![Preface
This book is devoted to linear and nonlinear non-Fredholm operators and
their applications. Fredholm operators are named in honour of Erik Ivar
Fredholm and constitute one of the most important classes of linear maps
in mathematics. They were introduced around 1900 in the study of integral
operators and, by definition, they share many properties with linear maps
between finite dimensional spaces (i.e. matrices). The Fredholm property im-
plies the solvability condition: the nonhomogeneous operator equation Lu = f
is solvable if and only if the right-hand side f is orthogonal to all solutions of
the homogeneous adjoint problem L∗
v = 0. The orthogonality is understood
in the sense of duality in the corresponding spaces. Indeed, solvability con-
ditions play an important role in the analysis of nonlinear problems.These
properties of Fredholm operators are widely used in many methods of linear
and nonlinear analysis. At present linear and nonlinear Fredholm operator
theory and solvability of corresponding equations are quite well understood
(see the book of [20] and the references therein).
However, in the case of non-Fredholm operators the usual solvability con-
ditions may not be applicable and solvability relations are, in general, not
known. In spite of some progress on linear/nonlinear non-Fredholm operator
theory these questions and related topics are not systematically studied in
the mathematical literature and to the best of our knowledge are not well
understood. The aim of this book is to attempt to close such a gap and ini-
tiate as well as stimulate readers to make contributions to this fascinating
subject.
My work on this subject, that is on solvability and well-posedness of a lin-
ear/nonlinear non-Fredholm operator equation, started in 2015 during my
visit to York University (Canada) as an Alexander von Humboldt Fellow
for experienced scientists. One of the seminars there was given by Dr. V.
Vougalther (University of Toronto). Both during the talk and afterwards, I
asked him many questions to most of them I received the answer either not
known or not studied. This was a point of departure for my study of this and
vii](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-10-320.jpg)
![viii Preface
related questions. Indeed, I found this subject very fascinating and taking into
account my ”alte liebe” on Fredholm operator theory (see my book [20]), I
started with much enthusiasm to work on this topic. Later in my visit to the
Fields Institute and the University of Toronto as a Deans Distinguished Vis-
iting Professor (2018) and as a James D. Murray Distinguished Professor at
the University of Waterloo (2019) together with my colleagues of the Univer-
sities mentioned above working on ”What has Biology and Medicine done for
Mahematics”, I also continued to work with Dr.V. Vougalther on linear and
nonlinear non-Fredholm operators and its applications to the various classes
of non-Fredholm elliptic equations.
One of the main question which will be considered in this book is the solv-
ability of linear and nonlinear equations related to non-Fredholm operators.
We present the explicit form of the solvability conditions and establish the
existence of solutions of the non-Fredholm equations considered in this book.
In particular, we address it in the following setting. Let A : E → F be the
operator corresponding to the left side of equation Au = f. Assume that this
operator fails to satisfy the Fredholm property (throughout this book in each
chapter starting from the second one we will present a quite large class of
pseudo-differential elliptic equations for which this will be the case).
Let fn be a sequence of functions in the image of the operator A, such that
fn → f in F as n → ∞. Denote by un a sequence of functions from E such
that Aun = fn, n ∈ N. Since the operator A does not satisfy the Fredholm
property, the sequence un may not be convergent. Let us call a sequence un
the solution in the sense of sequences of the equation Au = f if Aun → f. If
such a sequence converges to a function u∗
in the norm of the space E, then
u∗
is a solution of this equation in the usual sense. A solution in the sense of
sequences in this case is equivalent to the usual solution. However, in the case
of non-Fredholm operators, this convergence may not hold or it can occur in
some weaker sense. In such a case, a solution in the sense of sequences may
not imply the existence of the usual solution.
In this book we find the sufficient conditions for the equivalence of solutions
in the sense of sequences and the usual solutions, that is, the conditions
on sequences fn under which the corresponding sequences un are strongly
convergent. In the case of elliptic integro-differential equations that fail to
satisfy the Fredholm property (which can also be formulated in term of the
location of the essential spectrum), we prove existence of solution in the sense
of sequences in terms of the kernel of the given integro-differential elliptic
operator.
This book consists of five chapters and in particular includes our results that
have been published in the leading journals of mathematical societies of the
world.
Chapter 1 has more of a teaching aid character and consists of eight sections
and is dedicated, in particular, to some basic concepts concerning Sobolev](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-11-320.jpg)
![1.1 Functional spaces and embedding theorems 3
The norms Ck,γ
(Tn
) and Wl,p
(Tn
) are defined by (1.4) and (1.9) with Ω =
(]0, 2π[). The scalar product and the norm in Hl
(Tn
), which are equivalent
to those defined by (1.10), are defined in terms of Fourier coefficients,
(u, v)l =
X
û(ξ) · v̂(ξ) · (1 + |ξ|2
)l
; ∥u∥2
l = ⟨u, u⟩l, (1.11)
where the summation is over ξ ∈ Zn
; the bar denotes complex conjugation;
û(ξ) and v̂(ξ) are the Fourier coefficients,
û(ξ) = (2π)−n
Z
u(x)e−ix·ξ
dx. (1.12)
Here the integral is over [0, 2π]n
;
x · ξ = x1ξ1 + · · · + xnξn. (1.13)
The formula (1.11) defines the norm in Hl
(Tn
) for l ∈ R as well as l ∈ Z+.
We denote by C∞
(Ω̄) the space
k≥0
Ck
(Ω̄); by C∞
0 (Ω) the set of functions
from C∞
(Ω̄) which vanish on a neighbourhood of the boundary ∂Ω. We shall
use also spaces of functions which vanish on ∂Ω. In this case we shall denote
the corresponding space as follows:
Ck,γ
(Ω̄) ∩ {u|∂Ω = 0}, Wl,p
(Ω) ∩ {u|∂Ω = 0}. (1.14)
We denote the completion of C∞
0 (Ω) with respect to the norm of Hl
(Ω) by
Hl
0(Ω) and with respect to the norm of W1,p
(Ω) by W1,p
0 (Ω). It is well-known
that
H1
0 (Ω) = H1(Ω) ∩ {u|∂Ω = 0}; W1,p
0 (Ω) = W1,p
∩ {u|∂Ω = 0}. (1.15)
The Sobolev spaces Hρ
(Ω) with noninteger ρ ≥ 0, ρ = k+β, k ∈ Z, 0 ≤ β 1
are endowed with the norm
∥u∥2
ρ = ∥u∥2
k +
Z
|y|≤δ
∥u(x + y) − u(x)∥2
k · |y|−n−2β
dy (1.16)
(u(x) is extended over a δ-neighbourhood of the boundary, see [50]).
By S(Rn
) we denote the class of rapidly decreasing (at ∞) functions u(x) ∈
C∞
(Rn
), with
(1 + |x|)k
|Dα
u(x)| ≤ Ck,α
for each α = (α1, · · · , αn) ∈ Zn
+ and k ∈ Z+, where Ck,α are constants.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-19-320.jpg)
![4 1 Auxiliary Materials
Recall that an operator j : X → Y between Banach spaces with X ⊆ Y is
an embedding iff j(x) = x for all x ∈ X. The operator j is continuous iff
∥x∥Y ≤ constant ∥x∥X for all x ∈ X.
Further, j is compact iff j is continuous, and every bounded set in X is
relatively compact in Y . If the embedding X ,→ Y is compact, then each
bounded sequence {xn} in X has a subsequence {xn′ } which is convergent in
Y .
We shall widely use Sobolev’s embedding theorems formulated below.
Theorem 1.1 Let Ω be a bounded domain in Rn
, with smooth boundary ∂Ω
and 0 ≤ k ≤ m − 1 (See [50]). Then
Wm,p
(Ω) ,→ Wk,q
(Ω), if
1
q
≥
1
p
−
m − k
n
0, (1.17)
Wm,p
(Ω) ,→ Wk,q
(Ω), if q ∞,
1
p
=
m − k
n
, (1.18)
Wm,p
(Ω) ,→ Ck,δ
(Ω̄), if
n
p
m − (k + δ), 0 δ 1. (1.19)
The first embedding is compact if 1
q 1
p − m−k
n . The last two embeddings are
compact.
Theorem 1.2 Let 0 ≤ β α ≤ 1 or α, β ∈ Z with 0 ≤ β α (see [63]).
Then the embedding
Cα
(Ω̄) ,→ Cβ
(Ω̄) is compact (1.20)
and for k + β m + α, with 0 ≤ α, β ≤ 1, m ≥ k ≥ 0 the embeddings
Cm,α
(Ω̄) ,→ Ck,β
(Ω̄) are compact. (1.21)
1.2 Linear Elliptic Boundary value problems
Notation. Let Ω be a bounded domain in Rn
.
For α = (α1, · · · , αn) an n-tuple of nonnegative integers, recall that Dα
=
n
Y
i=1
∂
∂xi
αi
, |α| =
n
X
i=1
αi and let ξα
=
n
Y
i=1
(ξi)αi
if ξ ∈ Cn
.
Every linear differential operator L of order 2m (m ∈ N) has the form
Lu =
X
|α|≤2m
aα(x) · Dα
u. (1.22)](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-20-320.jpg)
![6 1 Auxiliary Materials
D(x, ξ, N) = det ∥bik(x, ξ, ⃗
N)∥ ̸= 0 (1.26)
for all x ∈ ∂Ω, and for all ⃗
N ∈ Rn
{0} normal to ∂Ω at x and ξ ∈ Rn
{0}
with ⟨ξ, ⃗
N⟩ = 0.
Definition 1.1 We say that (L, B1, · · · , Bm) defines an elliptic boundary
value problem of order (2m, m1, · · · , mm) if L given by (1.22), is uniformly
elliptic and of order 2m, each Bi given by (1.24) has order mi, 0 ≤ mi ≤
2m − 1, the mi’s are distinct, ∂Ω is non characteristic to Bi at each point
and {Bi}m
i=1 satisfy the Shapiro-Lopatinski condition with respect to L (see
[47]).
We have the following lemma (see [4, 33]).
Lemma 1.1 Let (L, B1, · · · , Bm) define an elliptic boundary value problem
of order (2m, m1, · · · , mm). Then
L ◦ △l
, B1 ◦ ˜
△l
, · · · , Bm ◦ ˜
△l
, L ◦
∂u
∂ ⃗
N
defines an elliptic boundary value problem of order (2k+2l, m1+2l, · · · , mm+
2l, 2m + 1) where ˜
△ is the Laplace-Beltrami operator, l ∈ N.
Proof. The principal symbol of L ◦ △l
is |ξ|2l
· p0(x, ξ), so it is clear that
L ◦ △l
is uniformly elliptic.
Let x ∈ ∂Ω and ξ, ⃗
N ∈ Rn
{0}, with ⟨ξ, ⃗
N⟩ = 0 and ⃗
N normal to ∂Ω at x.
It is obvious that the principal symbol operators Bi ◦ ˜
△l
and L◦ ∂
∂ ⃗
N
at ξ+τ ⃗
N
are ψl(ξ) · p0i(x, ξ + τ ⃗
N) and τp0(x, ξ + τ ⃗
N) respectively, where ψl(ξ) ̸= 0.
If τ+
1 , · · · , τ+
m are the m roots of p0(x, ξ + τ ⃗
N) = 0 having positive imaginary
part, then the m + 1 roots of |ξ + τ ⃗
N|2
· p0(x, ξ + τ ⃗
N) = 0 with positive
imaginary part are given by τ+
1 , · · · , τ+
m, i · | ⃗
N|
|ξ| .
We must show that if λ1, · · · , λm+1 ∈ C and h(τ) is a polynomial with
ψl(ξ)
m
X
i=1
λi · p0i(x, ξ + τ ⃗
N) + λm+1τp0(x, ξ + τ ⃗
N)
= h(τ) · τ −
i| ⃗
N|
|ξ|
!
·
m
Y
i=1
(τ − τ+
i ) (1.27)
then λi = 0, 1 ≤ i ≤ m + 1 and h(τ) ≡ 0. Due to the assumption that
(B1, · · · , Bm) satisfy the covering condition it is not difficult to see that
λ1 = · · · = λm = 0.
But then the right-hand side of (1.27) has more roots with positive imaginary
part than does the left-hand side, so that λm+1 = 0 and h(τ) ≡ 0. ⊓
⊔](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-22-320.jpg)
![8 1 Auxiliary Materials
∥u∥′
m−δ,p,∂Ω =
X
i∈I′
X
|α|≤m−1
′
Z
Ṽ1
|Dα
y (λi(y) · ui(y))|p
dy′
+
X
|α| =m−1
′
Z
Ṽ1
Z
Ṽ1
|Dα
y (λi(y) · ui(y)) − Dα
z (λi(z) · ui(z)) |p
·
dy′
dz′
|y′ − z′|n+p−1−δp
1
p
, (1.28)
where ui(y) = u(φ−1
i (y)), y′
= (y1, · · · , yn−1), I′
⊂ {1, · · · , l} such that Ui ∩
∂Ω ̸= ∅ and
P′
implies that the sum is taken over those α for which αn =
0, α = (α1, · · · , αn).
By definition, the norm in Wm− 1
p ,p
(∂Ω), p 1 is defined as the norm
∥ · ∥′
m− 1
p ,p,∂Ω
. For more details see [50].
Let us return to the discussion of elliptic boundary value problems . We first
recall some results regarding linear Fredholm operators. Let X and Y be real
Banach spaces. By L(X, Y ) we denote the Banach space of bounded linear
operators from X to Y . An operator T in L(X, Y ) is called Fredholm if the
kernel (nullspace) Ker T := {x ∈ X | Tx = 0} has finite dimension and the
image (range) of T, R(T) := {Tx | x ∈ X} is of finite codimension in Y , that
is codim R(T) = dim Y/R(T) ∞. For a Fredholm operator T : X → Y , the
numerical Fredholm index of T, ind(T) is defined by
ind(T) = dim Ker T − codim(R(T)).
Lemma 1.2 Let Ω ⊂ Rn
be open and bounded with ∂Ω smooth. Suppose
that s n/2, aα ∈ Hs
(Ω) if |α| ≤ 2m, while bα,i ∈ Hs+2m−mi
(∂Ω) and
i = 1, · · · m. Then the following three assertions are equivalent:
(i) The operator A = (L, B1, · · · , Bm)
A : Hs+2m
(Ω) −→ Hs
(Ω) ×
m
Y
i=1
Hs+2m−mi
(∂Ω) (1.29)
is an elliptic boundary value problem of order (2m, m1, · · · , mm)
(ii) The operator A = (L, B1, · · · Bm) is Fredholm
(iii) There is some c 0, such that if u ∈ Hs+2m
(Ω), then
∥u∥2m+s ≤ c
∥Lu∥s +
m
X
i=1
∥Bi(x, D)u∥2m+s−mi− 1
2
+ ∥u∥s
#
. (1.30)
Proof. If each aα ∈ Cs
(Ω) and each bα,i ∈ C2k+s−mi
(Ω), then a priori
estimate (1.30) is contained in [1]. It is not difficult to see that (1.30) also](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-24-320.jpg)
![1.3 Superposition operators 9
holds under the present smoothness conditions. Thus, in fact a priori estimate
(1.30) and equivalence (i) and (iii) follows from [1]. Equivalence (i) and (iii)
to (ii) can be proved analogously to [2]. ⊓
⊔
Remark 1.1 Of course, the Fredholm index of (L, B1, · · · , Bm) need not
be equal to 0. If L is uniformly elliptic and Biu(x) =
∂
∂ ⃗
N
i−1
u(x) for
1 ≤ i ≤ m, then the index
A = (L, B1, · · · , Bm) : H2m+s
(Ω) → Hs
(Ω) ×
m
Y
i=1
H2m+s−mi− 1
2 (∂Ω)
is 0 (see [4, 46]).
Remark 1.2 (Cγ
-theory) The a priori estimates(1.30) remain valid if we
choose the following B- spaces for 0 γ 1:
X = C2m+s,γ
(Ω̄), Y = Cs,γ
(Ω̄), Z = C(Ω̄), Yj = C2m+s−mi,γ
i.e.
∥u∥X ≤ constant(∥Lu∥Y +
m
X
j=1
∥Bju∥Yj + ∥u∥Z). (1.31)
Remark 1.3 The important fact is that the index of corresponding opera-
tors is the same in both theories.
Remark 1.4 As shown in [1, 2] the terms ∥u∥s and ∥u∥Z in (1.30), (1.31)
disappear if dim Ker A = {0}, where Au = (Lu, B1u, · · · , Bmu).
1.3 Superposition operators
The investigation of nonlinear equations in the following chapters relies on
properties of mappings of the form u 7→ f(u) in the spaces Cα
(Ω̄) and
Lp
(Ω), Hl
(Ω).
Definition 1.2 Let Ω ⊂ Rn
be a domain. We say that a function
Ω × Rm
∋ (x, u) 7−→ f(x, u) ∈ R
satisfies the Carathéodory conditions if
u 7−→ f(x, u) is continuous for almost every x ∈ Ω
and](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-25-320.jpg)
![10 1 Auxiliary Materials
x 7−→ f(x, u) is measurable for every u ∈ Ω.
Given any f satisfying the Carathéodory conditions and a function u : Ω →
Rm
, we can define another function by composition
Fu(x) := f(x, u(x)). (1.32)
The composed operator F is called a Nemytskii operator. In this section we
state some important results on the composition of Cα
(Ω̄), Lp
(Ω), Hl
(Ω)
with nonlinear functions (some of them without proof [42, 63]).
Proposition 1.1 Let Ω ⊂ Rn
be a bounded domain and
Ω × Rm
∋ (x, u) 7−→ f(x, u) ∈ R
satisfy the Carathéodory conditions. In addition, let
|f(x, u)| ≤ f0(x) + c(1 + |u|)r
(1.33)
where f0 ∈ Lp0
(Ω), p0 ≥ 1, and rp0 ≤ p1. Then the Nemytskii operator F
defined by (1.32) is bounded from Lp1
(Ω) into Lp0
(Ω), and satisfies
∥F(u)∥0,p0 ≤ C1 · (1 + ∥u∥r
p1
) (1.34)
Proof. By (1.33) and (1.7)
∥F(u)∥0,p0
≤ ∥f0(x)∥0,p0
+ C∥1∥0,p0
+ C∥|u|r
∥0,p0
≤ C′
+ C
Z
Ω
|u|rp0
dx
1
p0
= C′
+ ∥u∥r
0,p0r. (1.35)
Since Ω is bounded, then by Hölder’s inequality
∥v∥0,q ≤ C(Ω)∥v∥0,p when 1 ≤ q ≤ p, v ∈ Lp
(Ω) (1.36)
where C(Ω) = mes(Ω)
1
q − 1
p .
Inequalities (1.35) and (1.36) with q = rp0 and p = p1 imply (1.34). ⊓
⊔
It is well-known that the notions of continuity and boundedness of a nonlinear
operator are independent of one another ([42]). It turns out that the following
is valid.
Theorem 1.3 Let Ω ⊂ Rn
be a bounded domain and let
Ω × Rm
∋ (x, u) 7−→ f(x, u) ∈ R
satisfy the Carathéodory conditions. In addition, let p ∈ (1, ∞) and g ∈
Lq
(Ω) (where 1
p + 1
q = 1) be given, and let f satisfy](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-26-320.jpg)
![1.3 Superposition operators 11
|f(x, u)| ≤ C|u|p−1
+ g(x).
Then the Nemytskii operator F defined by (1.32) is a bounded and continuous
map from Lp
(Ω) to Lq
(Ω).
For a more detailed treatment, the reader can consult [42, 63].
Theorem 1.4 Let Ω be a bounded domain in Rn
with smooth boundary and
let
Ω × R ∋ (x, u) 7→ f(x, u) ∈ R
satisfy the Carathéodory conditions. Then for s n/2, f induces
1) a continuous mapping from Hs
(Ω) into Hs
(Ω) if f ∈ Cs
,
2) a continuously differentiable mapping from Hs
(Ω) into Hs
(Ω) if f ∈ Cs+1
.
Proof. First we consider the simplest case, that is f = f(u) is independent
of x.
By the Sobolev embedding theorem, we have Hs
(Ω) ⊂ C(Ω̄). Hence we have
f(u) ∈ C(Ω̄) for everyu ∈ Hs
(Ω). Moreover, if u is in C(s)
(Ω̄), we can obtain
the derivatives of f(u) by the chain rule, and in the general case, we can use
approximation by smooth functions. Note that all derivatives of f(u) have
the form of a product involving a derivative of f and derivatives of u. The
first factor is in C(Ω̄), while any l-th derivative of u lies in Hs−l
(Ω), which
imbeds into L2n/(n−2(s−l))
(Ω) if s − l n
2 .
We can use this fact and Hölder’s inequality to show that all derivatives of
f(u) up to order s are in L2
(Ω); moreover, it is clear from this argument that
f is actually continuous from Hs
(Ω) into Hs
(Ω). A proof of the differentia-
bility in this special case is that f = f(u) is based on the relation
f(u) − f(v) =
Z 1
0
f′
u(v + θ(u − v))(u − v)dθ
and the same arguments as before.
Let us now consider the general case, that is f = f(x, u). Let |α| ≤ s. We
must show that
u 7−→ Dα
F(u) (1.37)
defines a continuous map of Hs
(Ω) into L2
(Ω).
It is not difficult to see that (1.37) is a finite sum of operators of the form
u(x) 7−→ g(x, u(x)) · Dγ
u(x) (1.38)
where |γ| = γ1 +· · ·+γn ≤ s, while g is a partial derivative of f order at most
s. It is obvious that Dγ
is continuous from Hs
(Ω) into L2
(Ω) for |γ| ≤ s.
On the other hand, the continuous embedding of Hs
(Ω) inC(Ω̄) implies that](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-27-320.jpg)
![1.3 Superposition operators 15
∥Au∥C2p,α ≤ C(R) · ∥u∥Cα .
Moreover as it follows from Theorem 1.6 A ∈ C1
, A′
(0) = 0 and
∥A′
(u + h) − A′
(u)∥L(C2p,α,Cα) ≤ C · ∥h∥C2p,α(Ω̄).
Remark 1.5 As shown in the proofs of Theorems 1.5 and 1.6, continuity and
differentiability of the operator Au(x) = f(x, u(x), · · · , D2p
u(x)) between
C2p,α
(Ω̄) and Cα
(Ω̄) remains valid under slightly weaker conditions on a
given function f(x, y1, · · · , yp̃). We leave these as exercises for the reader.
In the investigation of nonlinear boundary value problems related to pseudod-
ifferential operators and in particular nonlinear Riemann-Hilbert problems we
need properties of the Nemytskii operators in the spaces Hs
(S1
) or Cp,α
(S1
),
where S1
is the unit circle. We recall some of the properties which will be
used often in the sequel.
The norm in Cα
(M) is given by
∥f∥Cα(M) = ∥f∥C + sup
x̸=y
|f(x) − f(y)|
|x − y|α
, M = S1
.
As before, by Ck,α
(M) we denote the space of Hölder continuous functions,
which have derivatives up to order k, with Dk
f ∈ Cα
(M). Let F be a super-
position operator defined by
Fu(x) = f(x, u(x)), x ∈ M.
The following theorems are not hard to prove (although not obvious).
