀n ≥ nE : (xn+1, xn) ∈ E.
Remark 28 Let (X, U) be a uniform space and Y a nonempty set. A regular quasiordering
on X × Y is partially antisymmetric. To see this, proceed as in Remark 24.
We state a result for complete uniform spaces parallel to Theorem 23.
(MP1) (X, U) is a complete uniform space, Y and M ⊆ X × Y are nonempty sets;
(MP2) is a quasiorder on X × Y ;
(MP3) The quasiorder is regular on M;
(MP4)If {(xα, yα)}α∈A ⊆ M is a decreasing net such that {xα}α∈A converges to x ∈ X,
then there exists y ∈ Y such that (x, y) ∈ M and
∀α ∈ A : (x, y) (xα, yα) .
Then, for each (x0, y0) ∈ M, there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) (x0, y0)
(ii) (x, y) ∈ M, (x, y) (x̄, ȳ) =⇒ x = x̄.
Proof. Consider the section S (x0, y0) := {(x, y) ∈ M : (x, y) (x0, y0)}. Let S0 ⊆
S (x0, y0) be a totally ordered subset of S (x0, y0), namely a decreasing net {(xα, yα)}α∈A
with some directed index set A. Then {xα}α∈A is a Cauchy net by (MP3). To see this,
one can argue in the same way as in the proof of Theorem 23. By completeness, {xα}α∈A
is convergent to some x̂ ∈ X, hence there is ŷ ∈ Y such that
∀α ∈ A : (x̂, ŷ) (xα, yα) .
This shows that S0 is bounded below in S (x0, y0). Zorn’s lemma, applied to the partially
antisymmetric quasiorder , ensures the existence of a partial minimal point (x̄, ȳ) ∈
S (x0, y0). This completes the proof.
Using Theorem 29 we may obtain results which are the analogues to those of Chapter
5 in complete uniform space. We do not go into the details, but switch to the case of
sequentially complete uniform spaces.

Chapter 7
Variational Principles on
Sequentially Complete Uniform
Spaces
Two observations gave rise to the developments of this chapter. First, there are two
proofs for the central result in Brønsted’s paper [8], the first one involves Zorn’s lemma,
the second one does not, but only a countable induction argument. The question arises,
in which cases the countable induction argument is sufficient. Secondly, minimal element
theorems on sequentially complete uniform spaces have been established by A. Löehne
and the author in [51] using a scalarization technique and the Brézis–Browder theorem
[6]. The proof of the latter involves a countable induction argument only. Again, the
question is under which assumptions this is sufficient. The results of this chapter show
that in presence of a monotone real valued function with suitable porperties linking the
order and the uniform structure it is not necessary to use full versions of Zorn’s lemma.
Since only sequences are involved, the completeness assumption of the last chapter can be
weakend to sequential completeness.
7.1 The basic theorem with sequential completeness
7.1.1 Preliminaries
Let (X, U) be uniform space. The quasiorder on X is called sequentially lower closed
iff the section S (x) = {x0 ∈ X : x0 x} is sequentially lower closed, i.e., if {xn}n∈IN ⊆
S (x) is decreasing with respect to and convergent to x̄ ∈ X, then x̄ ∈ S (x).
(M1) (X, U) is a separated, sequentially complete uniform space;
(M2) is a quasiorder on X;
123