Theorem 1.7 Let k ∈ R+. Then the superposition operator F : E1 → E2
defined by Fu(x) = f(x, u(x)) acts as a bounded operator in each of the
following cases (see also [62])
1) f ∈ C(S1
× R, R), E1 = C(S1
), E2 = C(S1
)
2) f ∈ C1
(S1
× R, R), E1 = Cα
(S1
), E2 = Cα
(S1
), 0 α 1
Theorem 1.8 Let k ∈ R+, 0 α 1. Then the superposition operator F :
E1 → E2 defined by Fu(x) = f(x, u(x)) is m times continuously differentiable
in each of the following cases
1) D0,j
f ∈ Ck
(S1
× R, R), E1 = Ck
(S1
), E2 = Ck
(S1
)
2) D0,j
f ∈ Ck+1
(S1
× R, R), E1 = Ck,α
(S1
), E2 = Ck,α
(S1
),
The j-th derivative of F is given by
D0,j
F(x, u(x))h1(x) . . . hj(x) = Dj
F(f)(h1, . . . hj)(x).](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-31-320.jpg)
![1.4 Pseudodifferential operators. Definitions and examples 17
∥f−1
(v)∥pq
W s/p,qp(Ω)
:=∥f−1
(v)∥pq
Lpq(Ω)
+
Z
Ω
Z
Ω
|f−1(v(x)) − f−1(v(y)|pq
|x − y|n+sq
dx dy
≤C∥v∥q
Lq(Ω)
+ C′
p
Z
Ω
Z
Ω
|v(x) − v(y)|q
|x − y|n+sq
dx dy = C′′
p ∥v∥q
W s,q(Ω)
,
where we have implicitly used that f−1
(v) ∼ sgn(v)|v|1/p
. Lemma 1.3 is
proved. ⊓
⊔
1.4 Pseudodifferential operators. Definitions and
examples
Let a(x, ξ) be a C∞
function for all x ∈ Rn
x and ξ ∈ Rn
ξ {0} which is positively
homogeneous of degree σ ≥ 0 in ξ:
a(x, tξ) = tσ
a(x, ξ), t 0. (1.45)
Assume that on the sphere |ξ| = 1, a(x, ξ) has a limit a(∞, ξ) as x → ∞ and
that
a′
(x, ξ) = a(x, ξ) − a(∞, ξ) ∈ S(Rn
x)
uniformly in ξ. In fact we assume that the same is true for all derivatives
with respect to ξ, i.e., for any integers p, α, β,
(1 + |x|)p
· ∂α
x Dβ
ξ (a(x, ξ) − a(∞, ξ)) → 0 as |x| → ∞
uniformly in ξ, for |ξ| = 1 where ∂x = 1
i
∂
∂x , Dξ = ∂
∂ξ . The function a(x, ξ) is
called the symbol. In sections 1.4 we will mainly follow [41] specifying some
details.
Definition 1.3 A (homogeneous) pseudodifferential operator A(x, D) with
the symbol a(x, ξ) is defined on functions u(x) ∈ S(Rn
x) in the following way:
A(x, D)u(x) = (2π)−n/2
Z
Rn
ξ
eix·ξ
· a(x, ξ)û(ξ)dξ. (1.46)
Recall that S(Rn
x) is the space of all functions on Rn
x which are of class C∞
and such that |x|k
|Dα
u(x)| is bounded for every k ∈ N and every multi-index
α.
The formula (1.46) can be rewritten as follows:](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-33-320.jpg)
![1.4 Pseudodifferential operators. Definitions and examples 19
a0(x) =
Z
|ξ′|=1
a(x, ξ′
)dξ′
.
Then one can show that
A(x, D)u(x) = a0(x)u(x) + lim
ϵ→0
Z
|x−y|ϵ
K(x, x − y)
|x − y|n
u(y) dy (1.49)
where the function K(x, z) is positive homogeneous of degree zero in z, such
that
(2π)−n/2 K(x, z)
|z|n
= F−1
ξ→x [a(x, ξ) − a0(x)].
Remark 1.7 By Hl
(Rn
x), −∞ l ∞ we denote the completion of S(Rn
x)
in the norm (1.50)
∥u∥2
l =
Z
(1 + |ξ|2
)l
· |û(ξ)|2
dξ. (1.50)
Order. A linear operator L : S(Rn
x) → S(Rn
x) is said to have order σ, if for
each real s there exists a constant Cs such that
∥Lu∥s−σ ≤ Cs∥u∥s. (1.51)
The infimum of the set of σ′
of L is called the true order of L. In the sequel
we will abbreviate pseudodifferential operators by ψDO.
Boundedness of ψDO with homogeneous symbol of degree σ.
Lemma 1.4 Let A(x, D) be a homogeneous pseudodifferential operator with
homogeneous symbol of degree σ. Then A(x, D) has order σ, i.e.
∥Au∥l−σ ≤ const. · ∥u∥l. (1.52)
Proof. It is not difficult to estimate the norm of the operator
A1u = F−1
a(∞, ξ)Fu (1.53)
Since
|a(∞, ξ)| ≤ C · |ξ|σ
,
then](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-35-320.jpg)
![24 1 Auxiliary Materials
ĥρ(ξ − η, η, τ) =
X
|β|=ρ
1
β!
∂β
η ĥ(ξ − η, τ + θ(η − τ)) · (η − τ)β
,
where 0 θ 1. Hence for σ ≥ ρ (ρ sufficiently large) we have
|ĥρ(ξ − η, η, τ)| ≤ const.
|η − τ|ρ
(1 + |ξ − η|2)ρ
(1 + |τ|2
)
σ−ρ
2 + (1 + |η|2
)
σ−ρ
2
.
(1.70)
An analogous estimate holds for σ ρ. This is not difficult and is left to the
reader. For the function ĝ we have
|ĝ(η − τ, τ)| ≤ const.
(1 + |τ|2
)
λ
2
(1 + |η − τ|2)ρ
, (1.71)
for sufficiently large ρ. From (1.70) and (1.71) it follows that
(1 + |ξ|2
)
l−(σ+λ−ρ)
2
Z
|ĥρ(ξ − η, η, τ)| · |ĝ(η − τ, τ)|dη ≤ C
(1 + |τ|2
)l/2
(1 + |ξ − τ|2)
n+1
2
.
(1.72)
Inequality (1.69) follows from (1.72). This proves Theorem 1.10. ⊓
⊔
Corollary 1.3 Let A(x, D), B(x, D) be pseudodifferential operators of order
σ and λ respectively. Let C0(x, D) be a pseudodifferential operator of order
σ + λ with symbol a(x, ξ) · b(x, ξ). Then
A(x, D) ◦ B(x, D) − C0(x, D)
has order σ + λ − 1.
Corollary 1.4 Let A(x, D), B(x, D) be pseudodifferential operators of order
σ and λ respectively. Then [A, B] = AB − BA has order σ + λ − 1.
In particular, if A(x, D) has order σ, then ∂jA − A∂j has order σ, where
∂j = 1
i · ∂
∂xj
; moreover, if φ(x), ψ(x) ∈ C∞
0 (Rn
), such that φ(x) · ψ(x) = 0,
then φAψ has order σ − 1.
Corollary 1.5 Let A(x, D) be a pseudodifferential operator of order σ and
let Ωi, i = 1, 2 be bounded domains in Rn
with Ω1 ∩ Ω2 = ∅. Suppose
u(x) ∈ Hl
(Rn
) for some l, with supp u ∈ Ω2. Then outside the support
of u, A(x, D)u ∈ C∞
.
Proof. Let φ1(x), φ2(x) ∈ C∞
0 (Rn
) be such that φi(x) = 1 in Ωi, i = 1, 2
and supp φ1 ∩ supp φ2 = ∅. For each x ∈ Ω1 we have
Au(x) = φ1(x)A [φ2(x)u(x)] .](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-40-320.jpg)
![26 1 Auxiliary Materials
and Tρ has order σ − ρ.
We omit the details of the proof which is based on the Taylor expansions
of the function ã′∗
(ξ − η, ξ) with respect to ξ, when ξ belongs to a small
neighborhood of η.
Remark 1.10 The lemma is obvious if A(x, D)u =
X
|α|=σ
aα(x)Dα
u (here we
take ζ(ξ) ≡ 1). Indeed, then
A∗
(x, D)u(x) =
X
|α|=σ
Dα
[a∗
α(x)u(x)]
where a∗
α is conjugate to aα(x).
Hence
A∗
(x, D)u(x) −
X
|α|=σ
a∗
α(x)Dα
u
has order σ − 1.
In section 1.4 ψDO was defined in a canonical way by formulas (1.46) and
(1.58). Theorem 1.10 and Lemma 1.5 showed us, that the operations A, B 7→
A ◦ B and A 7→ A∗
do not lie in the class of ψDO which are canonically
defined by (1.46) or (1.58). In other words, it is not an algebra. In order
to avoid this drawback, we have to add a number of remainder terms to
the definition of ψDO. Of course, we have a degree of freedom to choose
these additional terms. A reasonable way to proceed is to add terms of ψDO
operators decreasing orders. More precisely:
Definition 1.4 Let r0, r1, r2, . . . be a sequence (finite or infinite), such that
r0 r1 . . . rn, . . ., and rk → −∞ as k → ∞ (if the sequence is infinite).
Let Ak(x, D) be a ψDO of order rk defined in a canonical way. Let A be an
operator defined on S(Rn
x), such that
A −
N
X
k=0
Ak(x, D)
for all N has order less than rN . Then by definition A is
pseudodifferential operator with asymptotic expansion
A ∼
∞
X
k=0
Ak(x, D). (1.75)
By definition the symbol of A, i.e. a(x, ξ), is the formal series](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-42-320.jpg)
![1.4 Pseudodifferential operators. Definitions and examples 27
∞
X
k=0
ak(x, ξ). (1.76)
If the sequence {rk} is finite, then ψDO is defined by A =
ρ
X
k=0
Ak(x, D) + T∞
where T∞ has true order equal to −∞.
From the definition it follows that, in both cases ψDO has order r0. Fur-
thermore, ψDO with symbol a(x, ξ) ≡ 0 has true order equal to −∞, since
ak(x, ξ) ≡ 0, for k = 1, 2, · · · implies Ak(x, D) ≡ 0, k = 1, 2 . . ., and vice
versa; if a pseudodifferential operator has order equal to −∞, then its sym-
bol is equal to zero (see [41]). Thus a ψDO is defined by its symbol modulo
operators which have orders equal to −∞. Let Ω be an open set of Rn
. The
following theorem is given in [39].
Theorem 1.11 Let aj(x, ξ) be C∞
functions in Ω × (Rn
ξ {0}) which are
positively homogeneous of degree rj ↘ −∞. Then there exists a ψDO with
symbol
∞
X
k=0
ak(x, ξ).
Proof. Let φ(ξ) be a C∞
function in Rn
which equals 0 when |ξ| ≤ 1
2 and
equals 1 when |ξ| ≥ 1. We can choose a sequence tj ↗ +∞ increasing so
rapidly that
Dβ
ξ φ
ξ
tj
aj(x, ξ) + (1 + |x|k
) ∂α
x Dβ
ξ φ
ξ
tj
aj(x, ξ)
≤ 2−j
· (1 + |ξ|)rj−1−|β|
(1.77)
for all |α| + |β| + k ≤ j and |α| 0. In fact, Dβ
ξ [φ
ξ
tj
aj(x, ξ)] can be
computed by Leibniz’s formula. It is obvious that a term in which φ is not
differentiated is homogeneous of order rj −|β| in ξ when |ξ| tj and is equal
to 0 when |ξ| ≤
tj
2 . In other terms we have t−q
j , when 0 q ≤ |β| and these
terms are equal to 0 both when |ξ| ≤
tj
2 and |ξ| tj. Hence, the left hand
side of (1.77) can be estimated by
const. ·
1
t
rj−1−rj
j
· |ξ|rj−1−|β|
and it is obvious that we can choose a sequence tj, such that
Dβ
ξ φ
ξ
tj
aj(x, ξ) +(1+|x|k
) ∂α
x Dβ
ξ φ
ξ
tj
aj(x, ξ) ≤ 2−j
·(1+|ξ|)rj−1−|β|
.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-43-320.jpg)
![1.4 Pseudodifferential operators. Definitions and examples 29
AB =
N
X
j=0
Aj + TN,1
N
X
k=0
Bk + TN,2
!
=
N
X
j,k=0
AjBk + TN,3
where the order of TN,3 is less than q = max(σ0 +λN , λ0 +σN ). The assertion
of Theorem 1.12 follows from Theorem 1.10 for Aj ·Bk. Analogously, one can
show that if A ∈ L, then A∗
∈ L.
Elliptic ψDO. Let A(x, D) be a ψDO with symbol
a(x, ξ) ∼
∞
X
j=0
aj(x, ξ).
By definition, A is elliptic, if a0(x, ξ) ̸= 0 when ξ ̸= 0. We denote by Id the
identity operator.
Theorem 1.13 Let A(x, D) be an elliptic ψDOs. Then there exists
B(x, D) ∈ L, such that each operator
AB − Id and BA − Id
has order equal to −∞.
We omit the proof (see [2, 41]). In the end of this section, let us discuss some
particular class of pseudo-differential operator, so-called fractional Laplacian
which will play an important role in the following chapters. It is worth to note
that, many models arising in Biology, Medicine, Ecology and Finance lead
to nonlinear problems driven by fractional Laplace-type operators. There are
several ways of defining this operator in the whole space Rn
, at least to my
knowledge ten in the literature, most of them are equivalent (see [44]). I will
give here that definitions which are more relevant for the content of this book.
Let s ∈ (0, 1). For the convenience of the reader (see also section 1.1), we
define the fractional Sobolev space Hs
(Rn
) := W2,s
(Rn
);
Hs
(Rn
) :=
u ∈ L2
(Rn
)
Z
Rn
Z
Rn
|u(x) − u(y)|2
|x − y|n+2s
dxdy +∞
.
For u, v ∈ Hs
(Rn
), we set
⟨u, v⟩Hs(Rn) :=
Z
Rn
u(x)v(x) dx +
Z
Rn
Z
Rn
(u(x) − u(y))(v(x) − v(y))
|x − y|n+2s
dxdy.
One can easily see that ⟨·, ·⟩Hs(Rn) is an inner product on Hs
(Rn
) and that
the space Hs
(Rn
) endowed with this inner product is a Hilbert space . It is
well-known that the space Hs
(Rn
), s ∈ (0, 1) can be defined alternatively via
the Fourier transform, i.e.,](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-45-320.jpg)
![30 1 Auxiliary Materials
Hs
(Rn
) :=
u ∈ L2
(Rn
)
Z
Rn
(1 + |ξ|2s
)|Fx→ξu(ξ)|2
| dξ +∞
,
where by Fx→ξu(ξ) is defined the Fourier transform of u(x), x ∈ Rn
(some-
times we used instead of û(ξ) by the notation Fx→ξu(ξ).
The first definition of the fractional Laplacian (−∆x)s
, s ∈ (0, 1) is expressed
via the Fourier transform as a pseudo-differential (nonlocal!) operator with
the symbol is equal to |ξ|2s
, where |ξ| = (ξ1 + · · · + ξn)
1
2 . More precisely, let
u ∈ S(Rn
) (for the definition of this space see previous section). Then we have
Fx→ξu ∈ S(Rn
), but the function ξ 7→ |ξ|2s
·Fx→ξu(ξ) ∈ S(Rn
), because |ξ|2s
creates singularity at ξ = 0. What is more important that, |ξ|2x
Fx→ξu(ξ) ∈
L1
(Rn
) ∩L2
(Rn
), so we can use the inverse Fourier transform. Consequently,
for any u ∈ S(Rn
) we define the operator (−∆x)s
: S(Rn
) → L2
(Rn
) by
(−∆x)s
u(x) := F−1
ξ→x(|ξ|2x
Fx→ξu(ξ))(x), ∀x ∈ Rn
.
This pseudo-differential operator is called the fractional Laplacian of order s.
It is easy to see that
(−∆x)0
u = u,
(−∆x)1
u = −∆xu,
for any 0 s1, s2 1
(−∆x)s1
◦ (−∆x)s2
= (−∆x)s2
◦ (−∆x)s1
for any multi-index β ∈ Rn
Dα
x (−∆x)s
u = (−∆x)s
Dα
x u,
so that it follows that (−∆x)s
u ∈ C∞
(Rn
) for any u ∈ S(Rn
).
Remark 1.11 Actually, we can extend this and other definitions given below
that is, (−∆x)s
u to Hs
(Rn
) using the fact that the space S(Rn
) is dense in
Hs
(Rn
) (see, e.g., [15]).
The following definition of the fractional Laplacian is often used on the lit-
erature, i.e., let u ∈ S(Rn
).
(−∆x)s
u(x) := C(n, s) lim
ε→0
Z
RnB(x,ε)
(u(x) − u(y))2
|x − y|n+2s
dy, ∀x ∈ Rn
,
where the normalized constant C(n, s) given by
C(n, s) :=
s · ns
· Γ(s + n
2 )
π
n
2 Γ(1 − s)
, s ∈ (0, 1)](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-46-320.jpg)
![1.5 Linear Fredholm operators 31
and Γ(·) is the Euler function.
Remark 1.12 It is not difficult to check that, for u ∈ S(Rn
) and x ∈ Rn
and s ∈ (0, 1
2 ) the integral
Z
u(x) − u(y)
|x − y|n+2s
dy
is absolutely convergent, so that for s ∈ (0, 1
2 )
(−∆x)s
u(x) := C(n, s)
Z
Rn
u(x) − u(y)
|x − y|n+2s
dy.
The following proposition shows the relation between the ”classical” Lapla-
cian and the fractional Laplacian (see [44]).
Proposition 1.2 Let u ∈ S(Rn
). Then
1) lim
s→0+
(−∆x)s
u = u
2) lim
s→1−
(−∆x)s
u = −∆xu
1.5 Linear Fredholm operators
Fredholm operators are one of the most important classes of linear maps
in mathematics which were introduced around 1900 in the study of integral
operators. As we will see below they share many properties with linear maps
between finite dimensional spaces.
Let X and Y be Banach spaces, either over R or C. By L(X, Y ) we denote
the space of bounded linear operators from X to Y , and consider L(X, Y ) as
a Banach space with the usual norm. As we already mentioned in Section 1.2,
an operator T is called Fredholm if the kernel (nullspace) Ker T = {x ∈ X |
Tx = 0} has finite dimension and the image (range) of T, R(T) := {Tx | x ∈
X} is of finite codimension in Y , that is, codim R(T) := dim(Y/R(T)) ∞.
For a Fredholm operator T : X → Y , the numerical Fredholm index of T,
ind T is defined by
ind T := dim Ker T − codim R(T).
We denote the set of Fredholm operator as well as Fredholm operator of index
m by Φ(X, Y ) and Φm(X, Y ) respectively. In the study of global solvability
of linear Fredholm operator equation the subset Φ0(X, Y ) of L(X, Y ) play an
important role. It is worth to note that for T ∈ Φ0(X, Y ) it follow that](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-47-320.jpg)
![32 1 Auxiliary Materials
dim Ker T = dim(Y/R(T)).
Hence for T ∈ Φ0(X, Y ) from global solvability Tx = y for any y ∈ Y it
follows uniqueness of soluation of Tx = y and vice-versa. Obviously, each
invertible linear operator T : X → Y belong to Φ0(X, Y ). The following
lemma holds:
Lemma 1.6 Let T ∈ L(X, Y ). Then the following assertions are equivalent:
1. The operator T is Fredholm of index 0.
2. There is a compact operator K ∈ L(X, Y ) such that L + K is invertible.
3. There is an isomorphism S ∈ L(Y, X) such that S ◦ L is a compact vector
field, that is, it is compact perturbation of identity.
Proof. Let T ∈ Φ0(X, Y ). Since Ker T is finite dimensional, there is a con-
tinuous projection P of X onto Ker T. Let W be a closed subspace of Y ,
W := Y/TX. By definition of T ∈ Φ0(X, Y ), both Ker T and W are of the
same finite dimension, so we may select S ∈ GL(Ker T, W). We set K := S◦P
and observe that K is compact and T +K is an isomorphism. Thus 1. implies
2. To verify that 2. implies 3., choose K : X → Y linear compact operator
so that T + K is invertible. We set S := (T + K)−1
. Then it is easy to see
that S ◦ T = I − (T + K)−1
◦ K. Hence 2. implies 3. Finally to show that 3.
implies 1. we have to remind Riesz-Schauder Theorem that states that com-
pact linear vector field is Fredholm of index 0. Indeed, let S ◦ T = Id + K.
Then T = S−1
(Id + K) and ind T = ind S−1
+ ind(Id + K) = 0. ⊓
⊔
Remark 1.13 An operator S : Y → X satisfy 3. is called a parametrix for
T.
The subset Φ0(X, Y ) possesses interesting biological properties. For simplicity
of presentation as well as for notational convenience we consider the case
when X = Y . We denote by GL(X) the group of linear operator A : X → X.
According to Kuiper’s [43], the group GL(X) on a separable Hilbert (infinite
dimensional real) space is connected and even is contractible. The analogous
result is also true for many Banach spaces [20]. It is not difficult to see that
Φ0(X) inherits connectedness from GL(X). Indeed, if T ∈ Φ0(X) then
t 7−→ tT + (1 − t)S−1
, 0 ≤ t ≤ 1,
where S is a parametrix for T, defines a path in Φ0(X) connecting T to
GL(X).
Remark 1.14 If X = Rn
, then Φ0(Rn
) = L(Rn
) is contractible and GL(Rn
)
has two connected components, which are labeled by the function sgndet. In
contrast to a finite dimensional case Φ0(X) is connected when X is infinite](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-48-320.jpg)
![1.5 Linear Fredholm operators 33
dimensional space but not contractible (see [33]). Moreover, the set Φ0(X)
is open in L(X). Below we recap following well-known properties of Φ0(X),
some of them without proof (see [20],[33] and references therein).
Proposition 1.3 Let A, B ∈ Φ0(X). Then
1) AB ∈ Φ0(X) and
ind AB = ind A + ind B.
2) Let A ∈ Φ0(X) and K : X → X is a compact linear operator. Then
A + K ∈ Φ0(X) and
ind(A + K) = ind A
Let X∗
and Y ∗
be two Banach spaces with X ⊂ X∗
and Y ⊂ Y ∗
, where X and
Y also are Banach spaces. We assume that X∗
= X⊕Rp
and Y ∗
= Y ⊕Rq
. By
definition à ∈ L(X∗
, Y ∗
) is an extension of A ∈ L(X, Y ) by {p, q} dimensions
if Ãx = Ax for x ∈ X.
Lemma 1.7 Let the operator à ∈ L(X∗
, Y ∗
) be an extension of A ∈ L(X, Y )
by {p, q} dimension. Then
ind A = ind à − p + q.
Proof. We define A∗ : X∗
→ Y ∗
A∗(X + Y ) := Ax for x ∈ X, y ∈ Rp
.
Obviously, A∗ : X∗
→ Y ∗
is an extension of A by {p, q} dimension and
for A∗ ∈ L(X∗
, Y ∗
) holds dim Ker A∗ = dim Ker A + p and codim A∗ =
codim A + q. Hence,
ind A = ind A∗ − p + q.
On the other hand,
dim Ker A∗ = dim Ker A + p, codim A∗ = codim A + q.
Since Ãx = A∗x for all x ∈ X (they are different only by a finite-dimensional
setting), we obtain
ind à = ind A∗.
This completes the proof. ⊓
⊔
The following Lemma 1.8 plays an important role and also holds for a quite
large class of a nonlinear operator (see [20],[51],[63] and references therein).](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-49-320.jpg)
![actually carry it out through all his future life. Such cases are often
to be found on record or in every-day life.
In other cases, this quiet, hidden, but controlling purpose seems to
be formed by unconscious and imperceptible influences, so that the
mind can not revert to the specific time or manner when it
originated. For example, a child who is trained from early life to
speak the truth, can never revert to any particular moment when
this generic purpose originated.
It is sometimes the case, also, that a person will contemplate some
generic purpose before it occurs, while [pg 079] the process of its
final formation seems almost beyond the power of scrutiny. For
example, a man may be urged to relinquish one employment and
engage in another. He reflects, consults, and is entirely uncertain
how he shall decide. As time passes, he gradually inclines toward the
proposed change, until, finally, he finds his determination fixed, he
scarcely knows when or how.
Thus it appears that generic purposes commence sometimes so
instantaneously and obviously that the time and influences
connected with them can be recognized. In other cases, the decision
seems to be a gradual one, while in some instances the process can
be traced, and in others it is entirely unnoticed or forgotten.
It is in reference to such generic purposes that the moral character
of men is estimated. An honest man is one who has a fixed purpose
to act honestly in all circumstances. A truthful man is one who has
such a purpose to speak the truth at all times.
In such cases, the degree in which such a purpose controls all others
is the measure of a man's moral character in the estimate of society.
The history of mankind shows a great diversity of moral character
dependent on such generic choices. Some men possess firm and
reliable moral principles in certain directions, while they are very
destitute of them in others.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-51-320.jpg)
![Thus it will be seen that some have formed a very decided purpose
in regard to honesty in business affairs, who yet are miserable
victims to intemperance. Others have cultivated a principle called
honor, that restrains them from certain actions regarded as mean,
[pg 080] and yet they may be frequenters of gambling saloons and
other haunts of vice.
In the religions world, too, it is the case that some who are very firm
and decided on all points of religious observances and in the
cultivation of devotional emotions, are guilty of very mean actions,
such as some worldly men of honor would not practice at the
sacrifice of a right hand.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-52-320.jpg)
![On a Ruling Purpose or Chief End.
The most important of all the voluntary phenomena is the fact that,
while there can be a multitude of these quiet and hidden generic
purposes in the mind, it is also possible to form one which shall be
the dominant or controlling one, to which all the others, both generic
and specific, shall become subordinate. In common parlance this
would be called the ruling passion. It is also called the ruling
purpose, or controlling principle. This consists in the permanent
choice of some one mode of securing happiness as the chief end or
grand object of life.
There is a great variety of sources of happiness and of suffering to
the human mind. Now in the history of our race we find that each
one of these modes of enjoyment has been selected by different
individuals as the chief end of their existence—as the mode of
seeking enjoyment to which they sacrifice every other. Some persons
have chosen the pleasures of eating, drinking, and the other grosser
enjoyments of sense. Others have chosen those more elevated and
refined pleasures that come indirectly from the senses in the
emotions of taste.
Others have devoted themselves to intellectual enjoyments [pg 081]
as their chief resource for happiness. Others have selected the
exercise of physical and moral power, as in the case of conquerors
and physical heroes, or of those who have sought to control by
moral power, as rulers and statesmen.
Others have made the attainment of the esteem, admiration, and
love of their fellow-creatures, their chief end. Others, still, have
devoted themselves to the promotion of happiness around them as](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-53-320.jpg)
![their chief interest. Others have devoted themselves to the service of
God, or what they conceived to be such, and sometimes by the most
miserable life of asceticism and self-torture.
Others have made it their main object in life to obey the laws of
rectitude and virtue.
In all these cases, the moral character of the person, in the view of
all observers, has been decided by this dominant volition, and
exactly in proportion to the supremacy with which it has actually
controlled all other purposes.
Some minds seem to have no chief end of life. Their existence is a
succession of small purposes, each of which has its turn in
controlling the life. Others have a strong, defined and all-controlling
principle.
Now experience shows that both of these classes are capable, the
one of forming and the other of changing such a purpose. For
example, in a time of peace and ease there is little to excite the
mind strongly; but let a crisis come, where fortune, reputation, and
life are at stake, and men and women are obliged to form generic
decisions involving all they hold dear, and many minds that have no
controlling purpose immediately originate one, while those whose
former ruling [pg 082] aims were in one direction change them
entirely to another.
This shows how it is that days of peril create heroes, statesmen and
strong men and women. The hour of danger calls all the energies of
the soul into action. Great purposes are formed with the strongest
desire and emotion. Instantly the whole current of thought, and all
the coexisting desires and emotions, are conformed to these
purposes.
The experience of mankind proves that a dominant generic purpose
may extend to a whole life, and actually control all other generic and
specific volitions.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-54-320.jpg)
![How the Thoughts, Desires and Emotions are
controlled by the Will.
We will now consider some of the modes by which the will controls
the thoughts, desires and emotions.
We have seen, in previous pages, the influence which desire and
emotion exert in making both our perceptions and conceptions more
vivid. Whatever purpose or aim in life becomes an object of strong
desire, is always distinctly and vividly conceived, while all less
interesting objects are more faint and indistinct.
We have also seen that whenever any conception arises it always
brings connected objects, forming a new and complex picture.
Whenever the mind is under the influence of a controlling purpose,
the object of pursuit is always more interesting than any other. This
interest always fastens on those particulars in any mental
combination that are connected with the ruling purpose and seem
fitted to promote it, making them more vivid. Around these selected
objects their past associated ideas begin [pg 083] to cluster, forming
other complex pictures. In all these combinations, those ideas most
consonant with the leading interest of the mind become most vivid,
and the others fade away.
The grand method, then, for regulating the thoughts is by the
generic decisions of the mind as to the modes of seeking enjoyment.
In regard to the power of the mind over its own desires and
emotions, it is very clear that these sensibilities can not be regulated
by direct specific volitions. Let any person try to produce love, fear,](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-55-320.jpg)
![joy, hope or gratitude by simply choosing to have them arise, and it
is soon perceived that no such power exists.
But there are indirect modes by which the mind can control its
susceptibilities. The first method is by directing attention to those
objects of thought which are fitted to call forth such emotions. For
example, if we wish to awaken the emotion of fear, we can place
ourselves in circumstances of danger, or call up ideas of horror and
distress. If we wish to call forth emotions of gratitude, we can direct
attention to acts of kindness to ourselves calculated to awaken such
feelings. If we wish to excite desire for any object, we can direct
attention to those qualities in that object that are calculated to excite
desire. In all these cases the mind can, by an act of will, direct its
attention to subjects calculated to excite emotion and desire.
The other mode of regulating the desires and emotions is by the
direction of our generic volitions. For example, let a man of business,
who has never had any interest in commerce, decide to invest all his
property in foreign trade. As soon as this is done, the name of the
ship that bears his all can never be heard [pg 084] or seen but it
excites some emotion. A storm, that before would go unnoticed,
awakens fear; the prices in the commercial markets, before
unheeded, now awaken fear or afford pleasure. And thus multitudes
of varied desires and emotions are called into existence by this one
generic volition.
One result of a purpose to deny an importunate propensity is
frequently seen in the immediate or gradual diminution of that
desire. For example, if a person is satisfied that a certain article of
food is injurious and resolves on total abstinence, it will be found
that the desire for it is very much reduced, far more so than when
the effort is to diminish the indulgence.
When a generic purpose is formed that involves great interests, it is
impossible to prevent the desires and emotions from running
consonant with this purpose. The only mode of changing this current](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-56-320.jpg)
![is to give up this generic purpose and form another. Thus, if a man
has devoted his whole time and energies to money-making, it is
impossible for him to prevent his thoughts and feelings from running
in that direction. He must give up this as his chief end, and take a
nobler object, if he would elevate the whole course of his mental
action.
These are the principal phenomena of the grand mental faculty
which is the controlling power of the mind, and on the regulation of
which all its other powers are dependent.
The nature of regeneration, and the question whether it is
instantaneous or gradual or both, all are intimately connected with
the subject of this chapter.
[pg 085]](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-57-320.jpg)
![Chapter XVI. Constitutional Varieties of
the Human Mind.
In the preceding chapters have been presented the most important
mental faculties which are common to the race. There are none of
the powers and attributes of the mind as yet set forth which do not
belong to every mind which is regarded as rational and complete.
But, though all the race have these in common, yet we can not but
observe an almost endless variety of human character, resulting from
the diverse proportions and combinations of these several faculties.
These constitutional differences may be noticed, first, in regard to
the intellectual powers. Some minds are naturally predisposed to
exercise the reasoning powers. Others, with precisely the same kind
of culture, have little relish for this, and little power of appreciating
an argument.
In other cases, the imagination seems to be the predominating
faculty. In other minds there seems to be an equal balance of
faculties, so that no particular power predominates.
Next we see the same variety in reference to the susceptibilities. In
some minds, the desire for love and admiration is the predominating
principle. In others, the love of power takes the lead. Some are
eminently sympathizing. Others have a strong love of rectitude, or
natural conscience. In some, the principle of justice predominates.
In others, benevolence is the leading impulse.
[pg 086]](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-58-320.jpg)
![Finally, in regard to the power of volition, as has been before
indicated, there are some that possess a strong will that is decisive
and effective in regulating all specific volitions, while others possess
various and humbler measures of this power.
According to the science of Phrenology, some of these peculiarities
of mind are indicated by the size and shape of different portions of
the brain, and externally indicated on the skull.
That these differences are constitutional, and not the result of
education, is clear from the many facts showing that no degree of
care or training will serve to efface these distinctive traits of the
mind. To a certain degree they may be modified by education, and
the equal balance of the faculties be promoted, but never to such a
degree as to efface very marked peculiarities.
In addition to the endless diversities that result from these varied
proportions and combinations, there is a manifest variety in the
grades of mind. Some races are much lower in the scale of being
every way than others, while the same disparity exists in individuals
of the same race.
The wisdom and benevolence of this arrangement is very manifest
when viewed in reference to the interests of a commonwealth.
Where some must lead and others follow, it is well that some have
the love of power strong, and others have it less. Where some must
be rulers, to inflict penalties as well as to apportion rewards, it is
well that there be some who have the sense of justice as a leading
principle. And so in the developments of intellect. Some men are to
follow callings where the reasoning powers are most needed. Others
are to adopt pursuits in which taste [pg 087] and imagination are
chiefly required; and thus the varied proportions of these faculties
become serviceable.
And if it be true that the exercise of the social and moral faculties
secures the highest degree of enjoyment, those disparities in mental](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-59-320.jpg)
![Chapter XVII. Nature of Mind.—Habit.
This chapter is introduced because some theologians claim that the
depravity of man consists either in a habit or in something like a
habit.
[pg 088]
Habit is a facility in performing physical or mental operations, gained
by the repetition of such acts. As examples of this in physical
operations may be mentioned the power of walking, which is
acquired only by a multitude of experiments; the power of speech,
secured by a slow process of repeated acts of imitation; and the
power of writing, gained in the same way. Success in every pursuit
of life is attained by oft-repeated attempts, which finally induce a
habit.
As examples of the formation of intellectual habits, may be
mentioned the facility gained in acquiring knowledge by means of
repeated efforts, and the accuracy and speed with which the process
of reasoning is performed after long practice in this art.
As examples of moral habits may be mentioned those which are
formed by the oft-repeated exercise of self-government, justice,
veracity, obedience, and industry. The will, as has been shown, gains
a facility in controlling specific volitions and in yielding obedience to
the laws of right action by constant use, as really as do all the other
mental powers.
The happiness of man, in the present state of existence, depends
not so much upon the circumstances in which he is placed, or the
capacities with which he is endowed, as upon the formation of his](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-61-320.jpg)
![habits. A man might have the organ of sight, and be surrounded
with all the beauties of nature, and yet, if he did not form the habit
of judging of the form, distance and size of bodies, most of the
pleasure and use from this sense would be wanting. The world and
all its beauties would be a mere confused mass of colors.
If the habits of walking and of speech were not acquired, these
faculties and the circumstances for employing [pg 089] them would
not furnish the enjoyment they were designed to secure.
It is also the formation of intellectual habits by mental discipline and
study, which opens vast resources for enjoyment that otherwise
would be for ever closed. And it is by practicing obedience to parents
that moral habits of subordination are formed, which are
indispensable to our happiness as citizens, and as subjects of
government. There is no enjoyment which can be pointed out, which
is not, to a greater or less extent, dependent upon this principle.
The influence of habit in regard to the law of sacrifice is especially
interesting. The experience of multitudes of our race shows that
such tastes and habits may be formed in obeying this law, that what
was once difficult and painful becomes easy and pleasant.
But this ability to secure enjoyment through habits of self-control
and self-denial, induced by long practice, so far as experience
shows, could never be secured by any other method.
That the highest kinds of happiness are to be purchased by more or
less voluntary sacrifice and suffering to procure good for others,
seems to be a part of that nature of things which we at least may
suppose has existed from eternity. We can conceive of the eternal
First Cause only as we imagine a mind on the same pattern as our
own in constitutional capacities, but indefinitely enlarged in extent
and action. Knowledge, wisdom, power, justice, benevolence and
rectitude, must be the same in the Creator as in ourselves, at least
so far as we can conceive; and, as the practice of self-sacrifice and](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-62-320.jpg)
![suffering for the good of others is our highest conception of virtue, it
is impossible to regard [pg 090] the Eternal Mind as all-perfect
without involving this idea.
The formation of the habits depends chiefly upon the leading desire
or governing purpose, because whatever the mind desires the most
it will act the most to secure, and thus by repeated acts will form its
habits. The character of every individual, therefore, as before
indicated, depends upon the mode of seeking happiness selected by
the will. Thus the ambitious man has selected the attainment of
power and admiration as his leading purpose, and whatever modes
of enjoyment interfere with this are sacrificed. The sensual man
seeks his happiness from the various gratifications of sense, and
sacrifices other modes of enjoyment that interfere with this. The
man devoted to intellectual pursuits, and to seeking reputation and
influence through this medium, sacrifices other modes of enjoyment
to secure this gratification. The man who has devoted his affections
and the service of his life to God and the good of his fellow-men,
sacrifices all other enjoyments to secure that which results from the
fulfillment of such obligations. Thus a person is an ambitious man, a
sensual man, a man of literary ambition, or a man of piety and
benevolence, according to the governing purpose or leading choice
of his mind.
There is one fact in regard to the choice of the leading object of
desire, or the governing purpose of life, which is very important.
Certain modes of enjoyment, in consequence of repetition, increase
the desire, but lessen the capacity of happiness from this source;
while, in regard to others, gratification increases the [pg 091] desire,
and at the same time increases the capacity for enjoyment.
The enjoyments through the senses are of the first kind. It will be
found, as a matter of universal experience, that where this has been
chosen as the main purpose of life, though the desire for such
pleasures is continually increased, yet, owing to the physical effects
of excessive indulgence, the capacity for enjoyment is decreased.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-63-320.jpg)
![Thus the man who so degrades his nature as to make the pleasures
of eating and drinking the great pursuit of life, while his desires
never abate, finds his zest for such enjoyments continually
decreasing, and a perpetual need for new devices to stimulate
appetite and awaken the dormant capacities. The pleasures of sense
always pall from repetition—grow “stale, flat and unprofitable,”
though the deluded being who has thus slavishly yielded to such
appetites feels himself bound by chains of habit, which, even when
enjoyment ceases, seldom are broken.
The pleasures derived from the exercise of power, when its
attainment becomes the master passion, are also of this description.
The statesman, the politician, the conqueror, are all seeking for this,
and desire never abates while any thing of the kind remains to be
attained. We do not find that enjoyment increases in proportion as
power is secured. On the contrary, it seems to cloy in possession.
Alexander, the conqueror of the world, when he had gained all, wept
that objects of desire were extinct, and that possession could not
satisfy.
But there are other sources of happiness for which the desire ever
continues, and possession only increases the capacity for enjoyment.
Of this class is [pg 092] the susceptibility of happiness from giving
and receiving affection. Here, the more there is given and received,
the more is the power of giving and receiving increased. We find
that this principle outlives every other, and even the decays of
nature itself. When tottering age on the borders of the grave is just
ready to resign its wasted tenement, often from its dissolving ashes
the never-dying spark of affection has burst forth with new and
undiminished luster. This is that immortal fountain of happiness
always increased by imparting, never surcharged by receiving.
Another principle, which increases both desire and capacity by
exercise, is the power of enjoyment from being the cause of
happiness to others. Never was an instance known of regret for
devotion to the happiness of others. On the contrary, the more this](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-64-320.jpg)
![holy and delightful principle is in exercise, the more the desires are
increased, and the more are the susceptibilities for enjoyment from
this source enlarged. While the votaries of pleasure are wearing
down with the exhaustion of abused nature, and the votaries of
ambition are sighing over its thorny wreath, the benevolent spirit is
exulting in the success of its plans of good, and reaching forth to still
purer and more perfect bliss.
This principle is especially true in regard to the practice of rectitude.
The more the leading aim of the mind is devoted to right feeling and
action, or to obedience to all the laws of God, the more both the
desire and the capacity of enjoyment from this source are increased.
But there is another fact in regard to habit, which has an immense
bearing on the well-being of our race. [pg 093] When a habit of
seeking happiness in some one particular mode is once formed, the
change of this habit becomes difficult just in proportion to the
degree of repetition which has been practiced. A habit once formed,
it is no longer an easy matter to choose between the mode of
securing happiness chosen and another which the mind may be led
to regard as much superior. Thus, in gratifying the appetite, a man
may feel that his happiness is continually diminishing, and that, by
sacrificing this passion, he may secure much greater enjoyment from
another source; yet the force of habit is such, that decisions of the
will perpetually yield to its power.
Thus, also, if a man has found his chief enjoyment in that admiration
and applause of men so ardently desired, even after it has ceased to
charm, and seems like emptiness and vanity, still, when nobler
objects of pursuit are offered, the chains of habit bind him to his
wonted path. Though he looks and longs for the one that his
conscience and his intellect assure him is brightest and best, the
conflict with bad habit ends in fatal defeat and ruin. It is true that
every habit can be corrected and changed, but nothing requires
greater firmness of purpose and energy of will; for it is not one](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-65-320.jpg)
![resolution of mind that can conquer habit: it must be a constant
series of long-continued efforts.
The influence of habit in reference to emotions deserves special
attention as having a direct influence upon character and happiness.
All pleasurable emotions of mind, being grateful, are indulged and
cherished, and are not weakened by repetition unless they become
excessive. If the pleasures of sense are indulged beyond a certain
extent, the bodily system is [pg 094] exhausted, and satiety is the
consequence. If the love of power and admiration is indulged to
excess, so as to become the leading purpose of life, they are found
to be cloying. But, within certain limits, all pleasurable emotions do
not seem to lessen in power by repetition.
But in regard to painful emotions the reverse is true. The mind
instinctively resists or flies from them, so that often a habit of
suppressing such emotions is formed, until the susceptibility
diminishes, and sometimes appears almost entirely destroyed. Thus
a person often exposed to danger ceases to be troubled by fear,
because he forms a habit of suppressing it. A person frequently in
scenes of distress and suffering learns to suppress the emotions of
painful sympathy. The surgeon is an example of the last case,
where, by repeated operations, he has learned to suppress emotions
until they seldom recur. A person inured to guilt gradually deadens
the pangs of remorse, until the conscience becomes “seared as with
a hot iron.” Thus, also, with the emotion of shame. After a person
has been repeatedly exposed to contempt, and feels that he is
universally despised, he grows callous to any such emotions.
The mode by which the mind succeeds in forming such a habit
seems to be by that implanted principle which makes ideas that are
most in consonance with the leading desire of the mind become
vivid and distinct, while those that are less interesting fade away.
Now no person desires to witness pain except from the hope of
relieving it, unless it be that, in anger, the mind is sometimes
gratified with the infliction of suffering. But, in ordinary cases, the](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-66-320.jpg)
![sight of suffering [pg 095] is avoided except where relief can be
administered. In such cases, the desire of administering relief
becomes the leading one, so that the mind is turned off from the
view of the suffering to dwell on conceptions of modes of relief.
Thus the surgeon and physician gradually form such habits that the
sight of pain and suffering lead the mind to the conception of modes
of relief, whereas a mind not thus interested dwells on the more
painful ideas.
The habits of life are all formed either from the desire to secure
happiness or to avoid pain, and the fear of suffering is found to be a
much more powerful principle than the desire of happiness. The soul
flies from pain with all its energies, even when it will be inert at the
sight of promised joy. As an illustration of this, let a person be fully
convinced that the gift of two new senses would confer as great an
additional amount of enjoyment as is now secured by the eye and
ear, and the promise of this future good would not stimulate with
half the energy that would be caused by the threat of instant and
entire blindness and deafness.
If, then, the mind is stimulated to form good habits and to avoid the
formation of evil ones most powerfully by painful emotions, when
their legitimate object is not effected they continually decrease in
vividness, and the designed benefit is lost. If a man is placed in
circumstances of danger, and fear leads to habits of caution and
carefulness, the object of exciting this emotion is accomplished, and
the diminution of it is attended with no evil. But if fear is continually
excited, and no such habits are formed, then the susceptibility is
lessened, while the good to be secured by it [pg 096] is lost. So,
also, with emotions of sympathy. If we witness pain and suffering,
and it induces habits of active devotion to the good of those who
suffer, the diminution of the susceptibility is a blessing and no evil.
But if we simply indulge emotions, and do not form the habits they
were intended to secure, the power of sympathy is weakened, and
the designed benefit is lost. Thus, again, with shame: if this painful
emotion does not lead us to form habits of honor and rectitude, it is](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-67-320.jpg)
![continually weakened by repetition, and the object for which it was
bestowed is not secured. And so with remorse: if this emotion is
awakened without leading to habits of benevolence and virtue, it
constantly decays in power, and the good it would have secured is
for ever lost.
It does not appear, however, that the power of emotion in the soul is
thus destroyed. This is evident from the fact that the most hardened
culprits, when brought to the hour of death, where all plans of
future good cease to charm the mental eye, are often overwhelmed
with the most vivid emotions of sorrow, shame, remorse and fear.
And often, in the course of life, there are seasons when the soul
returns from its pursuit of deluding visions to commune with itself in
its own secret chambers. At such seasons, shame, remorse and fear
take up their abode in their long-deserted dwelling, and ply their
scorpion whips till they are obeyed, and the course of honor and
virtue is resumed, or till the distracted spirit again flies abroad for
comfort and relief.
There is a great diversity in human character, resulting from the
diverse proportions and combinations of those powers of mind which
the race have in common. [pg 097] At the same time, there is a
variety in the scale of being, or relative grade of each mind. While all
are alike in the common faculties of the human mind, some have
every faculty on a much larger scale than others, while some are of
a very humble grade.
The principle of habit has very great influence in modifying and
changing these varieties. Thus, by forming habits of intellectual
exercise, a mind of naturally humble proportions can be elevated
considerably above one more highly endowed by natural
constitution. So the training of some particular intellectual faculty,
which by nature is deficient, can bring it up nearer to the level of
other powers less disciplined by exercise.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-68-320.jpg)
![In like manner, the natural susceptibilities can be increased,
diminished or modified by habit. Certain tastes, that had little power,
can be so cultivated as to overtop all others.
So of the moral nature: it can be so exercised that a habit will be
formed which will generate a strength and prominency that nature
did not impart.
One of the most important results of habit is its influence on faith or
belief. Those persons who practice methods of false reasoning, who
turn away from evidence and follow their feelings in forming
opinions, eventually lose the power of sure, confiding belief.
On the contrary, an honest, conscientious steadiness in seeking the
truth and in yielding to evidence, secures the firmest and most
reliable convictions, and that peace of mind which alone results from
believing the truth.
The will itself is also subject to this same principle. A strong will, that
is trained to yield obedience to law [pg 098] in early life, acquires an
ease and facility in doing it which belongs ordinarily to weak minds,
and yet can retain all its vigor. And a mind that is trained to bring
subordinate volitions into strict and ready obedience to a generic
purpose, acquires an ease and facility in doing this which was not a
natural endowment.
Thus it appears that by the principle of habit every mind is furnished
with the power of elevating itself in the scale of being, and of so
modifying and perfecting the proportions and combinations of its
constitutional powers, that often the result is that there is no mode
of distinguishing between the effects of habit and those of natural
organization.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-69-320.jpg)
![Chapter XVIII. The Nature of Mind Our
Guide to the Natural Attributes of God.
The natural attributes of any mind are the powers and faculties to be
exercised, while it is the action or voluntary use of these faculties
that exhibits the moral attributes.
Having gained the existence of a Great First Cause by the use of one
principle of common sense, and the fact that this cause is an
intelligent mind by another, it has been shown that a third of these
principles leads to the belief that the natural attributes of God are
like our own. We can not conceive of any other kind of minds than
our own, because we have never had any past experience or
knowledge of any other.
[pg 099]
But while we thus conclude that the mind of the Creator is, so far as
we can conceive, precisely like our own in constitutional
organization, we are as necessarily led to perceive that the extent of
these powers is far beyond our own. A mind with the power, wisdom
and goodness exhibited in the very small portion of his works
submitted to our inspection, who has inhabited eternity, and
matured through everlasting ages—our minds are lost in attempting
any conception of the extent of such infinite faculties!
Thus we are necessarily led to conceive of the Creator as possessing
the intellectual powers described in previous pages. He perceives,
conceives, imagines, judges and remembers just as we do.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-70-320.jpg)
![So also all our varied susceptibilities to pleasure and pain exist in the
Eternal Mind. The desire of good and the fear of evil which are the
motive power in the human mind, exist also in the divine. Thus by
the light of nature we settle the question that the existence of
susceptibilities to pain and evil are not the results of the Creator's
will, but are a part of the eternal nature of things which he did not
originate or control.
All the minds we ever knew or heard of are moved to action by
desire to gain happiness and escape pain, and as we can conceive of
no other kind of mind than our own, we must attribute to the
Creator this foundation element of mental activity.
Thus we are led to attribute to the Creator all those susceptibilities
included in the moral sense, as described in previous pages. His
mind, like ours, feels that whatever makes the most happiness with
the least [pg 100] evil is right; that is to say, it is fitted to the eternal
nature of things, of which his own mind is a part.
So also the Creator possesses that sense of justice implanted in our
own minds, which involves the desire of good to those who make
happiness, and of evil to those who destroy happiness; and which
also demands that such retributions be proportioned to the good and
evil done, and to the power of the agent.
So also we must conceive of the Creator as possessing the
susceptibility of conscience, which includes in the very constitution
of mind retributions for right and wrong action.
Again, we are led to conceive of God as a rational free agent, with
power to choose either that which excites the strongest desire or
that which is perceived to be best on the whole for all concerned,
even if it does not excite the strongest desire.