124 Chapter 7. Variational Principles on Sequentially Complete Uniform Spaces
(M3) The function f : X → IR∪{+∞} is proper, bounded from below on X and monotone
with respect to , i.e.
x1 x2 =⇒ f (x1) ≤ f (x2) ,
moreover, for each E ∈ U, there is δ 0 such that
x1 x2, f (x2) − f (x1) δ =⇒ (x1, x2) ∈ E;
(M4) The quasiorder is sequentially lower closed.
Then, for each x0 ∈ dom f there exists x̄ ∈ dom f such that
x̄ ∈ S (x0) and {x̄} = S (x̄) .
Proof. Starting with x0 we choose a sequence according to
xn+1 ∈ S (xn) , f (xn+1) ≤ inf
x∈S(xn)
f (x) +
1
n
.
The monotonicity of f implies f (xn+1) ≤ f (xn) for all n ∈ IN. Since f is bounded from
below, the sequence {f (xn)}n∈IN converges to some r ∈ IR.
Take E ∈ U and choose nE ∈ IN such that f (xnE ) r + δ with δ from (M3). This
implies for n ≥ nE
f (xnE ) − f (xn) r + δ − r = δ.
Hence, for all m ≥ n ≥ nE
f (xm) − f (xn) δ
holds true implying (xn, xm) ∈ E. Therefore, {xn}n∈IN is a Cauchy sequence. Since X
is sequentially complete, it converges to some x̄ ∈ X and by (M4), x̄ ∈ S (xn) for each
n ∈ IN. Especially, x̄ ∈ S (x0) holds true.
Let x x̄, x 6= x̄. Since f is monotone, the yields f (x) ≤ f (x̄). On the other hand,
the transitivity of implies x x̄ xn for each n ∈ IN. The rules for the choice of xn+1
yield
f (x̄) ≤ f (xn+1) ≤ f (x) +
1
n
.
This yields f (x̄) ≤ r ≤ f (x), hence f (x̄) = r = f (x).
Again, take an arbitrary E ∈ U. Since {f (xn)}n∈IN converges to f (x), there is nE ∈ IN
such that
∀n ≥ nE : f (x) − f (xn) δ
This implies (xn, x) ∈ E. Since (X, U) is separated, we can conclude x = x̄.

7.1. The basic theorem with sequential completeness 125
Without serious difficulties it is possible to transform the equivalent formulations of the
basic minimal element theorem for metric spaces to the case of sequentially complete
uniform spaces. We shall give the statements and refer for the details of the proofs to
Section 4.1.3 and 7.1.3.
X → P (X) be a set-valued mapping. If T satisfies
∀x ∈ X, ∃x0
∈ T (x) : x0
x, (WC)
then there is x̄ ∈ X such that x̄ ∈ T (x̄), i.e., x̄ is a fixed point of T. If T satisfies
∀x ∈ X, ∀x0
∈ T (x) : x0
x, (SC)
then there is x̄ ∈ X such that {x̄} = T (x̄), i.e., x̄ is an invariant point of T.
∀x ∈ S (x0) M ∃x0
∈ S (x) {x} .
(M5), x̄ ∈ M, hence x̄ ∈ M ∩ S (x0).
Again, Theorem 30 can be derived from Theorem 31 and Theorem 32 in the same way as
Theorem 16 from Theorem 17 and Theorem 18, respectively.
7.1.4 Set relation ordering principle
In this section, the analogues to Theorems 21 and 22 shall be established.
Let X, Y as well as M ⊆ X × Y be nonempty sets. Again, we define as in Sec-
tion 4.1.6, M (x) := {(x0, y) ∈ X × Y : x0 = x, (x0, y) ∈ M} ∈ b
P (X × Y ) and MY (x) :=
{y ∈ Y : (x, y) ∈ M} ∈ b
P (Y ) for x ∈ X.
Let be a quasiorder on M. Then, ({M (x) : x ∈ X} , 4) as well as ({M (x) : x ∈ X} , 2)
is quasiordered, compare Section 4.1.6.
(M1’) (X, U) is a separated, sequentially complete uniform space and Y as well as M ⊆
X × Y are nonempty sets;
(M2’) is a quasiorder on X × Y ;

(M3’) The function g : Y → IR ∪ {+∞} is proper, bounded from below on Y and satisfies
the monotonicity condition
(x1, y1) (x2, y2) =⇒ g (y1) ≤ g (y2) ,
(x1, y1) (x2, y2) , g (y2) − g (y1) δ =⇒ (x1, x2) ∈ E;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a decreasing sequence with respect to such that {xn}n∈IN
is converges to x ∈ X, then there is y ∈ Y such that (x, y) ∈ M and
∀n ∈ IN : (x, y) (xn, yn) .
Then, for each x0 ∈ X with MY (x0)∩dom g 6= ∅, there exists x̄ ∈ X such that MY (x̄) 6= ∅
and
(i) M (x̄) 4 M (x0)
(ii) M (x) 4 M (x̄) =⇒ x = x̄.
Proof. Define a binary relation on X be setting
x0
X x ⇐⇒ M x0