Again, we are to conceive of the Creator as possessing a belief in
those principles of reason which he has implanted in our minds, and](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-71-320.jpg)
![made our guide in all matters, both of temporal and religious
concern.
Again, our experience of the nature and history of mind, leads to the
inference that no being has existed from all eternity in solitude, but
that there is more than one eternal, uncreated mind, and that all
their powers of enjoyment from giving and receiving happiness in
social relations have been in exercise from eternal ages. This is the
just and natural deduction of reason and experience, as truly as the
deduction that there is at least one eternal First Cause.
Again, all our experience of mind involves the idea of the mutual
relation of minds. We perceive that minds are made to match to
other minds, so that there [pg 101] can be no complete action of
mind, according to its manifest design, except in relation to other
beings. A mind can not love till there is another mind to call forth
such emotion. A mind can not bring a tithe of its power into
appropriate action except in a community of minds. The conception
of a solitary being, with all the social powers and sympathies of the
human mind infinitely enlarged, and yet without any sympathizing
mind to match and meet them, involves the highest idea of unfitness
and imperfection conceivable, while it is contrary to our uniform
experience of the nature and history of mind.
It has been argued that the unity of design in the works of nature
proves that there is but one creating mind. This is not so, for in all
our experience of the creations of finite beings no great design was
ever formed without a combination of minds, both to plan and to
execute. The majority of minds in all ages, both heathen and
Christian, have always conceived of the Creator as in some way
existing so as to involve the ideas of plurality and of the love and
communion of one mind with another.
And yet the unity and harmony of all created things as parts of one
and the same design, teach a degree of unity in the authorship of
the universe never known in the complex action of finite minds.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-72-320.jpg)
![Thus a unity and plurality in the Creator of all things is educed by
reason and experience from the works of nature.
[pg 102]](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-73-320.jpg)
![Chapter XIX. The Nature of Mind Our
Guide to the Moral Attributes of God.
Having employed the principles of common sense to gain a
knowledge of the natural attributes of God, we are next to employ
the same principles to gain his moral character; or those attributes
which are exhibited in willing. In other words, we are to seek the
character of God as expressed in his works or deeds.
In our experience of the moral character of minds in this world, we
find that some of the highest grades as to intellect and
susceptibilities, are lowest as to good-willing. How is it, then, with
the highest mind of all? Does he so prefer evil to good, that he
deliberately plans for the production of evil when he has power to
produce happiness in its place? Or does he sometimes prefer evil
and sometimes good, with the variable humors of the human race?
Or does he always prefer good when it costs him no trouble or
sacrifice, but never when it does? Or is he one who invariably
chooses what is best for all, even when it involves painful sacrifices
to himself?
In seeking a reply to these momentous questions, we return once
more to the principle of common sense before stated, i.e., the
nature of any work or contrivance is proof of the character and
design of the author.
In examining the works of the Creator, we find that the material
world impresses us as wisely adjusted and good in construction, only
as it is fitted to give enjoyment to sentient beings. It is the
intelligent, [pg 103] feeling, acting minds that give the value to](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-74-320.jpg)
![every other existence. If there were no minds, all perception of
beauty, fitness and goodness would perish.
It is minds, therefore, which are the chief works of the Creator's
hand, and which give value to all others.
If the nature of these minds is evil, then the author of them is
proved to be evil by his works. If their nature is good and perfect,
then their author is proved to be good and perfect.
Here again we are driven back to our own minds to gain the only
conceptions possible to us, not only of wisdom, but of goodness or
benevolence.
On examination, we shall find that we can form no idea of these
qualities which does not involve a limitation of power.
Our idea of power is that which we gain when we will to move our
bodies or to make any other change, and this change ensues. Our
only idea of a want of power is gained when the choice or willing of
a change or event does not produce it. Whenever, therefore, it shall
appear that the Creator wills or wishes a thing to exist or to be
changed, and that change or existence does not follow his so willing,
we can not help believing that he has not the power to produce it?
Again; our idea of perfectness always has reference to power; for a
thing is regarded as perfect in construction only when there is no
power in God or man to make it better. When any arrangement is as
good as it can be, so that neither God nor man has power to make it
better, we regard it as perfect, even when there is some degree of
evil involved.
We are now prepared to define what is included in [pg 104] the
terms perfect wisdom and perfect benevolence, when applied to the
Creator or to any other being, thus: A perfectly wise being is one
who invariably wills the best possible ends and the best possible
means of accomplishing those ends.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-75-320.jpg)
![An imperfectly wise being is one who does not invariably do this.
A perfectly benevolent being is one who invariably wills the most
good and the least evil in his power. An imperfectly benevolent
being is one who does not invariably will thus.
The degree in which a being is ranked as wise and good is estimated
by the extent to which his willing good or evil corresponds with his
power.
Thus it appears that, in a system where evil exists, the very idea of
perfect benevolence and wisdom involves the supposition of a
limitation of power.
To return, then, to the question as proposed at the commencement
of the chapter—Is the Creator a being who prefers good to evil
invariably, or is he one who only sometimes prefers evil to good, and
at other times prefers good to evil, with the varying humors of man;
or does he invariably choose what is best for all, even in cases
where it may cost personal sacrifice and suffering to himself?
It will be the object of what follows to prove that the last supposition
is the true one.
In attempting this, we again take the principle of common sense,
that “the nature of any contrivance proves the design and character
of the author.” Then we proceed to a review of the nature, first of
mind, and next of the material world, to prove that the design or
chief end of the Creator is, not to make happiness [pg 105]
irrespective of the amount, but to produce the greatest possible
happiness with the least possible evil. In other words, we are to
seek for proof that God has done all things for the best, so that he
has no power to do better.
In still another form, we are to seek for evidence, in the nature of
God's works, that he has ever done the best he could, so that the
amount of evil that ever was or ever will exist, is not caused by his](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-76-320.jpg)
![willing it, but by his want of power to prevent it; so that any change
would be an increase of evil and a lessening of good to the universe
as a whole.
In pursuing this attempt, it will be needful to reproduce two or three
chapters of a work by the author, already before the public, entitled,
The Bible and the People; or, Common Sense applied to Religion.
In this work the nature of mind is presented very much more in
detail, for the same purpose as that here indicated. What will now
follow is a brief review of previous chapters in that work, as a
summary of the evidence there presented that the chief end of God
in all his works is to produce the greatest possible happiness with
the least possible evil.
Whenever we find any contrivances all combining to secure a certain
good result, which, at the same time, involve some degree of
inevitable evil, and then discover that there are contrivances to
diminish and avoid this evil, we properly infer that the author
intended to secure as much of the good with as little of the evil as
possible. For example, a traveler finds a deserted mine, and all
around he discovers contrivances for obtaining gold, and, at the
same time, other contrivances for getting rid of the earth mixed with
it. The [pg 106] inevitable inference would be that the author of
these contrivances designed to secure as much gold with as little
earth as possible; and should any one say that he could have had
more gold and less earth if he chose to, the answer would be that
there is no evidence of this assertion, but direct evidence against it.
Again: should we discover a piece of machinery in which every
contrivance tended to secure a speed in movement, produced by the
friction of wheels against a rough surface, and at the same time
other contrivances were found for diminishing all friction that was
useless, we should infer that the author designed to secure the
greatest possible speed with the least possible friction.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-77-320.jpg)
![In like manner, if we can show that mind is a contrivance that acts
by the influence of fear of evil, and that pain seems as indispensable
to the action of a free agent as friction is to motion; if we can show
that there is no contrivance in mind or matter which is designed to
secure suffering as its primary end; if we can, on the contrary, show
that the direct end of all the organizations of mind and matter is to
produce happiness; if we can show that it is only the wrong action of
mind that involves most of the pain yet known, so that right action,
in its place, would secure only happiness; if we can show
contrivances for diminishing pain, and also contrivances for
increasing happiness by means of the inevitable pain involved in the
system of things, then the just conclusion will be gained that the
Author of the system of mind and matter designed “to produce the
greatest possible happiness with the least possible evil.”
[pg 107]
In the pages which follow, we shall present evidence exhibiting all
these particulars.
The only way in which we learn the nature of a thing is to observe
its qualities and actions. This is true of mind as much as it is of
matter. Experience and observation teach that the nature of mind is
such, that the fear of suffering is indispensable to secure a large
portion of the enjoyment within reach of its faculties, and that the
highest modes of enjoyment can not be secured except by sacrifice,
and thus by more or less suffering.
This appears to be an inevitable combination, as much so as friction
is inevitable in machinery.
We have the evidence of our own consciousness that it is fear of evil
to ourselves or to others that is the strongest motive power to the
mind. If we should find that no pain resulted from burning up our
own bodies, or from drowning, or from any other cause; if every one
perceived that no care, trouble, or pain resulted from losing all kinds
of enjoyment, the effort to seek it would be greatly diminished.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-78-320.jpg)
![If we could desire good enough to exert ourselves to seek it, and yet
should feel no discomfort in failing; if we could lose every thing, and
feel no sense of pain or care, the stimulus to action which
experience has shown to be most powerful and beneficent would be
lost.
We find that abundance of ease and prosperity enervates mental
power, and that mind increases in all that is grand and noble, and
also in the most elevating happiness, by means of danger, care and
pain. We may properly infer, then, that evil is a necessary part of the
experience of a perfectly-acting mind.
[pg 108]
So strong is the conviction that painful penalties are indispensable,
that the kindest parents and the most benevolent rulers are the
most sure to increase rather than diminish those that are already
involved in the existing nature of things.
Again: without a revelation we have no knowledge of any kind of
mind but by inference from our experience in this state of being. All
we know of the Eternal First Cause is by a process of reasoning,
inferring that his nature must be like the only minds of which we
have any knowledge. We assume, then, that he is a free agent,
regulated by desire for happiness and fear of evil.
We thus come to the conclusion that this organization of mind is a
part of the fixed and eternal nature of things, and does not result
from the will of the Creator. His own is the eternal pattern of an all-
perfect mind, and our own are formed on this perfect model, with
susceptibilities to pain as an indispensable motive power in gaining
happiness.
We will now recapitulate some of the particulars in the laws and
constitution of mind which tend to establish the position that its
Creator's grand design is “to produce the greatest possible
happiness with the least possible evil.”](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-79-320.jpg)
![Intellectual Powers.
First, then, in reference to the earliest exercise of mind in sensation.
The eye might have been so made that light would inflict pain, and
the ear so that sound would cause only discomfort. And so of all the
other senses.
But the condition of a well-formed, healthy infant [pg 109] is a most
striking illustration of the adaptation of the senses to receive
enjoyment. Who could gaze on the countenance of such a little one,
as its various senses are called into exercise without such a
conviction? The delight manifested as the light attracts the eye, or
as pleasant sounds charm the ear, or as the limpid nourishment
gratifies its taste, or as gentle motion and soft fondlings soothe the
nerves of touch, all testify to the benevolent design of its Maker.
Next come the pleasures of perception as the infant gradually
observes the qualities of the various objects around, and slowly
learns to distinguish its mother and its playthings from the confused
mass of forms and colors. Then comes the gentle curiosity as it
watches the movement of its own limbs, and finally discovers that its
own volitions move its tiny fingers, while the grand idea that it is
itself a cause is gradually introduced.
Next come the varied intellectual pleasures as the several powers
are exercised in connection with the animate and material world
around, in acquiring the meaning of words, and in imitating the
sounds and use of language. The adult, in toiling over the dry
lexicon, little realizes the pleasure with which the little one is daily
acquiring the philosophy, grammar, and vocabulary of its mother
tongue.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-80-320.jpg)
![A child who can not understand a single complete sentence, or
speak an intelligible phrase, will sit and listen with long-continued
delight to the simple enunciation of words, each one of which
presents a picture to his mind of a dog, a cat, a cow, a horse, a
whip, a ride, and many other objects and scenes that have given
pleasure in the past; while the single words, [pg 110] without any
sentences, bring back, not only vivid conceptions of these objects,
but a part of the enjoyment with which they have been connected.
Then, as years pass by, the intellect more and more administers
pleasure, while the reasoning powers are developed, the taste
cultivated, the imagination exercised, the judgment employed, and
the memory stored with treasures for future enjoyment.
In the proper and temperate use of the intellectual powers, there is
a constant experience of placid satisfaction, or of agreeable and
often of delightful emotions, while no one of these faculties is
productive of pain, except in violating the laws of the mental
constitution.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-81-320.jpg)
![The Susceptibilities.
In regard to the second general class of mental powers—the
susceptibilities—the first particular to be noticed is the ceaseless and
all-pervading desire to gain happiness and escape pain. This is the
mainspring of all voluntary activity; for no act of volition will take
place till some good is presented to gain, or some evil to shun. At
the same time, as has been shown, the desire to escape evil is more
potent and effective than the desire for good. Thousands of minds
that rest in passive listlessness, when there is nothing to stimulate
but hope of enjoyment, will exert every physical and mental power
to escape impending evil. The seasons of long-continued prosperity
in nations always tend to a deterioration of intellect and manhood. It
is in seasons of danger alone that fear wakes up the highest
energies, and draws forth the heroes of the race.
Mind, then, is an existence having the power of that [pg 111] self-
originating action of choice which constitutes free agency, while this
power can only be exercised when desires are excited to gain
happiness or to escape pain. This surely is the highest possible
evidence that its Author intended mind should thus act.
But a mind may act to secure happiness and avoid pain to itself, and
yet may gain only very low grades of enjoyment, while much higher
are within reach of its faculties. So, also, it may act to gain
happiness for itself as the chief end in such ways as to prevent or
destroy the higher happiness of others around.
In reference to this, we find those susceptibilities which raise man to
the dignity of a rational and moral being.](https://image.slidesharecdn.com/75760-250528143805-7249ad05/85/Linear-and-Nonlinear-Non-fredholm-Operators-Theory-and-Applications-1st-Edition-Messoud-Efendiev-82-320.jpg)
Linear and Nonlinear Non-fredholm Operators: Theory and Applications 1st Edition Messoud Efendiev
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- 5. Linear and Nonlinear Non-Fredholm Operators Messoud Efendiev Theory and Applications
- 6. Linear and Nonlinear Non-Fredholm Operators
- 7. Messoud Efendiev Linear and Nonlinear Non-Fredholm Operators Theory and Applications 123
- 8. Messoud Efendiev Institute of Computational Biology Helmholtz Zentrum München Neuherberg, Bayern, Germany ISBN 978-981-19-9879-9 ISBN 978-981-19-9880-5 (eBook) https://doi.org/10.1007/978-981-19-9880-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 9. This book is dedicated to the memory of the late Heydar Aliyev, on the 100th anniversary of his birth, who played an important role both in educational and scientific careers of a whole generation and beyond.
- 10. Preface This book is devoted to linear and nonlinear non-Fredholm operators and their applications. Fredholm operators are named in honour of Erik Ivar Fredholm and constitute one of the most important classes of linear maps in mathematics. They were introduced around 1900 in the study of integral operators and, by definition, they share many properties with linear maps between finite dimensional spaces (i.e. matrices). The Fredholm property im- plies the solvability condition: the nonhomogeneous operator equation Lu = f is solvable if and only if the right-hand side f is orthogonal to all solutions of the homogeneous adjoint problem L∗ v = 0. The orthogonality is understood in the sense of duality in the corresponding spaces. Indeed, solvability con- ditions play an important role in the analysis of nonlinear problems.These properties of Fredholm operators are widely used in many methods of linear and nonlinear analysis. At present linear and nonlinear Fredholm operator theory and solvability of corresponding equations are quite well understood (see the book of [20] and the references therein). However, in the case of non-Fredholm operators the usual solvability con- ditions may not be applicable and solvability relations are, in general, not known. In spite of some progress on linear/nonlinear non-Fredholm operator theory these questions and related topics are not systematically studied in the mathematical literature and to the best of our knowledge are not well understood. The aim of this book is to attempt to close such a gap and ini- tiate as well as stimulate readers to make contributions to this fascinating subject. My work on this subject, that is on solvability and well-posedness of a lin- ear/nonlinear non-Fredholm operator equation, started in 2015 during my visit to York University (Canada) as an Alexander von Humboldt Fellow for experienced scientists. One of the seminars there was given by Dr. V. Vougalther (University of Toronto). Both during the talk and afterwards, I asked him many questions to most of them I received the answer either not known or not studied. This was a point of departure for my study of this and vii
- 11. viii Preface related questions. Indeed, I found this subject very fascinating and taking into account my ”alte liebe” on Fredholm operator theory (see my book [20]), I started with much enthusiasm to work on this topic. Later in my visit to the Fields Institute and the University of Toronto as a Deans Distinguished Vis- iting Professor (2018) and as a James D. Murray Distinguished Professor at the University of Waterloo (2019) together with my colleagues of the Univer- sities mentioned above working on ”What has Biology and Medicine done for Mahematics”, I also continued to work with Dr.V. Vougalther on linear and nonlinear non-Fredholm operators and its applications to the various classes of non-Fredholm elliptic equations. One of the main question which will be considered in this book is the solv- ability of linear and nonlinear equations related to non-Fredholm operators. We present the explicit form of the solvability conditions and establish the existence of solutions of the non-Fredholm equations considered in this book. In particular, we address it in the following setting. Let A : E → F be the operator corresponding to the left side of equation Au = f. Assume that this operator fails to satisfy the Fredholm property (throughout this book in each chapter starting from the second one we will present a quite large class of pseudo-differential elliptic equations for which this will be the case). Let fn be a sequence of functions in the image of the operator A, such that fn → f in F as n → ∞. Denote by un a sequence of functions from E such that Aun = fn, n ∈ N. Since the operator A does not satisfy the Fredholm property, the sequence un may not be convergent. Let us call a sequence un the solution in the sense of sequences of the equation Au = f if Aun → f. If such a sequence converges to a function u∗ in the norm of the space E, then u∗ is a solution of this equation in the usual sense. A solution in the sense of sequences in this case is equivalent to the usual solution. However, in the case of non-Fredholm operators, this convergence may not hold or it can occur in some weaker sense. In such a case, a solution in the sense of sequences may not imply the existence of the usual solution. In this book we find the sufficient conditions for the equivalence of solutions in the sense of sequences and the usual solutions, that is, the conditions on sequences fn under which the corresponding sequences un are strongly convergent. In the case of elliptic integro-differential equations that fail to satisfy the Fredholm property (which can also be formulated in term of the location of the essential spectrum), we prove existence of solution in the sense of sequences in terms of the kernel of the given integro-differential elliptic operator. This book consists of five chapters and in particular includes our results that have been published in the leading journals of mathematical societies of the world. Chapter 1 has more of a teaching aid character and consists of eight sections and is dedicated, in particular, to some basic concepts concerning Sobolev
- 12. Preface ix spaces and embedding theorems, linear elliptic boundary value problems, lin- ear Fredholm operators and its properties, properties of superposition oper- ators in Sobolev and Hölder spaces, the Fourier transform and related quan- tities, fractional Laplacian as a pseudo-differential operator, as well as the properties of generalized Fourier transform in terms of the functions of con- tinuous spectrum of the Schrödinger operators with shallow and short-range potential. Chapter 1 is not self-sufficient, since it is intended as auxiliary material for other chapters. Chapter 2 is devoted to the well-posedness of a class of stationary nonlinear integro-differential equations containing the classical Laplacian and a drift term for which the Fredholm property may not be satisfied. Here we formu- late solvability conditions in terms of iterated kernels of a nonlinear integral operator which is related to the equation under consideration. Chapter 2 consists of four subsections. In sections 2.1 and 2.2 we consider a class of stationary nonlinear integro-differential scalar equation containing classical Laplacian and drift term on the whole line and on a finite interval respectively. In sections 2.3 and 2.4 we consider the same questions for sys- tems of integro-differential equations containing classical Laplacian and drift term. We emphasize that the study of the system case (sections 2.3 and 2.4) is more difficult than of the scalar case (sections 2.1 and 2.2) and requires some more cumbersome technicalities to be overcome. In population dynamics the integro-differential equations describe models with intra-specific competition and nonlocal consumption of resources. On the other hand the studies of the solutions of the integro-differential equations with the drift term are relevant to the understanding of the emergence and propagation of patterns in the theory of speciation. Chapter 3 deals with the existence in the sense of sequences of solutions for some integro-differential type equations containing the drift term and the square root of the one dimensional negative Laplacian (so-called super- diffusion) on the whole real line, and on a finite interval with periodic bound- ary conditions in the corresponding H2 spaces. The argument for proving existence of solutions in the sense of sequences in this chapter relies on fixed point techniques when the elliptic equations involve first order pseudo- differential operators (nonlocal) with and without the Fredholm property. Chapter 3 consists of four subsections. Sections 3.1 and 3.2 deal with scalar equations on the whole real line and a finite interval respectively. In sections 3.3 and 3.4 we consider the analogous problem for a system of equations, the study of which has additional difficulties and needs new ideas compared with the scalar case. Superdiffusion can be described as a random process of particle motion characterized by the probability density distribution of the jump length. The moments of this density distribution are finite in the case of the normal diffusion, but this is not the case for superdiffusion. Asymptotic behavior at infinity of the probability density function determines the value of the power of the negative Laplace operator (for the details see chapter 3).
- 13. x Preface In chapter 4 we establish the existence in the sense of sequences of solutions for certain nonlinear integro-differential type equations in two dimensions involving normal diffusion in one direction and anomalous diffusion in the other direction in H2 (R2 ) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under some reasonable technical conditions, the convergence in L1 (R2 ) of the integral kernels implies the existence and convergence in H2 (R2 ) of the solutions. Such anisotropy in the diffusion term (local versus nonlocal) make our analysis extremely difficult because in order to derive the desired estimates requires new ideas and cumbersome techniques. Chapter 4 consists of two sections. Section 4.1 is devoted to scalar nonlinear equations in the presence of the mixed-diffusion type mentioned above. These models are new and not much is understood about them, especially in the context of the integro-differential equations. We use the explicit form of the solvability conditions and establish the existence of solutions of such nonlinear equation. In section 4.2 we consider the analogous problem for a system of equations. The novelty of this section is that in each diffusion term we add the standard negative Laplacian in the x1 variable to the minus Laplacian in the x2 variable raised to a fractional power. Such anisotropy coming from a different fractional order in each equation of the system make our analysis more difficult than in scalar case and requires both new ideas and requires rather sophisticated techniques. It is important to study the equations of this kind in unbounded domains from the point of view of the understanding of the spread of the viral infections , since many countries have to deal with pandemics. In chapter 5 we consider two classes of non-Fredholm (4th and 2nd order) Schrödinger type operators and establish the solvability conditions in the sense of sequences for the equations involving them. To this end, we use the methods of the spectral and scattering theory for Schrödinger type operators, the potential functions V (x) of which are assumed to be shallow and short- range with a few extra regularity conditions. In this chapter, in contrast to previous ones, the coefficients of the operators are no longer constants and we cannot use the Fourier transform directly to obtain solvability conditions similar to those for the operators considered in a previous chapters. Instead we use the generalized Fourier transform which is based on replacing the Fourier harmonics by the functions of the continuous spectrum of the operator −∆+V (x), which are the solutions of the Lippmann-Schwinger equation (for the details see chapter 5). Chapter 5 consists of four sections: 5.1-5.4. In section 5.1 we consider prob- lems which contain the squares of the sums of second order non-Fredholm differential operators of Schrödinger type, that is 4th order operators. In sec- tions 5.2 and 5.3 we deal with the solvability in the sense of sequences of the operator equation consisting of the squares of the sums of second order non- Fredholm differential operators of Schrödinger type with a single potential
- 14. Preface xi both in the regular and singular cases. The sum of free negative Laplacian and the Schrödinger type operator has the meaning of the cumulative Hamil- tonian of the two non-interacting quantum particles, one of these particles moves freely and the other interacts with an external potential. The last sec- tion of chapter 5, that is section 5.4, is devoted to the solvability of generalised Poisson type equations with a scalar potential. I would like to thank many friends and colleagues who gave me helpful sug- gestions, advice and support. In particular, I wish to thank G. Akagi, H. Berestycki, N. Dancer, Y. Du, Y. Enatsu, F. Hamel, F. Hamdullahpur, M. Otani, S. Sivagolonathan, C.A. Stuart, E. Valdinochi, V. Vougalther, J.R.L. Webb, W.L. Wendland, J. Wu, A. Zaidi. Furthermore, I am greatly indebted to my colleagues at the Institute of Computational Biology in the Helmholtz Center Munich and Technical University of Munich, Marmara University in Istanbul, Alexander von Humboldt Foundation, as well as the Springer book series for their efficient handling of publication. I started to write this book when I visited the Fields Institute with a Fields Research Fellowship. I would like to express my sincere gratitude to the Fields Institute for providing an excellent and unique scientific atmosphere. In par- ticular, my thanks go to my colleagues, friends and staffs in the Fields Insti- tute, namely to Kumar Murthy, Esther Berzunza, Miriam Schoeman, Bryan Eelhart and Tyler Wilson. Last but not least, I wish to thank my family for constantly encouraging me during the writing of this book.
- 15. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Auxiliary Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Functional spaces and embedding theorems . . . . . . . . . . . . . . . . 1 1.2 Linear Elliptic Boundary value problems . . . . . . . . . . . . . . . . . . 4 1.3 Superposition operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Pseudodifferential operators. Definitions and examples . . . . . . 17 1.5 Linear Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Fourier transform and related topics . . . . . . . . . . . . . . . . . . . . . . 36 1.7 On the necessary conditions for preserving the nonnegative cone: double scale anomalous diffusion . . . . . . . . . . . . . . . . . . . . 44 1.8 The Lippman-Schwinger equation: the generalized Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2 Solvability in the sense of sequences: non-Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1 Non-Fredholm equations with normal diffusion and drift in the whole line: scalar case: scalar case . . . . . . . . . . . . . . . . . . . . . 66 2.2 Non-Fredholm equations in a finite interval with normal diffusion and drift: scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3 Non-Fredholm systems with normal diffusion and drift in the whole line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 Non-Fredholm systems in a finite interval with normal diffusion and drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 xiii
- 16. xiv Contents 3 Solvability of some integro-differential equations with drift and superdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1 The whole real line case: scalar equation. . . . . . . . . . . . . . . . . . . 100 3.2 The problem on the finite interval: scalar equation . . . . . . . . . . 104 3.3 The whole real line case: system case. . . . . . . . . . . . . . . . . . . . . . 123 3.4 The problem on the finite interval: system case . . . . . . . . . . . . . 127 4 Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion . . . . . . . . 143 4.1 Mixed-diffusion: scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Mixed-diffusion: system case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5 Non-Fredholm Schrödinger type operators . . . . . . . . . . . . . . . . 177 5.1 Solvability in the sense of sequences with two potentials . . . . . 183 5.2 Solvability in the sense of sequences with Laplacian and a single potential: regular casel: regular case . . . . . . . . . . . . . . . . . 187 5.3 Solvability in the sense of sequences with Laplacian and a single potential: singular case: singular case . . . . . . . . . . . . . . . . 190 5.4 Generalized Poisson type equation with a potential . . . . . . . . . 196 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
- 17. Chapter 1 Auxiliary Materials 1.1 Functional spaces and embedding theorems We shall use the following notation. We shall denote by R, C, Z and N the sets of real, complex, integer and natural numbers respectively; Z+ = {x ∈ Z | x ≥ 0} is the set of nonnegative integers. Rn is the standard real vector space of dimension n. We denote by Di the operator of partial differentiation with respect to xi, Diu = ∂u ∂xi (i = 1, . . . n). (1.1) As usual, we use multi-index notation to denote higher order partial deriva- tives, Dγ = Dγ1 1 · · · Dγn n , |γ| = γ1 + · · · + γn (1.2) is a partial derivatives of order |γ|, for a given γ = (γ1, · · · , γn), γi ∈ Z+. Let u : Ω ⊂ Rn be a real function defined on a bounded domain Ω. The space of continuous functions over the bounded domain Ω̄ is denoted by C(Ω̄); the norm in C(Ω̄) is defined in a standard way: ∥u∥C(Ω̄) = sup{|u(x)| | x ∈ Ω̄}. (1.3) The space Cm (Ω) consists of all real functions on Ω which have continuous partial derivatives up to order m. By definition, u belongs to Cm (Ω̄) iff (ab- breviation for if and only if) u ∈ Cm (Ω) and u and all its partial derivatives up to order m can be extended continuously to Ω̄. Let 0 < γ < 1 and k ∈ Z+. By definition Ck,γ (Ω̄) denotes the Hölder space of functions u : Ω → R such that, Dα u : Ω → R exists and is uniformly continuous when |α| = k and such that 1 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Efendiev, Linear and Nonlinear Non-Fredholm Operators, https://doi.org/10.1007/978-981-19-9880-5_1
- 18. 2 1 Auxiliary Materials |u|k,γ ≡ sup |Dα u(x) − Dα u(y)| |x − y|γ x, y ∈ Ω, x ̸= y, |α| ≤ k (1.4) is finite. For u ∈ Ck,γ (Ω̄), we set ∥u∥k,γ = |u|k,γ + X |α|≤k max{|Dα u(x)| | x ∈ Ω̄}. (1.5) We also have Ck,γ (∂Ω) = {φ : ∂Ω → R | there exists u ∈ Ck,γ (Ω̄) with u|∂Ω = φ}, and for φ ∈ Ck,γ (∂Ω) we set ∥φ∥k,γ = inf{∥u∥k,γ : u|∂Ω = φ; u ∈ Ck,γ (Ω̄)}. (1.6) In cases when it is clear from the context where the function under con- sideration is defined, we shall sometimes simply write u ∈ Ck instead of, for example, u ∈ Ck (Rn ). In several examples we shall use the spaces of functions that are 2π-periodic in every variable xi (i = 1, · · · , n). We shall consider such functions as being defined on the n-dimensional torus Tn = Rn /(2πZ)n . We denote by Lp (Ω), 1 ≤ p ∞, the space of measurable functions with the finite norm ∥u∥0,p = ∥u∥Lp = Z Ω |u(x)|p dx 1/p . (1.7) We denote by L∞ (Ω) the space of almost everywhere bounded functions, ∥u∥0,∞ = ∥u∥L∞ = vrai sup{|u(x)| | x ∈ Ω} (1.8) (for continuous functions this norm coincides with the norm of C(Ω̄)). The norm in the Sobolev space Wl,p (Ω), l ∈ Z+, 1 ≤ p ∞, is defined by the formula ∥u∥l,p = X |α|≤l ∥Dα u∥p Lp 1/p . (1.9) In the case p = 2 this Sobolev space is a Hilbert space and is denoted by Hl (Ω), Hl (Ω) = Wl,2 (Ω). The scalar product in Hl (Ω) is defined by the formula (u, v)l = X |α|≤l Z Ω Dα u(x) · Dα v(x)dx. (1.10) The space Wl,p (Ω) is the completion of Cl (Ω) with respect to the norm (1.9).
- 19. 1.1 Functional spaces and embedding theorems 3 The norms Ck,γ (Tn ) and Wl,p (Tn ) are defined by (1.4) and (1.9) with Ω = (]0, 2π[). The scalar product and the norm in Hl (Tn ), which are equivalent to those defined by (1.10), are defined in terms of Fourier coefficients, (u, v)l = X û(ξ) · v̂(ξ) · (1 + |ξ|2 )l ; ∥u∥2 l = ⟨u, u⟩l, (1.11) where the summation is over ξ ∈ Zn ; the bar denotes complex conjugation; û(ξ) and v̂(ξ) are the Fourier coefficients, û(ξ) = (2π)−n Z u(x)e−ix·ξ dx. (1.12) Here the integral is over [0, 2π]n ; x · ξ = x1ξ1 + · · · + xnξn. (1.13) The formula (1.11) defines the norm in Hl (Tn ) for l ∈ R as well as l ∈ Z+. We denote by C∞ (Ω̄) the space k≥0 Ck (Ω̄); by C∞ 0 (Ω) the set of functions from C∞ (Ω̄) which vanish on a neighbourhood of the boundary ∂Ω. We shall use also spaces of functions which vanish on ∂Ω. In this case we shall denote the corresponding space as follows: Ck,γ (Ω̄) ∩ {u|∂Ω = 0}, Wl,p (Ω) ∩ {u|∂Ω = 0}. (1.14) We denote the completion of C∞ 0 (Ω) with respect to the norm of Hl (Ω) by Hl 0(Ω) and with respect to the norm of W1,p (Ω) by W1,p 0 (Ω). It is well-known that H1 0 (Ω) = H1(Ω) ∩ {u|∂Ω = 0}; W1,p 0 (Ω) = W1,p ∩ {u|∂Ω = 0}. (1.15) The Sobolev spaces Hρ (Ω) with noninteger ρ ≥ 0, ρ = k+β, k ∈ Z, 0 ≤ β 1 are endowed with the norm ∥u∥2 ρ = ∥u∥2 k + Z |y|≤δ ∥u(x + y) − u(x)∥2 k · |y|−n−2β dy (1.16) (u(x) is extended over a δ-neighbourhood of the boundary, see [50]). By S(Rn ) we denote the class of rapidly decreasing (at ∞) functions u(x) ∈ C∞ (Rn ), with (1 + |x|)k |Dα u(x)| ≤ Ck,α for each α = (α1, · · · , αn) ∈ Zn + and k ∈ Z+, where Ck,α are constants.
- 20. 4 1 Auxiliary Materials Recall that an operator j : X → Y between Banach spaces with X ⊆ Y is an embedding iff j(x) = x for all x ∈ X. The operator j is continuous iff ∥x∥Y ≤ constant ∥x∥X for all x ∈ X. Further, j is compact iff j is continuous, and every bounded set in X is relatively compact in Y . If the embedding X ,→ Y is compact, then each bounded sequence {xn} in X has a subsequence {xn′ } which is convergent in Y . We shall widely use Sobolev’s embedding theorems formulated below. Theorem 1.1 Let Ω be a bounded domain in Rn , with smooth boundary ∂Ω and 0 ≤ k ≤ m − 1 (See [50]). Then Wm,p (Ω) ,→ Wk,q (Ω), if 1 q ≥ 1 p − m − k n 0, (1.17) Wm,p (Ω) ,→ Wk,q (Ω), if q ∞, 1 p = m − k n , (1.18) Wm,p (Ω) ,→ Ck,δ (Ω̄), if n p m − (k + δ), 0 δ 1. (1.19) The first embedding is compact if 1 q 1 p − m−k n . The last two embeddings are compact. Theorem 1.2 Let 0 ≤ β α ≤ 1 or α, β ∈ Z with 0 ≤ β α (see [63]). Then the embedding Cα (Ω̄) ,→ Cβ (Ω̄) is compact (1.20) and for k + β m + α, with 0 ≤ α, β ≤ 1, m ≥ k ≥ 0 the embeddings Cm,α (Ω̄) ,→ Ck,β (Ω̄) are compact. (1.21) 1.2 Linear Elliptic Boundary value problems Notation. Let Ω be a bounded domain in Rn . For α = (α1, · · · , αn) an n-tuple of nonnegative integers, recall that Dα = n Y i=1 ∂ ∂xi αi , |α| = n X i=1 αi and let ξα = n Y i=1 (ξi)αi if ξ ∈ Cn . Every linear differential operator L of order 2m (m ∈ N) has the form Lu = X |α|≤2m aα(x) · Dα u. (1.22)
- 21. 1.2 Linear Elliptic Boundary value problems 5 All coefficients aα(x) are assumed to be real. The partial differential operator defined by (1.22) is called elliptic of order 2m if its principal symbol , p0(x, ξ) = X |α|=2m aα(x) · ξα has the property that p0(x, ξ) ̸= 0 for all x ∈ Ω, ξ ∈ Rn {0}. The differential operator L defined by (1.22) is called uniformly elliptic in Ω, if there is some c 0, such that (−1)m X |α|=2m aα(x)ξα ≥ C|ξ|2m for every x ∈ Ω, ξ ∈ Rn {0}. (1.23) Throughout we assume that ∂Ω is a smooth (n − 1)-manifold. Suppose now that L is elliptic and of order 2m. Let {mi, 1 ≤ i ≤ m} be distinct integers with 0 ≤ mi ≤ 2m − 1, and suppose that for 1 ≤ i ≤ m we prescribe a differential operator Bi of order mi on ∂Ω, by Biu(x) = X |α|≤mi bα,i(x)Dα u(x), i = 1, · · · , m. (1.24) The family of boundary operators B = {B1, · · · , Bm} is said to satisfy the Shapiro-Lopatinski covering condition with respect to L provided that the following algebraic condition is satisfied. For each x ∈ ∂Ω, ⃗ N ∈ Rn {0} normal to ∂Ω at x and ξ ∈ Rn {0} with ⟨ξ, ⃗ N⟩ = 0, consider the (m + 1) polynomials of a single complex variable τ 7−→ p0(x, ξ + τ ⃗ N), τ 7−→ X |α|=mi bα,i(x) · (ξ + τ ⃗ N)α ≡ p0,i(x, ξ, τ), 1 ≤ i ≤ m. (1.25) Let τ+ 1 , · · · , τ+ m be the m complex zeros of p0(x, ξ + τ ⃗ N) which have positive imaginary part. Then {p0,i(τ)}m i=1 are assumed to be linearly independent modulo m Y i=1 (τ − τ+ i ) = M+ (x, ξ, ⃗ N, τ), i.e., after division by M+ (x, ξ, N, τ) all the various remainders are linearly independent. In other words, let p′ 0,i(x, ξ, ⃗ N, τ) = m−1 X k=0 bi,k(x, ξ, ⃗ N) · τk , i = 1, · · · , m be the remainders after division by M+ (x, ξ, ⃗ N, τ). Then the condition of the Shapiro-Lopatinski implies that
- 22. 6 1 Auxiliary Materials D(x, ξ, N) = det ∥bik(x, ξ, ⃗ N)∥ ̸= 0 (1.26) for all x ∈ ∂Ω, and for all ⃗ N ∈ Rn {0} normal to ∂Ω at x and ξ ∈ Rn {0} with ⟨ξ, ⃗ N⟩ = 0. Definition 1.1 We say that (L, B1, · · · , Bm) defines an elliptic boundary value problem of order (2m, m1, · · · , mm) if L given by (1.22), is uniformly elliptic and of order 2m, each Bi given by (1.24) has order mi, 0 ≤ mi ≤ 2m − 1, the mi’s are distinct, ∂Ω is non characteristic to Bi at each point and {Bi}m i=1 satisfy the Shapiro-Lopatinski condition with respect to L (see [47]). We have the following lemma (see [4, 33]). Lemma 1.1 Let (L, B1, · · · , Bm) define an elliptic boundary value problem of order (2m, m1, · · · , mm). Then L ◦ △l , B1 ◦ ˜ △l , · · · , Bm ◦ ˜ △l , L ◦ ∂u ∂ ⃗ N defines an elliptic boundary value problem of order (2k+2l, m1+2l, · · · , mm+ 2l, 2m + 1) where ˜ △ is the Laplace-Beltrami operator, l ∈ N. Proof. The principal symbol of L ◦ △l is |ξ|2l · p0(x, ξ), so it is clear that L ◦ △l is uniformly elliptic. Let x ∈ ∂Ω and ξ, ⃗ N ∈ Rn {0}, with ⟨ξ, ⃗ N⟩ = 0 and ⃗ N normal to ∂Ω at x. It is obvious that the principal symbol operators Bi ◦ ˜ △l and L◦ ∂ ∂ ⃗ N at ξ+τ ⃗ N are ψl(ξ) · p0i(x, ξ + τ ⃗ N) and τp0(x, ξ + τ ⃗ N) respectively, where ψl(ξ) ̸= 0. If τ+ 1 , · · · , τ+ m are the m roots of p0(x, ξ + τ ⃗ N) = 0 having positive imaginary part, then the m + 1 roots of |ξ + τ ⃗ N|2 · p0(x, ξ + τ ⃗ N) = 0 with positive imaginary part are given by τ+ 1 , · · · , τ+ m, i · | ⃗ N| |ξ| . We must show that if λ1, · · · , λm+1 ∈ C and h(τ) is a polynomial with ψl(ξ) m X i=1 λi · p0i(x, ξ + τ ⃗ N) + λm+1τp0(x, ξ + τ ⃗ N) = h(τ) · τ − i| ⃗ N| |ξ| ! · m Y i=1 (τ − τ+ i ) (1.27) then λi = 0, 1 ≤ i ≤ m + 1 and h(τ) ≡ 0. Due to the assumption that (B1, · · · , Bm) satisfy the covering condition it is not difficult to see that λ1 = · · · = λm = 0. But then the right-hand side of (1.27) has more roots with positive imaginary part than does the left-hand side, so that λm+1 = 0 and h(τ) ≡ 0. ⊓ ⊔
- 23. 1.2 Linear Elliptic Boundary value problems 7 With appropriate smoothness conditions on the coefficients (see Lemma 2.2 below), elliptic boundary value problems induce linear Fredholm operators in Sobolev spaces. Here the spaces W2m+k−mi−1/p,p (∂Ω) with the fractional differentiation order 2m + k − mi − 1 p play a decisive role. Before giving a precise definition we wish to point out a priori the most important property of these spaces, i.e. the surjective boundary operator T : C∞ (Ω̄) → C∞ (∂Ω) which assigns to each function u ∈ C∞ (Ω̄) its classical boundary value Tu on ∂Ω, can be extended uniquely to a continuous linear surjective operator T : W2m+k,p (Ω) → W2m+k−mi−1/p,p (∂Ω). Here k ≥ 0 and m ≥ 1 are integers, and 1 p ∞ (we are mainly interested in the case p = 2, W2m,2 (Ω) = H2m (Ω)). Then Tu is described naturally as the generalized boundary value of u ∈ W2m+k,p (Ω). These functions u have generalized derivatives Dα u up to order 2m + k on Ω. The functions Dα u with|α| ≤ mi have generalized boundary values which all lie in W2m+k−mi−1/p,p (∂Ω), since mi 2m. Consequently, Biu ∈ W2m+k−mi−1/p,p (∂Ω) also. The differential operators L and the boundary operator Bi are thus to be understood in the space of generalized derivatives on Ω and as generalized boundary values respectively. Definition of the space Wm−1/p,p (∂Ω). Let Ω be an open subset of Rn with sufficiently smooth boundary and {Ui}l i=1 be an open covering of Ω̄ with diffeomorphisms φi : Ui → Rn , φi ∈ Cm (Ui), such that φi(Ui) = V1 = {y ∈ Rn | |y| 1} if Ui ⊂ Ω, and φi(Ui ∩ Ω̄) = V + 1 = {y ∈ Rn | |y| 1, yn ≥ 0}, φi(Ui ∩ ∂Ω) = Ṽ1 = {y ∈ Rn | |y| 1, yn = 0} if Ui ∩ ∂Ω ̸= ∅. Let χi(x) be a partition of unity subordinated to {Ui}l i=1 and let λi(y) := χi(φ−1 i (y)). For each u(x) ∈ Cm (∂Ω), 0 δ 1, p 1 we define the norm:
- 24. 8 1 Auxiliary Materials ∥u∥′ m−δ,p,∂Ω = X i∈I′ X |α|≤m−1 ′ Z Ṽ1 |Dα y (λi(y) · ui(y))|p dy′ + X |α| =m−1 ′ Z Ṽ1 Z Ṽ1 |Dα y (λi(y) · ui(y)) − Dα z (λi(z) · ui(z)) |p · dy′ dz′ |y′ − z′|n+p−1−δp 1 p , (1.28) where ui(y) = u(φ−1 i (y)), y′ = (y1, · · · , yn−1), I′ ⊂ {1, · · · , l} such that Ui ∩ ∂Ω ̸= ∅ and P′ implies that the sum is taken over those α for which αn = 0, α = (α1, · · · , αn). By definition, the norm in Wm− 1 p ,p (∂Ω), p 1 is defined as the norm ∥ · ∥′ m− 1 p ,p,∂Ω . For more details see [50]. Let us return to the discussion of elliptic boundary value problems . We first recall some results regarding linear Fredholm operators. Let X and Y be real Banach spaces. By L(X, Y ) we denote the Banach space of bounded linear operators from X to Y . An operator T in L(X, Y ) is called Fredholm if the kernel (nullspace) Ker T := {x ∈ X | Tx = 0} has finite dimension and the image (range) of T, R(T) := {Tx | x ∈ X} is of finite codimension in Y , that is codim R(T) = dim Y/R(T) ∞. For a Fredholm operator T : X → Y , the numerical Fredholm index of T, ind(T) is defined by ind(T) = dim Ker T − codim(R(T)). Lemma 1.2 Let Ω ⊂ Rn be open and bounded with ∂Ω smooth. Suppose that s n/2, aα ∈ Hs (Ω) if |α| ≤ 2m, while bα,i ∈ Hs+2m−mi (∂Ω) and i = 1, · · · m. Then the following three assertions are equivalent: (i) The operator A = (L, B1, · · · , Bm) A : Hs+2m (Ω) −→ Hs (Ω) × m Y i=1 Hs+2m−mi (∂Ω) (1.29) is an elliptic boundary value problem of order (2m, m1, · · · , mm) (ii) The operator A = (L, B1, · · · Bm) is Fredholm (iii) There is some c 0, such that if u ∈ Hs+2m (Ω), then ∥u∥2m+s ≤ c ∥Lu∥s + m X i=1 ∥Bi(x, D)u∥2m+s−mi− 1 2 + ∥u∥s # . (1.30) Proof. If each aα ∈ Cs (Ω) and each bα,i ∈ C2k+s−mi (Ω), then a priori estimate (1.30) is contained in [1]. It is not difficult to see that (1.30) also
- 25. 1.3 Superposition operators 9 holds under the present smoothness conditions. Thus, in fact a priori estimate (1.30) and equivalence (i) and (iii) follows from [1]. Equivalence (i) and (iii) to (ii) can be proved analogously to [2]. ⊓ ⊔ Remark 1.1 Of course, the Fredholm index of (L, B1, · · · , Bm) need not be equal to 0. If L is uniformly elliptic and Biu(x) = ∂ ∂ ⃗ N i−1 u(x) for 1 ≤ i ≤ m, then the index A = (L, B1, · · · , Bm) : H2m+s (Ω) → Hs (Ω) × m Y i=1 H2m+s−mi− 1 2 (∂Ω) is 0 (see [4, 46]). Remark 1.2 (Cγ -theory) The a priori estimates(1.30) remain valid if we choose the following B- spaces for 0 γ 1: X = C2m+s,γ (Ω̄), Y = Cs,γ (Ω̄), Z = C(Ω̄), Yj = C2m+s−mi,γ i.e. ∥u∥X ≤ constant(∥Lu∥Y + m X j=1 ∥Bju∥Yj + ∥u∥Z). (1.31) Remark 1.3 The important fact is that the index of corresponding opera- tors is the same in both theories. Remark 1.4 As shown in [1, 2] the terms ∥u∥s and ∥u∥Z in (1.30), (1.31) disappear if dim Ker A = {0}, where Au = (Lu, B1u, · · · , Bmu). 1.3 Superposition operators The investigation of nonlinear equations in the following chapters relies on properties of mappings of the form u 7→ f(u) in the spaces Cα (Ω̄) and Lp (Ω), Hl (Ω). Definition 1.2 Let Ω ⊂ Rn be a domain. We say that a function Ω × Rm ∋ (x, u) 7−→ f(x, u) ∈ R satisfies the Carathéodory conditions if u 7−→ f(x, u) is continuous for almost every x ∈ Ω and
- 26. 10 1 Auxiliary Materials x 7−→ f(x, u) is measurable for every u ∈ Ω. Given any f satisfying the Carathéodory conditions and a function u : Ω → Rm , we can define another function by composition Fu(x) := f(x, u(x)). (1.32) The composed operator F is called a Nemytskii operator. In this section we state some important results on the composition of Cα (Ω̄), Lp (Ω), Hl (Ω) with nonlinear functions (some of them without proof [42, 63]). Proposition 1.1 Let Ω ⊂ Rn be a bounded domain and Ω × Rm ∋ (x, u) 7−→ f(x, u) ∈ R satisfy the Carathéodory conditions. In addition, let |f(x, u)| ≤ f0(x) + c(1 + |u|)r (1.33) where f0 ∈ Lp0 (Ω), p0 ≥ 1, and rp0 ≤ p1. Then the Nemytskii operator F defined by (1.32) is bounded from Lp1 (Ω) into Lp0 (Ω), and satisfies ∥F(u)∥0,p0 ≤ C1 · (1 + ∥u∥r p1 ) (1.34) Proof. By (1.33) and (1.7) ∥F(u)∥0,p0 ≤ ∥f0(x)∥0,p0 + C∥1∥0,p0 + C∥|u|r ∥0,p0 ≤ C′ + C Z Ω |u|rp0 dx 1 p0 = C′ + ∥u∥r 0,p0r. (1.35) Since Ω is bounded, then by Hölder’s inequality ∥v∥0,q ≤ C(Ω)∥v∥0,p when 1 ≤ q ≤ p, v ∈ Lp (Ω) (1.36) where C(Ω) = mes(Ω) 1 q − 1 p . Inequalities (1.35) and (1.36) with q = rp0 and p = p1 imply (1.34). ⊓ ⊔ It is well-known that the notions of continuity and boundedness of a nonlinear operator are independent of one another ([42]). It turns out that the following is valid. Theorem 1.3 Let Ω ⊂ Rn be a bounded domain and let Ω × Rm ∋ (x, u) 7−→ f(x, u) ∈ R satisfy the Carathéodory conditions. In addition, let p ∈ (1, ∞) and g ∈ Lq (Ω) (where 1 p + 1 q = 1) be given, and let f satisfy