4 M (x) .
in order to apply Theorem 30. From the definition of 4, one can see that X is reflexive
and transitive.
To check (M3) of Theorem 30, define a function f : X → IR ∪ {+∞} by
f (x) := inf
y∈MY (x)
g (y) .
Then f is proper and bounded below on X since g is proper and bounded below on
Y . Moreover, (x, y) ∈ M, y ∈ dom g implies x ∈ dom f. Assume that x1 X x2, i.e.,
M (x1) 4 M (x2). The definition of 4 and the monotonicity property of g yield
∀y2 ∈ MY (x2) ∃y1 ∈ MY (x1) : g (y1) ≤ g (y2)
implying f (x1) ≤ f (x2), i.e., f is monotone with respect to X. Fix E ∈ U and take
δ 0 from (M3’). Then there is y2 ∈ MY (x2) such that g (y2) ≤ f (x2) + δ
2 according to
the definition of f. Since for each y1 ∈ MY (x1) we have f (x1) ≤ g (y1), this implies
g (y2) − g (y1) ≤ f (x2) − f (x1) +
δ
2
.
Hence from x1 X x2, f (x2) − f (x1) δ
2 we may conclude (x1, x2) ∈ E.
It remains to show the sequential lower closedness of X. This is straightforward by
taking a X-decreasing and converging sequence and construct a -decreasing sequence in
the same way as in the proof of Theorem 21. Then (M4’) is sufficient for (M4) of Theorem
30.
Finally, an application of the latter theorem yields the desired results.
Again, the corresponding result for 2 has to be formulated as a maximal element result
for the same reasons as Theorem 22.

7.2. The basic theorem on a product space 127
(M1’) (X, U) is a separated, sequentially complete uniform space and Y as well as M ⊆
X × Y are nonempty sets;
(M2’) is a quasiorder on X × Y ;
(M3’) The function g : Y → IR ∪ {+∞} is proper, bounded from below on Y and satisfies
the monotonicity condition
(x1, y1) (x2, y2) =⇒ g (y1) ≥ g (y2) ,
(x1, y1) (x2, y2) , g (y1) − g (y2) δ =⇒ (x1, x2) ∈ E;
(M4’) If {(xn, yn)}n∈IN ⊆ M is a increasing sequence with respect to such that {xn}n∈IN
is converges to x ∈ X, then there is y ∈ Y such that (x, y) ∈ M and
∀n ∈ IN : (xn, yn) (x, y) .
Then, for each x0 ∈ X with MY (x0)∩dom g 6= ∅, there exists x̄ ∈ X such that MY (x̄) 6= ∅
and
(i) M (x0) 2 M (x̄)
(ii) M (x̄) 2 M (x) =⇒ x = x̄.
Proof. The proof is an application of Theorem 33 using the same arguments as in the
proof of Theorem 22 applying Theorem 21. Thus, the relation 0 and 20 are defined by
(x1, y1) 0
(x2, y2) ⇐⇒ (x2, y2) (x1, y1)
and
M (x1) 40
M (x2) ⇐⇒ M (x2) 2 M (x1) ,
respectively. Assumption (M3) of Theorem 33 is satisfied for g and 0. For more details
compare the proof of Theorem 22.
Of course, assumption (M3’) of Theorem 34 can be formulated using a function g : Y →
IR ∪ {−∞} being bounded from above and satisfying
(x1, y1) (x2, y2) =⇒ g (y1) ≤ g (y2) .
Then, one has to define g0 := −g using (−1) (−∞) = +∞ in order to apply Theorem 33.
7.2 The basic theorem on a product space
As in Chapter 5 for the case of a metric space X we establish a partial minimal element
theorem for subsets M ⊆ X × Y .