- 27. 1.3 Superposition operators 11 |f(x, u)| ≤ C|u|p−1 + g(x). Then the Nemytskii operator F defined by (1.32) is a bounded and continuous map from Lp (Ω) to Lq (Ω). For a more detailed treatment, the reader can consult [42, 63]. Theorem 1.4 Let Ω be a bounded domain in Rn with smooth boundary and let Ω × R ∋ (x, u) 7→ f(x, u) ∈ R satisfy the Carathéodory conditions. Then for s n/2, f induces 1) a continuous mapping from Hs (Ω) into Hs (Ω) if f ∈ Cs , 2) a continuously differentiable mapping from Hs (Ω) into Hs (Ω) if f ∈ Cs+1 . Proof. First we consider the simplest case, that is f = f(u) is independent of x. By the Sobolev embedding theorem, we have Hs (Ω) ⊂ C(Ω̄). Hence we have f(u) ∈ C(Ω̄) for everyu ∈ Hs (Ω). Moreover, if u is in C(s) (Ω̄), we can obtain the derivatives of f(u) by the chain rule, and in the general case, we can use approximation by smooth functions. Note that all derivatives of f(u) have the form of a product involving a derivative of f and derivatives of u. The first factor is in C(Ω̄), while any l-th derivative of u lies in Hs−l (Ω), which imbeds into L2n/(n−2(s−l)) (Ω) if s − l n 2 . We can use this fact and Hölder’s inequality to show that all derivatives of f(u) up to order s are in L2 (Ω); moreover, it is clear from this argument that f is actually continuous from Hs (Ω) into Hs (Ω). A proof of the differentia- bility in this special case is that f = f(u) is based on the relation f(u) − f(v) = Z 1 0 f′ u(v + θ(u − v))(u − v)dθ and the same arguments as before. Let us now consider the general case, that is f = f(x, u). Let |α| ≤ s. We must show that u 7−→ Dα F(u) (1.37) defines a continuous map of Hs (Ω) into L2 (Ω). It is not difficult to see that (1.37) is a finite sum of operators of the form u(x) 7−→ g(x, u(x)) · Dγ u(x) (1.38) where |γ| = γ1 +· · ·+γn ≤ s, while g is a partial derivative of f order at most s. It is obvious that Dγ is continuous from Hs (Ω) into L2 (Ω) for |γ| ≤ s. On the other hand, the continuous embedding of Hs (Ω) inC(Ω̄) implies that
- 28. 12 1 Auxiliary Materials u(x) 7−→ g(x, u(x)) is continuous from Hs (Ω) into C(Ω̄). Thus u(x) 7−→ g(x, u(x)) · Dγ u(x) defines a continuous map of Hs (Ω) into L2 (Ω) and hence so does u 7−→ Dα F(u). ⊓ ⊔ As before, let p ∈ N and p̃ denote the number of multi-indices with |α| ≤ p and let Ω be a bounded domain in Rn . Corollary 1.1 An analogous result is valid for a continuity of the operator Fu(x) = f(x, u(x), · · · , Dp u(x)) : Hs+p (Ω) → Hs (Ω) where p, s ∈ N with s n 2 and f : Ω × Rp̃ → R is Cs . Corollary 1.2 Let p, s ∈ N with s n 2 and f : Ω × Rp̃ → R be Cs+1 . Then the operator F : Hs+p (Ω) → Hs (Ω) defined by Fu(x) = f(x, u(x), · · · , Dp u(x)) is Fréchet differentiable from Hs+p (Ω) into Hs (Ω). We will show some continuity and C1 -differentiability results for a nonlinear differential operator of the form Au(x) = f(x, u(x), ..D2p u(x)) in Hölder spaces. They are based on the following Theorems 1.5 and 1.6. Theorem 1.5 Let the function f(x, y) = f(x, y1, · · · , yp̃) be defined on Ω̄ × Rp̃ and satisfy the following conditions: 1) f(x, 0) = 0 2) For any R 0, sup |y|≤R ∂2 f ∂yi∂yj ≤ C(R), sup |y|≤R ∥f∥C1,α(Ω̄) ≤ C(R), where C(R) is constant depending on R. Let u1(x), · · · , up̃(x) ∈ Cα (Ω̄), 0 α 1, ∥ui∥Cα(Ω̄) ≤ R, i = 1, . . . , p̃. Then ∥f(x, u1(x), · · · , up̃(x))∥Cα(Ω̄) ≤ C1(R) · p̃ X i=1 ∥ui∥Cα(Ω̄). (1.39) Proof. Obviously,
- 29. 1.3 Superposition operators 13 f(x, y, . . . , yp̃) = Z 1 0 d dt f(x, ty1, . . . , typ̃)dt = p̃ X j=1 yj Z 1 0 ∂f(x, ty1, . . . , typ̃) ∂yj dt = p̃ X j=1 φj(x, y1, . . . , yp̃) · yj where φj(x, y1, . . . , yp̃) = Z 1 0 ∂f(x, ty1, . . . , typ̃) ∂yj dt. Hence f(x, u1(x), · · · , up̃(x)) = p̃ X j=1 φj(x, u1(x), · · · , up̃(x)) · uj(x). Since Cα (Ω̄), 0 α 1 is a Banach algebra, we have ∥f(x, u1(x), · · · up̃(x))∥Cα ≤ p̃ X j=1 ∥φj(x, u1(x), · · · up̃(x))∥Cα · ∥uj∥Cα . Hence we have to prove that sup |y|≤R ∥φj(x, u1(x), · · · , up̃(x))∥Cα ≤ C1(R). Indeed |φj(x + ξ, u1(x + ξ), · · · , up̃(x + ξ)) − φj(x, u1(x), · · · , up̃(x))| ≤ |φj(x+ξ, u1(x+ξ), · · · , up̃(x+ξ))−φj(x, u1(x+ξ), · · · , up̃(x+ξ))| +|φj(x, u1(x + ξ), · · · , up̃(x + ξ)) − φj(x, u1(x), · · · , up̃(x))|. (1.40) The first term on the right-hand side of (1.40) is bounded by C(R)·|ξ|α . The second term is bounded by sup |y|≤R ∂φj ∂yk ·|φj(x, u1(x+ξ), · · · , up̃(x+ξ))−φj(x, u1(x), · · · , up̃(x))| ≤ CRR|ξ|α . (1.41) The estimates (1.40) and (1.41) yield (1.39). ⊓ ⊔ Theorem 1.6 Let the function f(x, y) = f(x, y1, · · · , yp̃) be defined on Ω̄ × Rp̃ satisfy the following conditions: 1) f(x, 0) = 0, gradyf(x, 0) = 0
- 30. 14 1 Auxiliary Materials 2) For any R 0, sup |y|≤R ∥f(x, y)∥C2,α(Ω̄) ≤ C(R) and sup |y|≤R ∂3 f ∂yi∂yj∂yk ≤ C(R), where C(R) is constant depending on R. Let as before, u1(x), · · · , up̃(x) ∈ Cα (Ω̄) with ∥ui∥Cα(Ω̄) ≤ R, i = 1, · · · , p̃. Then the following estimate holds. ∥f(x, u1(x), · · · , up̃(x))∥Cα(Ω̄) ≤ C2(R) · p̃ X i=1 ∥ui∥2 Cα . (1.42) Proof. Obviously we have f(x, y1, · · · , up̃) = p̃ X i,j=1 gij(x, y1, · · · , yp̃) · yi · yj so we can write f(x, u1(x), · · · , up̃(x)) = p̃ X i,j=1 gij(x, u1(x), . . . , up̃(x)) · ui(x) · uj(x) and we have ∥f(x, u1(x), · · · , up̃(x)∥Cα(Ω̄) ≤ p̃ X i,j=1 ∥gij(x, u1(x), · · · , up̃(x)∥Cα ∥ui∥Cα ∥uj∥Cα (1.43) Due to Theorem 1.5 we obtain ∥gij(x, u1(x), · · · , up̃(x)∥Cα(Ω̄) ≤ C0(R) (1.44) Hence the estimates (1.43) and (1.44) yield (1.42) ∥f(x, u1(x), · · · , up̃(x)∥Cα(Ω̄) ≤ C2(R) · p̃ X i=1 ∥ui∥2 Cα . This completes the proof. ⊓ ⊔ We apply Theorems 1.5 and 1.6 to the operator Au(x) = f(x, u(x), · · · , D2p u(x)) where the function f(x, y1, · · · , yp̃) satisfy conditions of Theorems 1.5 and 1.6, respectively. Hence we have
- 31. 1.3 Superposition operators 15 ∥Au∥C2p,α ≤ C(R) · ∥u∥Cα . Moreover as it follows from Theorem 1.6 A ∈ C1 , A′ (0) = 0 and ∥A′ (u + h) − A′ (u)∥L(C2p,α,Cα) ≤ C · ∥h∥C2p,α(Ω̄). Remark 1.5 As shown in the proofs of Theorems 1.5 and 1.6, continuity and differentiability of the operator Au(x) = f(x, u(x), · · · , D2p u(x)) between C2p,α (Ω̄) and Cα (Ω̄) remains valid under slightly weaker conditions on a given function f(x, y1, · · · , yp̃). We leave these as exercises for the reader. In the investigation of nonlinear boundary value problems related to pseudod- ifferential operators and in particular nonlinear Riemann-Hilbert problems we need properties of the Nemytskii operators in the spaces Hs (S1 ) or Cp,α (S1 ), where S1 is the unit circle. We recall some of the properties which will be used often in the sequel. The norm in Cα (M) is given by ∥f∥Cα(M) = ∥f∥C + sup x̸=y |f(x) − f(y)| |x − y|α , M = S1 . As before, by Ck,α (M) we denote the space of Hölder continuous functions, which have derivatives up to order k, with Dk f ∈ Cα (M). Let F be a super- position operator defined by Fu(x) = f(x, u(x)), x ∈ M. The following theorems are not hard to prove (although not obvious). Theorem 1.7 Let k ∈ R+. Then the superposition operator F : E1 → E2 defined by Fu(x) = f(x, u(x)) acts as a bounded operator in each of the following cases (see also [62]) 1) f ∈ C(S1 × R, R), E1 = C(S1 ), E2 = C(S1 ) 2) f ∈ C1 (S1 × R, R), E1 = Cα (S1 ), E2 = Cα (S1 ), 0 α 1 Theorem 1.8 Let k ∈ R+, 0 α 1. Then the superposition operator F : E1 → E2 defined by Fu(x) = f(x, u(x)) is m times continuously differentiable in each of the following cases 1) D0,j f ∈ Ck (S1 × R, R), E1 = Ck (S1 ), E2 = Ck (S1 ) 2) D0,j f ∈ Ck+1 (S1 × R, R), E1 = Ck,α (S1 ), E2 = Ck,α (S1 ), The j-th derivative of F is given by D0,j F(x, u(x))h1(x) . . . hj(x) = Dj F(f)(h1, . . . hj)(x).
- 32. 16 1 Auxiliary Materials Analogous results are valid in Sobolev spaces : Theorem 1.9 Let X = Y = Hs (S1 )(s ≥ 1) be the Sobolev space of real functions x(τ) on the circumference of a circle, where 0 ≤ τ 2π; f(τ, x) is a smooth real function, x ∈ R, 0 ≤ τ 2π. Then the operator F : Hs (S1 ) → Hs (S1 ) defined by Fx(τ) = f(τ, x(τ)) is continuous. Proof. It is not difficult to see, that d dτ k f(τ, x(τ)) = X p+q≤k r1+···+rq=k−p rj ≥0 Cp,q,r1...rq ∂p+q f(τ, x(τ)) ∂τp · ∂xq x(r1) (τ) · · · x(rq) (τ) where Cp,q,r1...rq are some constants. If x(τ) ∈ Hs , then it follows that the derivatives {dl x(τ) dτl | 0 ≤ l ≤ s − 1} are continuous. Therefore in ds dτs f(τ, x(τ)) all terms without ones are continuous. The last term is equal to ds x(τ) dτs × Q(τ) where Q(τ) is a continuous function, hence also square integrable. As a consequence of these arguments we obtain continuity. ⊓ ⊔ Remark 1.6 An analogous result holds for vector functions, and also in the multidimensional case, for functions on arbitrary smooth compact manifold with boundary. Lemma 1.3 Let the function f ∈ C2 (R, R) satisfies C1|u|p−1 ≤ f′ (u) ≤ C1|u|p−1 , p 1, with C1 and C2 some positive constants. Then, for every s ∈ (0, 1) and 1 q ≤ ∞, we have ∥u∥W s/p,pq(Ω) ≤ Cp∥f(u)∥ 1/p W s,q(Ω) where the constant Cp is independent of u. Proof. Indeed, let f−1 be the inverse function to f. Then, due to conditions on f, the function G(v) := sgn(v)|f−1 (v)|p is nondegenerate and satisfies C2 ≤ G′ (v) ≤ C1, for some positive constants C1 and C2. Therefore, we have |f−1 (v1) − f−1 (v2)|p ≤ Cp|G(v1) − G(v2)| ≤ C′ p|v1 − v2|, for all v1, v2 ∈ R. Finally, according to the definition of the fractional Sobolev spaces ,
- 33. 1.4 Pseudodifferential operators. Definitions and examples 17 ∥f−1 (v)∥pq W s/p,qp(Ω) :=∥f−1 (v)∥pq Lpq(Ω) + Z Ω Z Ω |f−1(v(x)) − f−1(v(y)|pq |x − y|n+sq dx dy ≤C∥v∥q Lq(Ω) + C′ p Z Ω Z Ω |v(x) − v(y)|q |x − y|n+sq dx dy = C′′ p ∥v∥q W s,q(Ω) , where we have implicitly used that f−1 (v) ∼ sgn(v)|v|1/p . Lemma 1.3 is proved. ⊓ ⊔ 1.4 Pseudodifferential operators. Definitions and examples Let a(x, ξ) be a C∞ function for all x ∈ Rn x and ξ ∈ Rn ξ {0} which is positively homogeneous of degree σ ≥ 0 in ξ: a(x, tξ) = tσ a(x, ξ), t 0. (1.45) Assume that on the sphere |ξ| = 1, a(x, ξ) has a limit a(∞, ξ) as x → ∞ and that a′ (x, ξ) = a(x, ξ) − a(∞, ξ) ∈ S(Rn x) uniformly in ξ. In fact we assume that the same is true for all derivatives with respect to ξ, i.e., for any integers p, α, β, (1 + |x|)p · ∂α x Dβ ξ (a(x, ξ) − a(∞, ξ)) → 0 as |x| → ∞ uniformly in ξ, for |ξ| = 1 where ∂x = 1 i ∂ ∂x , Dξ = ∂ ∂ξ . The function a(x, ξ) is called the symbol. In sections 1.4 we will mainly follow [41] specifying some details. Definition 1.3 A (homogeneous) pseudodifferential operator A(x, D) with the symbol a(x, ξ) is defined on functions u(x) ∈ S(Rn x) in the following way: A(x, D)u(x) = (2π)−n/2 Z Rn ξ eix·ξ · a(x, ξ)û(ξ)dξ. (1.46) Recall that S(Rn x) is the space of all functions on Rn x which are of class C∞ and such that |x|k |Dα u(x)| is bounded for every k ∈ N and every multi-index α. The formula (1.46) can be rewritten as follows:
- 34. 18 1 Auxiliary Materials A(x, D)u(x) =(2π)−n/2 Z eix·ξ a(∞, ξ)û(ξ)dξ + (2π)−n/2 Z eix·η a′ (x, η)û(η) dη, (1.47) where a′ (x, ξ) = a(x, ξ)−a(∞, ξ). Then it is not difficult to see, that (Fubini’s formula) (Âu)(ξ) = a(∞, ξ)û(ξ) + (2π)−n/2 Z â′ (ξ − η, η)û(η) dη, where â′ (ξ, η) = Fx→ξa′ (x, η) and û(ξ) := Fx→ξu = (2π)−n/2 Z e−ix·ξ u(x)dx. The formula (1.47) is convenient from a computational point of view. Partic- ular cases are: 1) Homogeneous differential operator A(x, D) i.e. A(x, D)u(x) = X |α|=σ aα(x) · ∂α u(x) = X |α|=σ aα(x) · F−1 ξα Fu(ξ) = (2π)−n/2 Z eix·ξ X |α|=σ aα(x) · ξα û(ξ)dξ. (1.48) Here the symbol a(x, ξ) is the characteristic polynomial, i.e. a(x, ξ) = X |α|=σ aα(x) · ξα . If aα(x) = aα(∞) + a′ α(x), then Âu(ξ) = aα(∞, ξ)û(ξ) + (2π)−n/2 Z X |α|=σ â′ α(ξ − η) · ηα û(η) dη which is an expression of the differential operator (1.48) in the form (1.47). 2) The operator ∧u = F−1 |ξ|Fu. It is obvious that ∧2 u = − △ u is the Laplace operator . The symbol of ∧u is a(x, ξ) = |ξ|. 3) In the case when σ = 0 a homogeneous pseudodifferential operator is called a homogenous singular integral operator. We set
- 35. 1.4 Pseudodifferential operators. Definitions and examples 19 a0(x) = Z |ξ′|=1 a(x, ξ′ )dξ′ . Then one can show that A(x, D)u(x) = a0(x)u(x) + lim ϵ→0 Z |x−y|ϵ K(x, x − y) |x − y|n u(y) dy (1.49) where the function K(x, z) is positive homogeneous of degree zero in z, such that (2π)−n/2 K(x, z) |z|n = F−1 ξ→x [a(x, ξ) − a0(x)]. Remark 1.7 By Hl (Rn x), −∞ l ∞ we denote the completion of S(Rn x) in the norm (1.50) ∥u∥2 l = Z (1 + |ξ|2 )l · |û(ξ)|2 dξ. (1.50) Order. A linear operator L : S(Rn x) → S(Rn x) is said to have order σ, if for each real s there exists a constant Cs such that ∥Lu∥s−σ ≤ Cs∥u∥s. (1.51) The infimum of the set of σ′ of L is called the true order of L. In the sequel we will abbreviate pseudodifferential operators by ψDO. Boundedness of ψDO with homogeneous symbol of degree σ. Lemma 1.4 Let A(x, D) be a homogeneous pseudodifferential operator with homogeneous symbol of degree σ. Then A(x, D) has order σ, i.e. ∥Au∥l−σ ≤ const. · ∥u∥l. (1.52) Proof. It is not difficult to estimate the norm of the operator A1u = F−1 a(∞, ξ)Fu (1.53) Since |a(∞, ξ)| ≤ C · |ξ|σ , then
- 36. 20 1 Auxiliary Materials ∥A1u∥2 l−σ = Z 1 + |ξ|2 l−σ · |a(∞, ξ)û(ξ)|2 dξ ≤ C2 Z 1 + |ξ|2 l · |û(ξ)|2 dξ = C2 ∥u∥2 l . It remains to check (1.52) for the operator Âu(ξ) = (2π)−n/2 Z â′ (ξ − η, η)û(η) dη, where â′ (ξ, η) = Fx→ξ (a(x, η) − a(∞, η)) . For simplicity assume that a(∞, ξ) ≡ 0. Since |â′ (ξ − η, η)| ≤ Cp |η|σ (1 + |ξ − η|2)p for sufficiently large p, it follows that (1 + |ξ|2 )l−σ · |Âu|2 ≤ const. Z (1+|ξ|2 ) l−σ 2 (1+|η|2) l−σ 2 · (1+|η|2 )l/2 (1+|ξ−η|2)p · |û(η)| dη 2 . (1.54) Since |ξ|2 ≤ 2|ξ − η|2 + 2|η|2 it follows that 1 + |ξ|2 ≤ 2(1 + |ξ − η|2 )(1 + |η|)2 ; and analogously 1 + |η|2 ≤ 2(1 + |ξ − η|2 )(1 + |ξ|2 ). Hence, 1 + |ξ|2 1 + |η|2 k ≤ 2|k| (1 + |ξ − η|2 )|k| for each k ∈ R. (1.55) Choosing p = n+1 2 + |l−σ| 2 in (1.54) we have: (1 + |ξ|2 )l−σ |Âu|2 ≤ const. Z (1 + |η|2 )l/2 · |û(η)|2 (1 + |ξ − η|2) n+1 2 dη !2 . (1.56) From the Schwarz inequality it follows that
- 37. 1.4 Pseudodifferential operators. Definitions and examples 21 Z φ(ξ − η)v(η) dη 2 ≤ Z |φ(ξ − η)| dη · Z |φ(ξ − η)| · |v(η)|2 dη. Hence Z φ(ξ − η)v(η) dη 2 0 ≤ Z |φ(η)| dη · Z Z |φ(ξ − η)| · |v(η)|2 dη dξ = Z |φ(η)|dη 2 · ∥v∥2 0. (1.57) Therefore from (1.56) and (1.57) it follows that ∥Au∥2 l−σ ≤ const. ∥u∥2 l . ⊓ ⊔ Particular cases: 1) The differential operator A(x, D)u = P |α|=σ aα(x)Dα u has order σ. 2) The singular integral operator S(x, D) has order 0. 3) The operator A(x, D)u(x) = a(x) · u(x) has order 0. Pseudodifferential operators of negative order. Let a(x, ξ) be a function which is positive homogeneous of degree σ 0 in ξ. Then it has a singularity at ξ = 0, and the formula (1.46) has no meaning. By ζ(ξ) we denote a fixed C∞ non-negative function which equals one for |ξ| 1 and vanishes for |ξ| 1 2 . By definition, a pseudodifferential operator with symbol a(x, ξ), a(x, tξ) = tσ a(x, ξ), where σ 0 and t 1 is Aξ(x, D)u(x) = (2π)−n/2 Z eixξ · ζ(ξ) · a(x, ξ) · û(ξ)dξ. (1.58) It is not difficult to prove, that Lemma 1.4 is valid for pseudodifferential operators Aξ(x, D). Remark 1.8 Independent of the value of σ, a difference Aζ1 (x, D) − Aζ2 (x, D) with the same symbol has true order equal to −∞, since ζ1(ξ) − ζ2(ξ) ∈ C∞ 0 . Therefore sometimes we will also call the operator Aζ(x, D) a pseudodifferential operator of order σ, σ ≥ 0. Let A(x, D) and B(x, D) (or Aζ1 (x, D), Bζ2 (x, D)) be pseudodifferential op- erators with symbols a(ξ) and b(ξ). Obviously A ◦ B (or Aζ1 ◦ Bζ1 ) is a pseudodifferential operator with symbol a(ξ) · b(ξ). If A(x, D) and B(x, D) are differential operators, i.e. A(x, D)u = X |α|=σ aα(x)∂α u, B(x, D)u = X |β|=λ bβ(x) · ∂β u
- 38. 22 1 Auxiliary Materials then A(x, D) ◦ B(x, D) is not a homogeneous differential operator with the symbol a(x, ξ) · b(x, ξ): A ◦ Bu = X |α|=σ aα(x)∂α X |ρ|=λ bβ(x)∂β u(x) = X |γ|≤σ+λ Cγ(x, D)u; where the symbol Cγ(x, ξ) of Cγ(x, D) is defined by Cγ(x, ξ) = 1 γ! Dγ ξ a(x, ξ) · ∂γ x b(x, ξ) where |γ| ≤ σ + λ, γ! = γ1! . . . γn! and Dγ = ∂γ1+...+γn ∂ξγ1 1 . . . ∂ξγn n . An analogous result is valid for pseudodifferential operators. Theorem 1.10 Let A(x, D) and B(x, D) be homogeneous pseudodifferential operators of orders σ and λ with the symbols a(x, ξ) and b(x, ξ). Then the following is valid: A(x, D) ◦ B(x, D) = C0u + C1u + . . . + Cρ−1u + Tρu, (1.59) where Ci(x, D) are pseudodifferential operators of order λ + σ − i with sym- bols Ci(x, ξ) = P |α|=i 1 α! Dα ξ a(x, ξ) · ∂β x b(x, ξ), i = 0, . . . , ρ − 1 and Tρu is an operator of order σ + λ − ρ, such that ∥Tρu∥l−(σ+λ−ρ) ≤ const. ∥u∥l. (1.60) Proof. In order to consider operators of arbitrary order we introduce as before C∞ non-negative functions ζi(ξ), i = 1, 2 which are equal to one for |ξ| 1 and vanish for |ξ| 1 2 . We set h(x, ξ) = ζ1(ξ) · a(x, ξ), g(x, ξ) = ζ2(ξ) · b(x, ξ). Instead of Aζ1 and Aζ2 we will write A(x, D), B(x, D). If A(x, D) = A1 + A2, B(x, D) = B1 + B2, where A1(x, D) and B1(x, D) are operators with symbols a(∞, ξ) and b(∞, ξ), then AB = A1B1 + A2B1 + A1B2 + A2B2. (1.61) For simplicity we assume that a(∞, ξ) = b(∞, ξ) = 0. Then d AB(ξ) = (2π)−n · Z ĥ(ξ − η, η) Z ĝ(η − τ, τ)û(τ)dτdη = (2π)−n/2 · Z (2π)−n/2 Z ĥ(ξ − η, η) · ĝ(η − τ, τ)dη û(τ)dτ (1.62)
- 39. 1.4 Pseudodifferential operators. Definitions and examples 23 which follows from Fubini’s theorem. According to the Taylor formula (with respect to the second variable) ĥ(ξ − η, η) = X |α|≤ρ−1 1 α! ∂α ĥ(ξ − η, τ) · (η − τ)α + ĥρ(ξ − η, η, τ) (1.63) where ĥρ is the remainder. Hence, it follows from (1.63) that AB = X |α|≤ρ−1 Φα + Tρ,1 where Φα(x, D) = (2π)−n/2 Z eixξ 1 α! Dα ξ h(x, ξ) · ∂α x g(x, ξ) û(ξ)dξ (1.64) and d Tρ,1u(ξ) = (2π)−n/2 Z Z hρ(ξ − η, η, τ) · ĝ(η − τ, τ)dηû(τ)dτ. (1.65) Consider the difference of Ci(x, D)u − P |α|=i Φα(x, D)u, where Ci(x, D)u = (2π)−n/2 Z eix·ξ ζ(ξ) X |α|=i 1 α! Dα ξ a(x, ξ) · ∂β x b(x, ξ) û(ξ)dξ (1.66) and ζ(ξ) is a C∞ non-negative function which equals one for |ξ| 1 and vanishes for |ξ| 1 2 . The difference of corresponding symbols is ζ(ξ) · Ci(x, ξ) − X |α|=i fα(x, ξ) = 1, |ξ| ≤ r/2 0, |ξ| r (1.67) where r is some positive number. Hence Tρ,2(x, D) := Ci(x, D)u− P |α|=i Φα(x, D) has true order equal to −∞. Therefore AB = ρ−1 X i=0 Ci + Tρ,1 + Tρ,2. (1.68) In order to complete the proof of Theorem 1.10, we need the estimate ∥Tρ,1(x, D)u∥l−(σ+λ−ρ) ≤ const. ∥u∥l. (1.69) Recall that
- 40. 24 1 Auxiliary Materials ĥρ(ξ − η, η, τ) = X |β|=ρ 1 β! ∂β η ĥ(ξ − η, τ + θ(η − τ)) · (η − τ)β , where 0 θ 1. Hence for σ ≥ ρ (ρ sufficiently large) we have |ĥρ(ξ − η, η, τ)| ≤ const. |η − τ|ρ (1 + |ξ − η|2)ρ (1 + |τ|2 ) σ−ρ 2 + (1 + |η|2 ) σ−ρ 2 . (1.70) An analogous estimate holds for σ ρ. This is not difficult and is left to the reader. For the function ĝ we have |ĝ(η − τ, τ)| ≤ const. (1 + |τ|2 ) λ 2 (1 + |η − τ|2)ρ , (1.71) for sufficiently large ρ. From (1.70) and (1.71) it follows that (1 + |ξ|2 ) l−(σ+λ−ρ) 2 Z |ĥρ(ξ − η, η, τ)| · |ĝ(η − τ, τ)|dη ≤ C (1 + |τ|2 )l/2 (1 + |ξ − τ|2) n+1 2 . (1.72) Inequality (1.69) follows from (1.72). This proves Theorem 1.10. ⊓ ⊔ Corollary 1.3 Let A(x, D), B(x, D) be pseudodifferential operators of order σ and λ respectively. Let C0(x, D) be a pseudodifferential operator of order σ + λ with symbol a(x, ξ) · b(x, ξ). Then A(x, D) ◦ B(x, D) − C0(x, D) has order σ + λ − 1. Corollary 1.4 Let A(x, D), B(x, D) be pseudodifferential operators of order σ and λ respectively. Then [A, B] = AB − BA has order σ + λ − 1. In particular, if A(x, D) has order σ, then ∂jA − A∂j has order σ, where ∂j = 1 i · ∂ ∂xj ; moreover, if φ(x), ψ(x) ∈ C∞ 0 (Rn ), such that φ(x) · ψ(x) = 0, then φAψ has order σ − 1. Corollary 1.5 Let A(x, D) be a pseudodifferential operator of order σ and let Ωi, i = 1, 2 be bounded domains in Rn with Ω1 ∩ Ω2 = ∅. Suppose u(x) ∈ Hl (Rn ) for some l, with supp u ∈ Ω2. Then outside the support of u, A(x, D)u ∈ C∞ . Proof. Let φ1(x), φ2(x) ∈ C∞ 0 (Rn ) be such that φi(x) = 1 in Ωi, i = 1, 2 and supp φ1 ∩ supp φ2 = ∅. For each x ∈ Ω1 we have Au(x) = φ1(x)A [φ2(x)u(x)] .