(A1) (X, U) is a sequentially complete, separated uniform space; Y is a nonempty set and
M ⊆ X × Y is also nonempty;
(A2) is a quasiorder on X × Y ;
(A3) The function f : Y → IR ∪ {+∞} is bounded from below on
YM := {y ∈ Y : ∃x ∈ X : (x, y) ∈ M}
and monotone with respect to , i.e.,
(x1, y1) (x2, y2) =⇒ f (y1) ≤ f (y2) ,
(x1, y1) (x2, y2) , f (y2) − f (y1) δ =⇒ (x1, x2) ∈ E;
(A4) If the sequence {(xn, yn)}n∈IN ⊆ M is decreasing with respect to and {xn}n∈IN
converges to some x ∈ X, then there is y ∈ Y such that (x, y) ∈ M and (x, y) (xn, yn)
for each n ∈ IN.
Then, for each (x0, y0) ∈ M with f (y0) ∈ IR, there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) (x0, y0)
(ii) (x, y) ∈ M, (x, y) (x̄, ȳ) =⇒ x = x̄, f (y) = f (ȳ) .
If, additionally, the assumption
(A5) (x1, y1) (x2, y2), y1 6= y2 implies f (y1) f (y2);
holds true, then (x̄, ȳ) can be chosen to be a minimal point of M with respect to , i.e.,
{(x̄, ȳ)} = S (x̄, ȳ) ∩ M.
Proof. Starting with (x0, y0) we choose a sequence according to
(xn+1, yn+1) ∈ S (xn, yn) ∩ M, f (yn+1) ≤ inf
(x,y)∈S(xn,yn)∩M
f (y) +
1
n
.
The monotonicity of f implies f (yn+1) ≤ f (yn) for all n ∈ IN. Since f is bounded from
below on YM , the sequence {f (yn)}n∈IN converges to some r ∈ IR.
Take E ∈ U and choose nE ∈ IN such that f (ynE ) r + δ with δ from (A3). This
implies for n ≥ nE
f (ynE ) − f (yn) r + δ − r = δ.
Hence, for all m ≥ n ≥ nE
f (ym) − f (yn) δ
holds true implying (xn, xm) ∈ E. Therefore, {xn}n∈IN is a Cauchy sequence converging
to some x̄ ∈ X by completeness. By (A4), there is ȳ ∈ Y such that (x̄, ȳ) ∈ M and
∀n ∈ IN : (x̄, ȳ) (xn, yn) .

Let (x, y) (x̄, ȳ). The monotonicity property of f implies f (y) ≤ f (ȳ). On the other
hand, the transitivity of implies (x, y) (x̄, ȳ) (xn, yn) for each n ∈ IN. The rules for
the choice of yn+1 gives
f (ȳ) ≤ f (yn+1) ≤ f (y) +
1
n
.
This yields f (ȳ) ≤ r ≤ f (y), hence f (ȳ) = r = f (y).
Again, take an arbitrary E ∈ U. Since {f (yn)}n∈IN converges to f (y), there is nE ∈ IN
such that
∀n ≥ nE : f (y) − f (yn) δ
implying (xn, x) ∈ E. Since (X, U) is separated, we may conclude x = x̄. Since f (ȳ) =
f (y), (x̄, ȳ) ∈ M is minimal with repect to if (A5) is satisfied. This completes the proof
of the theorem.
Before investigating some special cases of Theorem 35, we mention a version where the
order relation is defined in terms involving the function f. Consider X, Y , M, , f as in
Theorem 35 and define an order relation by
(x1, y1) f (x2, y2) :⇐⇒
(
(x1, y1) = (x2, y2) or
(x1, y1) (x2, y2) and f (y1) f (y2) .
Obviously, (x1, y1) f (x2, y2) implies (x1, y1) (x2, y2). Therefore, it is easily seen that
the assumptions (A1) to (A5) of Theorem 35 are satisfied for f if (A1) to (A4) are
satisfied for . This is the idea of the proof of the following theorem.
Theorem 36 Let the assumptions (A1) through (A4) of Theorem 35 be in force. Then,
for each (x0, y0) ∈ M with y0 ∈ dom f there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) f (x0, y0)
(ii) (x, y) ∈ M, (x, y) f (x̄, ȳ) =⇒ (x, y) = (x̄, ȳ) ,
i.e., (x̄, ȳ) is a minimal point of M with respect to f .
Proof. According to the remarks above, an obvious application of Theorem 35.
Next, we produce a series of corollaries from Theorem 35 by special choices of Y and
the order relation . Thereby, many recent results can be proven, e.g. minimal point
theorems from [47] and [44] as well as results from [51] and [50].
The first case involves a locally convex space Y and f is replaced by a continuous linear
functional.
(A1) (X, U) is a sequentially complete, separated uniform space and {pλ}λ∈Λ a family of
pseudometrics generating the uniformity; Y is a locally convex space and M ⊆ X × Y is
a nonempty set;