- 41. 1.4 Pseudodifferential operators. Definitions and examples 25 Let a(x, ξ) be the symbol of A(x, D). Since Dα ξ φ1 = 0 for |α| 0 it follows from Theorem 1.10, that φ1(x)A(x, D) = A1(x, D) + T1, where A1(x, D) is a pseudodifferential operator with symbol a1(x, ξ) = φ1(x) · a(x, ξ), and T1 has true order equal to −∞. Analogously, since Dα ξ a1(x, ξ)∂α x φ2(x) ≡ 0 for all α, then T2φ := A1(x, D)φ has true order equal to −∞. Thus φ1(x)A(x, D)φ2(x) = φ1(x)T1 + T2 has order equal to −∞. Consequently v(x) = φ1(x)A(φ2(x)u(x)) belongs to T l Hl(Rn ). Hence due to the Sobolev embedding theorem we obtain v(x) ∈ C∞ (Rn ). ⊓ ⊔ In the literature Corollary 1.5 is called the pseudo-local property. Remark 1.9 In the case of A(x, D)u = P |α|=σ aα(x)Dα u, we have Au(x) ≡ 0 in Ω1. Conjugate operator. Let A(x, D) be a pseudodifferential operator of order σ defined by A(x, D)u = (2π)−n/2 Z eix·ξ ζ(ξ)a(x, ξ)û(ξ)dξ. Let (u, v) = Z u(x)v(x)dx (1.73) and a∗ (x, ξ) = a(x, ξ) the complex conjugate. Then one can show that A∗ (x, D) defined by (1.74) is the conjugate operator to A(x, D) with respect to the scalar product (1.73), d A∗v(ξ) = (2π)−n/2 Z e−ix·ξ a∗ (x, ξ)ζ(ξ)v(x)dx. (1.74) Lemma 1.5 Let A(x, D) be a pseudodifferential operators of order σ with symbol a(x, ξ) and let A∗ (x, D) be conjugate to A(x, D). Then, for all ρ 0, ρ an integer, A∗ (x, D) = ρ−1 X i=0 Bi(x, D) + Tρ where Bi(x, D) are ψDO with symbol X |α|=i 1 α! ∂α x · Dα ξ a∗ (x, ξ)
- 42. 26 1 Auxiliary Materials and Tρ has order σ − ρ. We omit the details of the proof which is based on the Taylor expansions of the function ã′∗ (ξ − η, ξ) with respect to ξ, when ξ belongs to a small neighborhood of η. Remark 1.10 The lemma is obvious if A(x, D)u = X |α|=σ aα(x)Dα u (here we take ζ(ξ) ≡ 1). Indeed, then A∗ (x, D)u(x) = X |α|=σ Dα [a∗ α(x)u(x)] where a∗ α is conjugate to aα(x). Hence A∗ (x, D)u(x) − X |α|=σ a∗ α(x)Dα u has order σ − 1. In section 1.4 ψDO was defined in a canonical way by formulas (1.46) and (1.58). Theorem 1.10 and Lemma 1.5 showed us, that the operations A, B 7→ A ◦ B and A 7→ A∗ do not lie in the class of ψDO which are canonically defined by (1.46) or (1.58). In other words, it is not an algebra. In order to avoid this drawback, we have to add a number of remainder terms to the definition of ψDO. Of course, we have a degree of freedom to choose these additional terms. A reasonable way to proceed is to add terms of ψDO operators decreasing orders. More precisely: Definition 1.4 Let r0, r1, r2, . . . be a sequence (finite or infinite), such that r0 r1 . . . rn, . . ., and rk → −∞ as k → ∞ (if the sequence is infinite). Let Ak(x, D) be a ψDO of order rk defined in a canonical way. Let A be an operator defined on S(Rn x), such that A − N X k=0 Ak(x, D) for all N has order less than rN . Then by definition A is pseudodifferential operator with asymptotic expansion A ∼ ∞ X k=0 Ak(x, D). (1.75) By definition the symbol of A, i.e. a(x, ξ), is the formal series
- 43. 1.4 Pseudodifferential operators. Definitions and examples 27 ∞ X k=0 ak(x, ξ). (1.76) If the sequence {rk} is finite, then ψDO is defined by A = ρ X k=0 Ak(x, D) + T∞ where T∞ has true order equal to −∞. From the definition it follows that, in both cases ψDO has order r0. Fur- thermore, ψDO with symbol a(x, ξ) ≡ 0 has true order equal to −∞, since ak(x, ξ) ≡ 0, for k = 1, 2, · · · implies Ak(x, D) ≡ 0, k = 1, 2 . . ., and vice versa; if a pseudodifferential operator has order equal to −∞, then its sym- bol is equal to zero (see [41]). Thus a ψDO is defined by its symbol modulo operators which have orders equal to −∞. Let Ω be an open set of Rn . The following theorem is given in [39]. Theorem 1.11 Let aj(x, ξ) be C∞ functions in Ω × (Rn ξ {0}) which are positively homogeneous of degree rj ↘ −∞. Then there exists a ψDO with symbol ∞ X k=0 ak(x, ξ). Proof. Let φ(ξ) be a C∞ function in Rn which equals 0 when |ξ| ≤ 1 2 and equals 1 when |ξ| ≥ 1. We can choose a sequence tj ↗ +∞ increasing so rapidly that Dβ ξ φ ξ tj aj(x, ξ) + (1 + |x|k ) ∂α x Dβ ξ φ ξ tj aj(x, ξ) ≤ 2−j · (1 + |ξ|)rj−1−|β| (1.77) for all |α| + |β| + k ≤ j and |α| 0. In fact, Dβ ξ [φ ξ tj aj(x, ξ)] can be computed by Leibniz’s formula. It is obvious that a term in which φ is not differentiated is homogeneous of order rj −|β| in ξ when |ξ| tj and is equal to 0 when |ξ| ≤ tj 2 . In other terms we have t−q j , when 0 q ≤ |β| and these terms are equal to 0 both when |ξ| ≤ tj 2 and |ξ| tj. Hence, the left hand side of (1.77) can be estimated by const. · 1 t rj−1−rj j · |ξ|rj−1−|β| and it is obvious that we can choose a sequence tj, such that Dβ ξ φ ξ tj aj(x, ξ) +(1+|x|k ) ∂α x Dβ ξ φ ξ tj aj(x, ξ) ≤ 2−j ·(1+|ξ|)rj−1−|β| .
- 44. 28 1 Auxiliary Materials With t0 = 1 set a(x, ξ) = ∞ X j=0 φ ξ tj aj(x, ξ). (1.78) Because of our choice of the tj, the series and the differentiated series converge absolutely. We set A(x, D)u(x) = (2π)−n/2 Z eix·ξ a(x, ξ)û(ξ)dξ and Aj(x, D)u(x) = (2π)−n/2 Z eix·ξ φ ξ tj aj(x, ξ)û(ξ)dξ. Therefore instead of the asymptotic expansion A ∼ ∞ X j=0 Aj we have A(x, D)u = ∞ X j=0 Aj(x, D)u. It is not difficult to see that A(x, D) − k X j=0 Aj(x, D)u has order rk+1. We denote by L the class ψDO defined by asymptotic expansions (1.75). Theorem 1.12 L is an algebra with involution . Proof. Let A(x, D) ∼ P∞ j=0 Aj(x, D) and B(x, D) ∼ P∞ k=0 Bk(x, D) where Aj(x, D) and Bk(x, D) are canonical ψDO’s defined by (1.46) with orders σj and λk. We have to show that A(x, D) ◦ B(x, D) ⊂ L. It is sufficient to show that for any number q there is a finite sum of canonical operators P Cj(x, D), whose symbols have strictly decreasing orders, and which differ from A(x, D) ◦ B(x, D) by an operator of order q. Suppose the symbols of the Aj(x, D) and the Bk(x, D) has degrees r0 r1 . . . → −∞, and s0 s1 . . . → −∞, respectively. Let N be so large that σ0 + sN q and λ0 + σN q, and set A = N X j=0 Aj + TN,1, B = N X k=0 Bk + TN,2 where the orders TN,1 and TN,2 are less than σN and sN respectively. Hence
- 45. 1.4 Pseudodifferential operators. Definitions and examples 29 AB = N X j=0 Aj + TN,1 N X k=0 Bk + TN,2 ! = N X j,k=0 AjBk + TN,3 where the order of TN,3 is less than q = max(σ0 +λN , λ0 +σN ). The assertion of Theorem 1.12 follows from Theorem 1.10 for Aj ·Bk. Analogously, one can show that if A ∈ L, then A∗ ∈ L. Elliptic ψDO. Let A(x, D) be a ψDO with symbol a(x, ξ) ∼ ∞ X j=0 aj(x, ξ). By definition, A is elliptic, if a0(x, ξ) ̸= 0 when ξ ̸= 0. We denote by Id the identity operator. Theorem 1.13 Let A(x, D) be an elliptic ψDOs. Then there exists B(x, D) ∈ L, such that each operator AB − Id and BA − Id has order equal to −∞. We omit the proof (see [2, 41]). In the end of this section, let us discuss some particular class of pseudo-differential operator, so-called fractional Laplacian which will play an important role in the following chapters. It is worth to note that, many models arising in Biology, Medicine, Ecology and Finance lead to nonlinear problems driven by fractional Laplace-type operators. There are several ways of defining this operator in the whole space Rn , at least to my knowledge ten in the literature, most of them are equivalent (see [44]). I will give here that definitions which are more relevant for the content of this book. Let s ∈ (0, 1). For the convenience of the reader (see also section 1.1), we define the fractional Sobolev space Hs (Rn ) := W2,s (Rn ); Hs (Rn ) := u ∈ L2 (Rn ) Z Rn Z Rn |u(x) − u(y)|2 |x − y|n+2s dxdy +∞ . For u, v ∈ Hs (Rn ), we set ⟨u, v⟩Hs(Rn) := Z Rn u(x)v(x) dx + Z Rn Z Rn (u(x) − u(y))(v(x) − v(y)) |x − y|n+2s dxdy. One can easily see that ⟨·, ·⟩Hs(Rn) is an inner product on Hs (Rn ) and that the space Hs (Rn ) endowed with this inner product is a Hilbert space . It is well-known that the space Hs (Rn ), s ∈ (0, 1) can be defined alternatively via the Fourier transform, i.e.,
- 46. 30 1 Auxiliary Materials Hs (Rn ) := u ∈ L2 (Rn ) Z Rn (1 + |ξ|2s )|Fx→ξu(ξ)|2 | dξ +∞ , where by Fx→ξu(ξ) is defined the Fourier transform of u(x), x ∈ Rn (some- times we used instead of û(ξ) by the notation Fx→ξu(ξ). The first definition of the fractional Laplacian (−∆x)s , s ∈ (0, 1) is expressed via the Fourier transform as a pseudo-differential (nonlocal!) operator with the symbol is equal to |ξ|2s , where |ξ| = (ξ1 + · · · + ξn) 1 2 . More precisely, let u ∈ S(Rn ) (for the definition of this space see previous section). Then we have Fx→ξu ∈ S(Rn ), but the function ξ 7→ |ξ|2s ·Fx→ξu(ξ) ∈ S(Rn ), because |ξ|2s creates singularity at ξ = 0. What is more important that, |ξ|2x Fx→ξu(ξ) ∈ L1 (Rn ) ∩L2 (Rn ), so we can use the inverse Fourier transform. Consequently, for any u ∈ S(Rn ) we define the operator (−∆x)s : S(Rn ) → L2 (Rn ) by (−∆x)s u(x) := F−1 ξ→x(|ξ|2x Fx→ξu(ξ))(x), ∀x ∈ Rn . This pseudo-differential operator is called the fractional Laplacian of order s. It is easy to see that (−∆x)0 u = u, (−∆x)1 u = −∆xu, for any 0 s1, s2 1 (−∆x)s1 ◦ (−∆x)s2 = (−∆x)s2 ◦ (−∆x)s1 for any multi-index β ∈ Rn Dα x (−∆x)s u = (−∆x)s Dα x u, so that it follows that (−∆x)s u ∈ C∞ (Rn ) for any u ∈ S(Rn ). Remark 1.11 Actually, we can extend this and other definitions given below that is, (−∆x)s u to Hs (Rn ) using the fact that the space S(Rn ) is dense in Hs (Rn ) (see, e.g., [15]). The following definition of the fractional Laplacian is often used on the lit- erature, i.e., let u ∈ S(Rn ). (−∆x)s u(x) := C(n, s) lim ε→0 Z RnB(x,ε) (u(x) − u(y))2 |x − y|n+2s dy, ∀x ∈ Rn , where the normalized constant C(n, s) given by C(n, s) := s · ns · Γ(s + n 2 ) π n 2 Γ(1 − s) , s ∈ (0, 1)
- 47. 1.5 Linear Fredholm operators 31 and Γ(·) is the Euler function. Remark 1.12 It is not difficult to check that, for u ∈ S(Rn ) and x ∈ Rn and s ∈ (0, 1 2 ) the integral Z u(x) − u(y) |x − y|n+2s dy is absolutely convergent, so that for s ∈ (0, 1 2 ) (−∆x)s u(x) := C(n, s) Z Rn u(x) − u(y) |x − y|n+2s dy. The following proposition shows the relation between the ”classical” Lapla- cian and the fractional Laplacian (see [44]). Proposition 1.2 Let u ∈ S(Rn ). Then 1) lim s→0+ (−∆x)s u = u 2) lim s→1− (−∆x)s u = −∆xu 1.5 Linear Fredholm operators Fredholm operators are one of the most important classes of linear maps in mathematics which were introduced around 1900 in the study of integral operators. As we will see below they share many properties with linear maps between finite dimensional spaces. Let X and Y be Banach spaces, either over R or C. By L(X, Y ) we denote the space of bounded linear operators from X to Y , and consider L(X, Y ) as a Banach space with the usual norm. As we already mentioned in Section 1.2, an operator T is called Fredholm if the kernel (nullspace) Ker T = {x ∈ X | Tx = 0} has finite dimension and the image (range) of T, R(T) := {Tx | x ∈ X} is of finite codimension in Y , that is, codim R(T) := dim(Y/R(T)) ∞. For a Fredholm operator T : X → Y , the numerical Fredholm index of T, ind T is defined by ind T := dim Ker T − codim R(T). We denote the set of Fredholm operator as well as Fredholm operator of index m by Φ(X, Y ) and Φm(X, Y ) respectively. In the study of global solvability of linear Fredholm operator equation the subset Φ0(X, Y ) of L(X, Y ) play an important role. It is worth to note that for T ∈ Φ0(X, Y ) it follow that
- 48. 32 1 Auxiliary Materials dim Ker T = dim(Y/R(T)). Hence for T ∈ Φ0(X, Y ) from global solvability Tx = y for any y ∈ Y it follows uniqueness of soluation of Tx = y and vice-versa. Obviously, each invertible linear operator T : X → Y belong to Φ0(X, Y ). The following lemma holds: Lemma 1.6 Let T ∈ L(X, Y ). Then the following assertions are equivalent: 1. The operator T is Fredholm of index 0. 2. There is a compact operator K ∈ L(X, Y ) such that L + K is invertible. 3. There is an isomorphism S ∈ L(Y, X) such that S ◦ L is a compact vector field, that is, it is compact perturbation of identity. Proof. Let T ∈ Φ0(X, Y ). Since Ker T is finite dimensional, there is a con- tinuous projection P of X onto Ker T. Let W be a closed subspace of Y , W := Y/TX. By definition of T ∈ Φ0(X, Y ), both Ker T and W are of the same finite dimension, so we may select S ∈ GL(Ker T, W). We set K := S◦P and observe that K is compact and T +K is an isomorphism. Thus 1. implies 2. To verify that 2. implies 3., choose K : X → Y linear compact operator so that T + K is invertible. We set S := (T + K)−1 . Then it is easy to see that S ◦ T = I − (T + K)−1 ◦ K. Hence 2. implies 3. Finally to show that 3. implies 1. we have to remind Riesz-Schauder Theorem that states that com- pact linear vector field is Fredholm of index 0. Indeed, let S ◦ T = Id + K. Then T = S−1 (Id + K) and ind T = ind S−1 + ind(Id + K) = 0. ⊓ ⊔ Remark 1.13 An operator S : Y → X satisfy 3. is called a parametrix for T. The subset Φ0(X, Y ) possesses interesting biological properties. For simplicity of presentation as well as for notational convenience we consider the case when X = Y . We denote by GL(X) the group of linear operator A : X → X. According to Kuiper’s [43], the group GL(X) on a separable Hilbert (infinite dimensional real) space is connected and even is contractible. The analogous result is also true for many Banach spaces [20]. It is not difficult to see that Φ0(X) inherits connectedness from GL(X). Indeed, if T ∈ Φ0(X) then t 7−→ tT + (1 − t)S−1 , 0 ≤ t ≤ 1, where S is a parametrix for T, defines a path in Φ0(X) connecting T to GL(X). Remark 1.14 If X = Rn , then Φ0(Rn ) = L(Rn ) is contractible and GL(Rn ) has two connected components, which are labeled by the function sgndet. In contrast to a finite dimensional case Φ0(X) is connected when X is infinite
- 49. 1.5 Linear Fredholm operators 33 dimensional space but not contractible (see [33]). Moreover, the set Φ0(X) is open in L(X). Below we recap following well-known properties of Φ0(X), some of them without proof (see [20],[33] and references therein). Proposition 1.3 Let A, B ∈ Φ0(X). Then 1) AB ∈ Φ0(X) and ind AB = ind A + ind B. 2) Let A ∈ Φ0(X) and K : X → X is a compact linear operator. Then A + K ∈ Φ0(X) and ind(A + K) = ind A Let X∗ and Y ∗ be two Banach spaces with X ⊂ X∗ and Y ⊂ Y ∗ , where X and Y also are Banach spaces. We assume that X∗ = X⊕Rp and Y ∗ = Y ⊕Rq . By definition à ∈ L(X∗ , Y ∗ ) is an extension of A ∈ L(X, Y ) by {p, q} dimensions if Ãx = Ax for x ∈ X. Lemma 1.7 Let the operator à ∈ L(X∗ , Y ∗ ) be an extension of A ∈ L(X, Y ) by {p, q} dimension. Then ind A = ind à − p + q. Proof. We define A∗ : X∗ → Y ∗ A∗(X + Y ) := Ax for x ∈ X, y ∈ Rp . Obviously, A∗ : X∗ → Y ∗ is an extension of A by {p, q} dimension and for A∗ ∈ L(X∗ , Y ∗ ) holds dim Ker A∗ = dim Ker A + p and codim A∗ = codim A + q. Hence, ind A = ind A∗ − p + q. On the other hand, dim Ker A∗ = dim Ker A + p, codim A∗ = codim A + q. Since Ãx = A∗x for all x ∈ X (they are different only by a finite-dimensional setting), we obtain ind à = ind A∗. This completes the proof. ⊓ ⊔ The following Lemma 1.8 plays an important role and also holds for a quite large class of a nonlinear operator (see [20],[51],[63] and references therein).