(A2) ≤K is a quasiorder on Y with K ⊆ Y being a convex set containing θ ∈ Y and a
cone in P (Y ), further, let k ∈ K − cl K; a relation on X × Y is defined via
(x1, y1) (x2, y2) :⇐⇒ ∀λ ∈ Λ : y1 + pλ (x1, x2) k ≤K y2;
(A3) There is a bounded set W ⊆ Y such that
YM := {y ∈ Y : ∃x ∈ X : (x, y) ∈ M} ⊆ W ⊕ K;
(A4) If the sequence {(xn, yn)}n∈IN ⊆ M is decreasing with respect to and {xn}n∈IN
converges to some x ∈ X, then there is y ∈ YM such that (x, y) (xn, yn) for each n ∈ IN.
Then, for each (x0, y0) ∈ M there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) (x0, y0)
(ii) (x, y) ∈ M, (x, y) (x̄, ȳ) =⇒ x = x̄.
If, additionally,
(A5) K+ := {y∗ ∈ Y ∗ : ∀y ∈ K {0} : y∗ (y) 0} 6= ∅;
is satisfied, then (x̄, ȳ) can be chosen to be a minimal point of M with respect to , i.e.
{(x̄, ȳ)} = S (x̄, ȳ) ∩ M.
Proof. Of course, is a quasiorder. A standard separation argument, applied to {−k}
and K yields a continuous linear functional y∗ ∈ Y ∗ such that y∗ (k) = 1 and
∀y ∈ K : y∗
(y) ≥ 0.
We set f (y) := y∗ (y) and check assumption (M3) of Theorem 35. Since W is bounded
and f nonnegative on K, f is bounded below on W ⊕ K and all the more on YM . Take
E ∈ U. Then there are r 0, λ ∈ Λ such that
Er,λ := {(x1, x2) ∈ X × X : pλ (x1, x2) r} ⊆ E
since the sets Er,λ form a base of the uniformity U. If (x1, y1) (x2, y2) and f (y2) −
f (y1) r, then
y∗
(y1) + pλ (x1, x2) = y∗
(y1 + kpλ (x1, x2)) ≤ y∗
(y2)
since f is linear and nonnegative on K. Therefore
pλ (x1, x2) ≤ y∗
(y2) − y∗
(y1) ≤ f (y2) − f (y1) r,
hence (x1, x2) ∈ Er,λ ⊆ E as desired.
Since assumption (A4) coincides with assumption (M4) of Theorem 35 we can apply
the latter to get (i) and (ii). If, additionally, K+ 6= ∅, then simply take y∗ ∈ K+ to ensure
(M5). This completes the proof.

Corollary 47 is a generalization of Theorem 1 in [47] and Theorem 3.10.4 of [44]. Replacing
the relation by y∗ in the sense of Theorem 36, we obtain generalizations of Theorem
4 in [47] and Theorem 3.10.7 of [44].
The generalizations mainly concern the space X and the boundedness assumption:
We deal with sequentially complete uniform spaces instead of complete metric spaces.
Moreover, a single element b
y is replaced by a bounded set W in assumption (A2).
Note that Corollary 47 has a counterpart using quasimetrics instead of pseudometrics.
Also, a formulation with an order metric is possible.
The reach of Corollary 47 is limited by the appearence of a nontrivial continuous linear
functional on Y , i.e., as a rule, Y has to be a locally convex space. Moreover, it has been
observed in [47], [44] as well as in [51] and [50] that the boundedness assumption can be
relaxed by a weaker one. In the following corollary, a sublinear functional on a linear
space being linear only on a one dimensional subspace is used as a substitute for the
continuous linear functional in Corollary 47. This allows to deal with merely linear spaces
and to replace the boundedness assumption by a weaker one. Functionals of this type has
been introduced and investigated by C. Tammer and P. Weidner in [39], [40], [41] and
extensively in [125]. Compare also Theorem 2.3.1. in [44].
(A1) (X, U) is a sequentially complete, separated uniform space and {pλ}λ∈Λ a family
of pseudometrics generating the uniformity; Y is a linear space and M ⊆ X × Y is a
nonempty set;
(A2) ≤K is a quasiorder on Y with K ⊆ Y being a convex set containing θ ∈ Y and a
cone in P (Y ), further, let k ∈ K − K; a relation on X × Y is defined via
(x1, y1) (x2, y2) :⇐⇒ ∀λ ∈ Λ : y1 + pλ (x1, x2) k ≤K y2;
(A3) There exist b
y ∈ Y and b
t ∈ IR such that
{y ∈ Y : ∃x ∈ X : (x, y) ∈ M} ∩

b
y − b
tk ⊕ (−K)

= ∅;
(A4) If the sequence {(xn, yn)}n∈IN is decreasing with respect to and {xn}n∈IN converges
to some x ∈ X, then there is y ∈ Y such that (x, y) ∈ M and (x, y) (xn, yn) for each
n ∈ IN.
Then, for each (x0, y0) ∈ M with y0 ∈ IR {k} ⊕ (−K) there exists (x̄, ȳ) ∈ M such that
(i) (x̄, ȳ) (x0, y0)
(ii) (x, y) ∈ M, (x, y) (x̄, ȳ) =⇒ x = x̄.
Proof. We are going to apply Theorem 35. Of course, (M1) and (M2) of this theorem
are satisfied. We check (M3) using the function
f (y) := inf {t ∈ IR : y − b
y ∈ {tk} ⊕ (−K)} .

Since the function ϕ (y) := inf {t ∈ IR : y ∈ {tk} ⊕ (−K)} is monotone with respect to
≤K and subadditive, f is monotone as well. Note that ϕ satisfies
∀s ∈ IR, y ∈ Y : ϕ (y + sk) = ϕ (y) + s.
This property is called translation property, see [44], Section 2.3. Further, f is bounded
below on YM := {y ∈ Y : ∃x ∈ X : (x, y) ∈ M}. To see this, assume the contrary, i.e.,
there is a ỹ ∈ YM such that f (ỹ) −b
t and hence there is t̃ ∈ IR, t̃ −b
t such that
ỹ − b
y ∈ t̃ {k} ⊕ (−K). This implies
ỹ ∈

b
y + t̃k ⊕ (−K) =

b
y − b
tk ⊕

t̃ + b
t

k ⊕ (−K) ⊆

b
y − b
tk ⊕ (−K)
contradicting (A3). Hence f is bounded below on YM .
Take E ∈ U and r 0, λ ∈ Λ such that Er,λ ⊆ E. Assuming (x1, y1) (x2, y2) and
f (y2) − f (y1) r we obtain by monotonicity and the translation property of ϕ
pλ (x1, x2) ≤ ϕ (y2 − b
y) − ϕ (y2 − b
y) = f (y2) − f (y1) r,
hence (x1, x2) ∈ Er,λ ⊆ E. Therefore, the assumptions of Theorem 35 are satisfied. Note
that y0 ∈ IR {k} ⊕ (−K) implies f (y0) ∈ IR.
The conclusions (i) and (ii) of Theorem 35 yield (i) and (ii) above.
Corollary 48 produces a generalization of Corollary 47: Y can be replaced by a linear
space and the boundedness assumption can be weakened.
Note that within the setting of Corollary 48 it is difficult to give a sufficient condition
for (A5) of Theorem 35, i.e., for the existence of a minimal point with respect to .
Usually, topological properties are used as in part (g) of Theorem 2.3.1. in [44]. Note also
that f (y) = f (ȳ) can be added in (ii) of Corollary 48.
The set Y in Theorem 35 is arbitrary, hence the possibility of choosing Y ⊆ P (V ),
V being a quasiordered linear space, is not excluded. We turn to this case in order to
derive results similar to those of [50]. Since sets are compared, the order relations 4 and
2 appear.
(A1) (X, U) is a sequentially complete, separated uniform space and {pλ}λ∈Λ a family of
pseudometrics generating the uniformity; V is a linear space and M ⊆ X × b
P (V ) is a
nonempty set;
(A2) ≤ is a quasiorder on V with K ⊆ V being a convex set containing θ ∈ V and a cone
in P (V ) and k ∈ K − K; a relation is defined via
(x1, W1) (x2, W2) :⇐⇒ ∀λ ∈ Λ : W1 ⊕ {pλ (x1, x2) k} 4 W2;
(A3) There exist b
v ∈ V and b
t ∈ IR such that


[
(x,W)∈M
W



b
v − b
tk ⊕ (−K)

= ∅;

(A4) If the sequence {(xn, Wn)}n∈IN ⊆ M is decreasing with respect to and {xn}n∈IN
converges to some x ∈ X, then there is W ∈ b
P (V ) such that (x, W) ∈ M and (x, W)
(xn, Wn) for each n ∈ IN.
Then, for each (x0, W0) ∈ M with IR {k} ⊕ {b
v} ∩ (W0 ⊕ K) 6= ∅ there exists x̄, W̄

∈ M
such that
(i) x̄, W̄

(x0, W0)
(ii) (x, W) ∈ M, (x, W) x̄, W̄

=⇒ x = x̄.
Proof. Again, we wish to apply Theorem 35. It is not hard to verify that (M1) and
(M2) of this theorem are matched as well as (M4). To verify (M3), we define a function
f : b
P (V ) → IR ∪ {±∞} by
f (W) := inf {t ∈ IR : tk + b
v ∈ W ⊕ K} .
Using this definition, the monotonicity property of f and the translation property
∀W ∈ b
P (V ) , ∀s ∈ IR : f (W ⊕ {sk}) = f (W) + s
can be proven straighforward. Let us show that f is bounded below on
n
W ∈ b
P (V ) : ∃x ∈ X : (x, W) ∈ M
o
.
Assume the contrary. Then there is (x, W) ∈ M such that f (W) −b
t. Hence there is
s ∈ IR, s b
t such that
sk + b
v ∈ W ⊕ K.
Especially, W 6= ∅. Take w ∈ W. Then
w ∈ b
v + sk ⊕ (−K) = b
v + s + b
t

k − b
tk ⊕ (−K) ⊆ b
v − b
tk ⊕ (−K)
contradicting (A3). The last part of assumption (M3) can be proven as in the proof of
Corollary 48. We may apply Theorem 35 to obtain (i) and (ii) above from its conclusions.
Corollary 50 Let (A1) and (A2) of Corollar49 be satisfied with 4 replaced by 2. More-
over, assume:
(A3) There exist b
v ∈ V and b
t ∈ IR such that
(x, W) ∈ M =⇒ W *

b
v − b
tk ⊕ (−K)
(A4) If the sequence {(xn, Wn)}n∈IN ⊂ M is decreasing with respect to and {xn}n∈IN
converges to some x ∈ X, then there is W ∈ b
P (V ) such that (x, W) ∈ M and (x, W)
(xn, Wn) for each n ∈ IN.
Then, for each (x0, W0) ∈ M with W0 ⊆ {t0k + b
v} ⊕ (−K) for some t0 ∈ IR, there exists
x̄, W̄

∈ M such that
(i) x̄, W̄

(x0, W0)
(ii) (x, W) ∈ M, (x, W) x̄, W̄

=⇒ x = x̄.

Proof. As in the proof of Corollary 50, the only problem is to verify (M3) of Theorem
35 involving the function f : b
P (V ) → IR ∪ {±∞} defined by
f (W) := inf {t ∈ IR : W ⊆ {tk + b
v} ⊕ (−K)} .
Using this definition, the monotonicity property of f and the translation property
∀W ∈ b
P (V ) , ∀s ∈ IR : f (W ⊕ {sk}) = f (W) + s
can be proven straighforward. Let us show that f is bounded below on
n
W ∈ b
P (V ) : ∃x ∈ X : (x, W) ∈ M
o
.
Assume the contrary. Then there is (x, W) ∈ M such that f (W) −b
t. Hence there is
s ∈ IR, s b
t such that
W ⊆ {sk + b
v} ⊕ (−K) =

s + b
t

k − b
tk + b
v ⊕ (−K) ⊆

−b
tk + b
v ⊕ (−K)
contradicting (A3). The last part of assumption (M3) of Theorem 35 can be proven as in
the proof of Corollary 48. We may apply Theorem 35 to obtain (i) and (ii) above from its
conclusions.
Both of Corollary 49 and Corollary 50 imply Corollary 48 by setting
M = {(x, {v}) : (x, v) ∈ M} .
Note the complete symmetry of the constructions in Corollary 49 and 50, respectively.
Again, a sufficient condition for (A5) is a difficult task and requires additional topological
assumptions such as compactness. We refer to [52].

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Andreas Hamel
Oleariusstraße 1
D-06108 Halle (Saale)
Email: hamel@mathematik.uni-halle.de
Selbständigkeitserklärung
Hiermit erkäre ich, meine Habilitationsschrift mit dem Thema
Variational Principles on Metric and Uniform Spaces
selbständig und ohne fremde Hilfe verfaßt zu haben. Andere als die angegebenen Quellen
und Hilfsmittel habe ich nicht benutzt und die den benutzten Werken wörtlich oder in-
haltlich entnommenen Stellen habe ich als solche kenntlich gemacht.
Halle, 03.02.2005 Dr. Andreas Hamel

Andreas Hamel
Oleariusstraße 1
D-06108 Halle (Saale)
Email: hamel@mathematik.uni-halle.de
Lebenslauf
Name Andreas Heinrich Hamel
Adresse Oleariusstraße 1, 06108 Halle (Saale)
Geburtstag/-ort 8. September 1965, Naumburg (Saale)
Familienstand geschieden, eine Tochter
Schulbildung 1972–1980 Polytechnische Oberschule Bismark/Loburg
1980–1984 Erweiterte Oberschule Zerbst, Abitur 1984
Wehrdienst 1984–1986 als Bausoldat (entspricht Zivildienst)
Studium 1986–1991 Mathematik an der Technischen Hochschule ”Carl
Schorlemmer” Merseburg
1991 Diplom mit der Arbeit ”Gradientenberechnung für eine
Klasse von Aufgaben der Optimalen Steuerung mit verteilten
Parametern”, Prädikat ”Mit Auszeichnung”
Betreuer: Prof. Dr. H. Benker
Tätigkeiten 9/1991–8/1994 wissenschaftlicher Assistent erst an der TH
Merseburg und seit deren Auflösung im März 1993 an der
Martin-Luther-Universität Halle-Wittenberg
9/1994–12/1994 Promotionsstipendium an der Martin-
Luther-Universität Halle-Wittenberg
1/1995–3/2004 und 10/2004–2/2005 wissenschaftlicher Assis-
tent an der Martin-Luther-Universität Halle-Wittenberg, ar-
beitslos von 4–9/2004
ab 2/2005 Forschungsstipendiat am Instituto Nacional de
Matemática Pura e Aplicada (IMPA) in Rio de Janeiro,
Brasilien

Dissertation 1996 mit dem Thema ”Anwendungen des Variationsprinzips
von Ekeland in der Optimalen Steuerung”, Prädikat ”summa
cum laude”
Betreuer: Prof. Dr. H. Benker
Preise 1997 Martin-Luther-Medaille und Dorothea-Erxleben-Preis
der Martin-Luther-Universität Halle-Wittenberg
Halle, 03.02.2005

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