- 50. Discovering Diverse Content Through Random Scribd Documents
- 51. actually carry it out through all his future life. Such cases are often to be found on record or in every-day life. In other cases, this quiet, hidden, but controlling purpose seems to be formed by unconscious and imperceptible influences, so that the mind can not revert to the specific time or manner when it originated. For example, a child who is trained from early life to speak the truth, can never revert to any particular moment when this generic purpose originated. It is sometimes the case, also, that a person will contemplate some generic purpose before it occurs, while [pg 079] the process of its final formation seems almost beyond the power of scrutiny. For example, a man may be urged to relinquish one employment and engage in another. He reflects, consults, and is entirely uncertain how he shall decide. As time passes, he gradually inclines toward the proposed change, until, finally, he finds his determination fixed, he scarcely knows when or how. Thus it appears that generic purposes commence sometimes so instantaneously and obviously that the time and influences connected with them can be recognized. In other cases, the decision seems to be a gradual one, while in some instances the process can be traced, and in others it is entirely unnoticed or forgotten. It is in reference to such generic purposes that the moral character of men is estimated. An honest man is one who has a fixed purpose to act honestly in all circumstances. A truthful man is one who has such a purpose to speak the truth at all times. In such cases, the degree in which such a purpose controls all others is the measure of a man's moral character in the estimate of society. The history of mankind shows a great diversity of moral character dependent on such generic choices. Some men possess firm and reliable moral principles in certain directions, while they are very destitute of them in others.
- 52. Thus it will be seen that some have formed a very decided purpose in regard to honesty in business affairs, who yet are miserable victims to intemperance. Others have cultivated a principle called honor, that restrains them from certain actions regarded as mean, [pg 080] and yet they may be frequenters of gambling saloons and other haunts of vice. In the religions world, too, it is the case that some who are very firm and decided on all points of religious observances and in the cultivation of devotional emotions, are guilty of very mean actions, such as some worldly men of honor would not practice at the sacrifice of a right hand.
- 53. On a Ruling Purpose or Chief End. The most important of all the voluntary phenomena is the fact that, while there can be a multitude of these quiet and hidden generic purposes in the mind, it is also possible to form one which shall be the dominant or controlling one, to which all the others, both generic and specific, shall become subordinate. In common parlance this would be called the ruling passion. It is also called the ruling purpose, or controlling principle. This consists in the permanent choice of some one mode of securing happiness as the chief end or grand object of life. There is a great variety of sources of happiness and of suffering to the human mind. Now in the history of our race we find that each one of these modes of enjoyment has been selected by different individuals as the chief end of their existence—as the mode of seeking enjoyment to which they sacrifice every other. Some persons have chosen the pleasures of eating, drinking, and the other grosser enjoyments of sense. Others have chosen those more elevated and refined pleasures that come indirectly from the senses in the emotions of taste. Others have devoted themselves to intellectual enjoyments [pg 081] as their chief resource for happiness. Others have selected the exercise of physical and moral power, as in the case of conquerors and physical heroes, or of those who have sought to control by moral power, as rulers and statesmen. Others have made the attainment of the esteem, admiration, and love of their fellow-creatures, their chief end. Others, still, have devoted themselves to the promotion of happiness around them as
- 54. their chief interest. Others have devoted themselves to the service of God, or what they conceived to be such, and sometimes by the most miserable life of asceticism and self-torture. Others have made it their main object in life to obey the laws of rectitude and virtue. In all these cases, the moral character of the person, in the view of all observers, has been decided by this dominant volition, and exactly in proportion to the supremacy with which it has actually controlled all other purposes. Some minds seem to have no chief end of life. Their existence is a succession of small purposes, each of which has its turn in controlling the life. Others have a strong, defined and all-controlling principle. Now experience shows that both of these classes are capable, the one of forming and the other of changing such a purpose. For example, in a time of peace and ease there is little to excite the mind strongly; but let a crisis come, where fortune, reputation, and life are at stake, and men and women are obliged to form generic decisions involving all they hold dear, and many minds that have no controlling purpose immediately originate one, while those whose former ruling [pg 082] aims were in one direction change them entirely to another. This shows how it is that days of peril create heroes, statesmen and strong men and women. The hour of danger calls all the energies of the soul into action. Great purposes are formed with the strongest desire and emotion. Instantly the whole current of thought, and all the coexisting desires and emotions, are conformed to these purposes. The experience of mankind proves that a dominant generic purpose may extend to a whole life, and actually control all other generic and specific volitions.
- 55. How the Thoughts, Desires and Emotions are controlled by the Will. We will now consider some of the modes by which the will controls the thoughts, desires and emotions. We have seen, in previous pages, the influence which desire and emotion exert in making both our perceptions and conceptions more vivid. Whatever purpose or aim in life becomes an object of strong desire, is always distinctly and vividly conceived, while all less interesting objects are more faint and indistinct. We have also seen that whenever any conception arises it always brings connected objects, forming a new and complex picture. Whenever the mind is under the influence of a controlling purpose, the object of pursuit is always more interesting than any other. This interest always fastens on those particulars in any mental combination that are connected with the ruling purpose and seem fitted to promote it, making them more vivid. Around these selected objects their past associated ideas begin [pg 083] to cluster, forming other complex pictures. In all these combinations, those ideas most consonant with the leading interest of the mind become most vivid, and the others fade away. The grand method, then, for regulating the thoughts is by the generic decisions of the mind as to the modes of seeking enjoyment. In regard to the power of the mind over its own desires and emotions, it is very clear that these sensibilities can not be regulated by direct specific volitions. Let any person try to produce love, fear,
- 56. joy, hope or gratitude by simply choosing to have them arise, and it is soon perceived that no such power exists. But there are indirect modes by which the mind can control its susceptibilities. The first method is by directing attention to those objects of thought which are fitted to call forth such emotions. For example, if we wish to awaken the emotion of fear, we can place ourselves in circumstances of danger, or call up ideas of horror and distress. If we wish to call forth emotions of gratitude, we can direct attention to acts of kindness to ourselves calculated to awaken such feelings. If we wish to excite desire for any object, we can direct attention to those qualities in that object that are calculated to excite desire. In all these cases the mind can, by an act of will, direct its attention to subjects calculated to excite emotion and desire. The other mode of regulating the desires and emotions is by the direction of our generic volitions. For example, let a man of business, who has never had any interest in commerce, decide to invest all his property in foreign trade. As soon as this is done, the name of the ship that bears his all can never be heard [pg 084] or seen but it excites some emotion. A storm, that before would go unnoticed, awakens fear; the prices in the commercial markets, before unheeded, now awaken fear or afford pleasure. And thus multitudes of varied desires and emotions are called into existence by this one generic volition. One result of a purpose to deny an importunate propensity is frequently seen in the immediate or gradual diminution of that desire. For example, if a person is satisfied that a certain article of food is injurious and resolves on total abstinence, it will be found that the desire for it is very much reduced, far more so than when the effort is to diminish the indulgence. When a generic purpose is formed that involves great interests, it is impossible to prevent the desires and emotions from running consonant with this purpose. The only mode of changing this current
- 57. is to give up this generic purpose and form another. Thus, if a man has devoted his whole time and energies to money-making, it is impossible for him to prevent his thoughts and feelings from running in that direction. He must give up this as his chief end, and take a nobler object, if he would elevate the whole course of his mental action. These are the principal phenomena of the grand mental faculty which is the controlling power of the mind, and on the regulation of which all its other powers are dependent. The nature of regeneration, and the question whether it is instantaneous or gradual or both, all are intimately connected with the subject of this chapter. [pg 085]
- 58. Chapter XVI. Constitutional Varieties of the Human Mind. In the preceding chapters have been presented the most important mental faculties which are common to the race. There are none of the powers and attributes of the mind as yet set forth which do not belong to every mind which is regarded as rational and complete. But, though all the race have these in common, yet we can not but observe an almost endless variety of human character, resulting from the diverse proportions and combinations of these several faculties. These constitutional differences may be noticed, first, in regard to the intellectual powers. Some minds are naturally predisposed to exercise the reasoning powers. Others, with precisely the same kind of culture, have little relish for this, and little power of appreciating an argument. In other cases, the imagination seems to be the predominating faculty. In other minds there seems to be an equal balance of faculties, so that no particular power predominates. Next we see the same variety in reference to the susceptibilities. In some minds, the desire for love and admiration is the predominating principle. In others, the love of power takes the lead. Some are eminently sympathizing. Others have a strong love of rectitude, or natural conscience. In some, the principle of justice predominates. In others, benevolence is the leading impulse. [pg 086]
- 59. Finally, in regard to the power of volition, as has been before indicated, there are some that possess a strong will that is decisive and effective in regulating all specific volitions, while others possess various and humbler measures of this power. According to the science of Phrenology, some of these peculiarities of mind are indicated by the size and shape of different portions of the brain, and externally indicated on the skull. That these differences are constitutional, and not the result of education, is clear from the many facts showing that no degree of care or training will serve to efface these distinctive traits of the mind. To a certain degree they may be modified by education, and the equal balance of the faculties be promoted, but never to such a degree as to efface very marked peculiarities. In addition to the endless diversities that result from these varied proportions and combinations, there is a manifest variety in the grades of mind. Some races are much lower in the scale of being every way than others, while the same disparity exists in individuals of the same race. The wisdom and benevolence of this arrangement is very manifest when viewed in reference to the interests of a commonwealth. Where some must lead and others follow, it is well that some have the love of power strong, and others have it less. Where some must be rulers, to inflict penalties as well as to apportion rewards, it is well that there be some who have the sense of justice as a leading principle. And so in the developments of intellect. Some men are to follow callings where the reasoning powers are most needed. Others are to adopt pursuits in which taste [pg 087] and imagination are chiefly required; and thus the varied proportions of these faculties become serviceable. And if it be true that the exercise of the social and moral faculties secures the highest degree of enjoyment, those disparities in mental
- 60. powers which give exercise to the virtues of compassion, self-denial, fortitude and benevolence in serving the weak, and the corresponding exercises of gratitude, reverence, humility and devotion in those who are thus benefited, then we can see the wisdom and benevolence of this gradation of mental capacity. Moreover, in a commonwealth perfectly organized, where the happiness of the whole becomes that of each part, whatever tends to the highest general good tends to the best interest of each individual member. This being so, the lowest and humblest in the scale of being, in his appropriate place, is happier than he could be by any other arrangement, and happier than he could be if all were equally endowed. This subject is very important, because some theologians present these disparities of mental organization as indications of the depravity consequent on Adam's sin.
- 61. Chapter XVII. Nature of Mind.—Habit. This chapter is introduced because some theologians claim that the depravity of man consists either in a habit or in something like a habit. [pg 088] Habit is a facility in performing physical or mental operations, gained by the repetition of such acts. As examples of this in physical operations may be mentioned the power of walking, which is acquired only by a multitude of experiments; the power of speech, secured by a slow process of repeated acts of imitation; and the power of writing, gained in the same way. Success in every pursuit of life is attained by oft-repeated attempts, which finally induce a habit. As examples of the formation of intellectual habits, may be mentioned the facility gained in acquiring knowledge by means of repeated efforts, and the accuracy and speed with which the process of reasoning is performed after long practice in this art. As examples of moral habits may be mentioned those which are formed by the oft-repeated exercise of self-government, justice, veracity, obedience, and industry. The will, as has been shown, gains a facility in controlling specific volitions and in yielding obedience to the laws of right action by constant use, as really as do all the other mental powers. The happiness of man, in the present state of existence, depends not so much upon the circumstances in which he is placed, or the capacities with which he is endowed, as upon the formation of his
- 62. habits. A man might have the organ of sight, and be surrounded with all the beauties of nature, and yet, if he did not form the habit of judging of the form, distance and size of bodies, most of the pleasure and use from this sense would be wanting. The world and all its beauties would be a mere confused mass of colors. If the habits of walking and of speech were not acquired, these faculties and the circumstances for employing [pg 089] them would not furnish the enjoyment they were designed to secure. It is also the formation of intellectual habits by mental discipline and study, which opens vast resources for enjoyment that otherwise would be for ever closed. And it is by practicing obedience to parents that moral habits of subordination are formed, which are indispensable to our happiness as citizens, and as subjects of government. There is no enjoyment which can be pointed out, which is not, to a greater or less extent, dependent upon this principle. The influence of habit in regard to the law of sacrifice is especially interesting. The experience of multitudes of our race shows that such tastes and habits may be formed in obeying this law, that what was once difficult and painful becomes easy and pleasant. But this ability to secure enjoyment through habits of self-control and self-denial, induced by long practice, so far as experience shows, could never be secured by any other method. That the highest kinds of happiness are to be purchased by more or less voluntary sacrifice and suffering to procure good for others, seems to be a part of that nature of things which we at least may suppose has existed from eternity. We can conceive of the eternal First Cause only as we imagine a mind on the same pattern as our own in constitutional capacities, but indefinitely enlarged in extent and action. Knowledge, wisdom, power, justice, benevolence and rectitude, must be the same in the Creator as in ourselves, at least so far as we can conceive; and, as the practice of self-sacrifice and
- 63. suffering for the good of others is our highest conception of virtue, it is impossible to regard [pg 090] the Eternal Mind as all-perfect without involving this idea. The formation of the habits depends chiefly upon the leading desire or governing purpose, because whatever the mind desires the most it will act the most to secure, and thus by repeated acts will form its habits. The character of every individual, therefore, as before indicated, depends upon the mode of seeking happiness selected by the will. Thus the ambitious man has selected the attainment of power and admiration as his leading purpose, and whatever modes of enjoyment interfere with this are sacrificed. The sensual man seeks his happiness from the various gratifications of sense, and sacrifices other modes of enjoyment that interfere with this. The man devoted to intellectual pursuits, and to seeking reputation and influence through this medium, sacrifices other modes of enjoyment to secure this gratification. The man who has devoted his affections and the service of his life to God and the good of his fellow-men, sacrifices all other enjoyments to secure that which results from the fulfillment of such obligations. Thus a person is an ambitious man, a sensual man, a man of literary ambition, or a man of piety and benevolence, according to the governing purpose or leading choice of his mind. There is one fact in regard to the choice of the leading object of desire, or the governing purpose of life, which is very important. Certain modes of enjoyment, in consequence of repetition, increase the desire, but lessen the capacity of happiness from this source; while, in regard to others, gratification increases the [pg 091] desire, and at the same time increases the capacity for enjoyment. The enjoyments through the senses are of the first kind. It will be found, as a matter of universal experience, that where this has been chosen as the main purpose of life, though the desire for such pleasures is continually increased, yet, owing to the physical effects of excessive indulgence, the capacity for enjoyment is decreased.
- 64. Thus the man who so degrades his nature as to make the pleasures of eating and drinking the great pursuit of life, while his desires never abate, finds his zest for such enjoyments continually decreasing, and a perpetual need for new devices to stimulate appetite and awaken the dormant capacities. The pleasures of sense always pall from repetition—grow “stale, flat and unprofitable,” though the deluded being who has thus slavishly yielded to such appetites feels himself bound by chains of habit, which, even when enjoyment ceases, seldom are broken. The pleasures derived from the exercise of power, when its attainment becomes the master passion, are also of this description. The statesman, the politician, the conqueror, are all seeking for this, and desire never abates while any thing of the kind remains to be attained. We do not find that enjoyment increases in proportion as power is secured. On the contrary, it seems to cloy in possession. Alexander, the conqueror of the world, when he had gained all, wept that objects of desire were extinct, and that possession could not satisfy. But there are other sources of happiness for which the desire ever continues, and possession only increases the capacity for enjoyment. Of this class is [pg 092] the susceptibility of happiness from giving and receiving affection. Here, the more there is given and received, the more is the power of giving and receiving increased. We find that this principle outlives every other, and even the decays of nature itself. When tottering age on the borders of the grave is just ready to resign its wasted tenement, often from its dissolving ashes the never-dying spark of affection has burst forth with new and undiminished luster. This is that immortal fountain of happiness always increased by imparting, never surcharged by receiving. Another principle, which increases both desire and capacity by exercise, is the power of enjoyment from being the cause of happiness to others. Never was an instance known of regret for devotion to the happiness of others. On the contrary, the more this
- 65. holy and delightful principle is in exercise, the more the desires are increased, and the more are the susceptibilities for enjoyment from this source enlarged. While the votaries of pleasure are wearing down with the exhaustion of abused nature, and the votaries of ambition are sighing over its thorny wreath, the benevolent spirit is exulting in the success of its plans of good, and reaching forth to still purer and more perfect bliss. This principle is especially true in regard to the practice of rectitude. The more the leading aim of the mind is devoted to right feeling and action, or to obedience to all the laws of God, the more both the desire and the capacity of enjoyment from this source are increased. But there is another fact in regard to habit, which has an immense bearing on the well-being of our race. [pg 093] When a habit of seeking happiness in some one particular mode is once formed, the change of this habit becomes difficult just in proportion to the degree of repetition which has been practiced. A habit once formed, it is no longer an easy matter to choose between the mode of securing happiness chosen and another which the mind may be led to regard as much superior. Thus, in gratifying the appetite, a man may feel that his happiness is continually diminishing, and that, by sacrificing this passion, he may secure much greater enjoyment from another source; yet the force of habit is such, that decisions of the will perpetually yield to its power. Thus, also, if a man has found his chief enjoyment in that admiration and applause of men so ardently desired, even after it has ceased to charm, and seems like emptiness and vanity, still, when nobler objects of pursuit are offered, the chains of habit bind him to his wonted path. Though he looks and longs for the one that his conscience and his intellect assure him is brightest and best, the conflict with bad habit ends in fatal defeat and ruin. It is true that every habit can be corrected and changed, but nothing requires greater firmness of purpose and energy of will; for it is not one
- 66. resolution of mind that can conquer habit: it must be a constant series of long-continued efforts. The influence of habit in reference to emotions deserves special attention as having a direct influence upon character and happiness. All pleasurable emotions of mind, being grateful, are indulged and cherished, and are not weakened by repetition unless they become excessive. If the pleasures of sense are indulged beyond a certain extent, the bodily system is [pg 094] exhausted, and satiety is the consequence. If the love of power and admiration is indulged to excess, so as to become the leading purpose of life, they are found to be cloying. But, within certain limits, all pleasurable emotions do not seem to lessen in power by repetition. But in regard to painful emotions the reverse is true. The mind instinctively resists or flies from them, so that often a habit of suppressing such emotions is formed, until the susceptibility diminishes, and sometimes appears almost entirely destroyed. Thus a person often exposed to danger ceases to be troubled by fear, because he forms a habit of suppressing it. A person frequently in scenes of distress and suffering learns to suppress the emotions of painful sympathy. The surgeon is an example of the last case, where, by repeated operations, he has learned to suppress emotions until they seldom recur. A person inured to guilt gradually deadens the pangs of remorse, until the conscience becomes “seared as with a hot iron.” Thus, also, with the emotion of shame. After a person has been repeatedly exposed to contempt, and feels that he is universally despised, he grows callous to any such emotions. The mode by which the mind succeeds in forming such a habit seems to be by that implanted principle which makes ideas that are most in consonance with the leading desire of the mind become vivid and distinct, while those that are less interesting fade away. Now no person desires to witness pain except from the hope of relieving it, unless it be that, in anger, the mind is sometimes gratified with the infliction of suffering. But, in ordinary cases, the
- 67. sight of suffering [pg 095] is avoided except where relief can be administered. In such cases, the desire of administering relief becomes the leading one, so that the mind is turned off from the view of the suffering to dwell on conceptions of modes of relief. Thus the surgeon and physician gradually form such habits that the sight of pain and suffering lead the mind to the conception of modes of relief, whereas a mind not thus interested dwells on the more painful ideas. The habits of life are all formed either from the desire to secure happiness or to avoid pain, and the fear of suffering is found to be a much more powerful principle than the desire of happiness. The soul flies from pain with all its energies, even when it will be inert at the sight of promised joy. As an illustration of this, let a person be fully convinced that the gift of two new senses would confer as great an additional amount of enjoyment as is now secured by the eye and ear, and the promise of this future good would not stimulate with half the energy that would be caused by the threat of instant and entire blindness and deafness. If, then, the mind is stimulated to form good habits and to avoid the formation of evil ones most powerfully by painful emotions, when their legitimate object is not effected they continually decrease in vividness, and the designed benefit is lost. If a man is placed in circumstances of danger, and fear leads to habits of caution and carefulness, the object of exciting this emotion is accomplished, and the diminution of it is attended with no evil. But if fear is continually excited, and no such habits are formed, then the susceptibility is lessened, while the good to be secured by it [pg 096] is lost. So, also, with emotions of sympathy. If we witness pain and suffering, and it induces habits of active devotion to the good of those who suffer, the diminution of the susceptibility is a blessing and no evil. But if we simply indulge emotions, and do not form the habits they were intended to secure, the power of sympathy is weakened, and the designed benefit is lost. Thus, again, with shame: if this painful emotion does not lead us to form habits of honor and rectitude, it is
- 68. continually weakened by repetition, and the object for which it was bestowed is not secured. And so with remorse: if this emotion is awakened without leading to habits of benevolence and virtue, it constantly decays in power, and the good it would have secured is for ever lost. It does not appear, however, that the power of emotion in the soul is thus destroyed. This is evident from the fact that the most hardened culprits, when brought to the hour of death, where all plans of future good cease to charm the mental eye, are often overwhelmed with the most vivid emotions of sorrow, shame, remorse and fear. And often, in the course of life, there are seasons when the soul returns from its pursuit of deluding visions to commune with itself in its own secret chambers. At such seasons, shame, remorse and fear take up their abode in their long-deserted dwelling, and ply their scorpion whips till they are obeyed, and the course of honor and virtue is resumed, or till the distracted spirit again flies abroad for comfort and relief. There is a great diversity in human character, resulting from the diverse proportions and combinations of those powers of mind which the race have in common. [pg 097] At the same time, there is a variety in the scale of being, or relative grade of each mind. While all are alike in the common faculties of the human mind, some have every faculty on a much larger scale than others, while some are of a very humble grade. The principle of habit has very great influence in modifying and changing these varieties. Thus, by forming habits of intellectual exercise, a mind of naturally humble proportions can be elevated considerably above one more highly endowed by natural constitution. So the training of some particular intellectual faculty, which by nature is deficient, can bring it up nearer to the level of other powers less disciplined by exercise.
- 69. In like manner, the natural susceptibilities can be increased, diminished or modified by habit. Certain tastes, that had little power, can be so cultivated as to overtop all others. So of the moral nature: it can be so exercised that a habit will be formed which will generate a strength and prominency that nature did not impart. One of the most important results of habit is its influence on faith or belief. Those persons who practice methods of false reasoning, who turn away from evidence and follow their feelings in forming opinions, eventually lose the power of sure, confiding belief. On the contrary, an honest, conscientious steadiness in seeking the truth and in yielding to evidence, secures the firmest and most reliable convictions, and that peace of mind which alone results from believing the truth. The will itself is also subject to this same principle. A strong will, that is trained to yield obedience to law [pg 098] in early life, acquires an ease and facility in doing it which belongs ordinarily to weak minds, and yet can retain all its vigor. And a mind that is trained to bring subordinate volitions into strict and ready obedience to a generic purpose, acquires an ease and facility in doing this which was not a natural endowment. Thus it appears that by the principle of habit every mind is furnished with the power of elevating itself in the scale of being, and of so modifying and perfecting the proportions and combinations of its constitutional powers, that often the result is that there is no mode of distinguishing between the effects of habit and those of natural organization.
- 70. Chapter XVIII. The Nature of Mind Our Guide to the Natural Attributes of God. The natural attributes of any mind are the powers and faculties to be exercised, while it is the action or voluntary use of these faculties that exhibits the moral attributes. Having gained the existence of a Great First Cause by the use of one principle of common sense, and the fact that this cause is an intelligent mind by another, it has been shown that a third of these principles leads to the belief that the natural attributes of God are like our own. We can not conceive of any other kind of minds than our own, because we have never had any past experience or knowledge of any other. [pg 099] But while we thus conclude that the mind of the Creator is, so far as we can conceive, precisely like our own in constitutional organization, we are as necessarily led to perceive that the extent of these powers is far beyond our own. A mind with the power, wisdom and goodness exhibited in the very small portion of his works submitted to our inspection, who has inhabited eternity, and matured through everlasting ages—our minds are lost in attempting any conception of the extent of such infinite faculties! Thus we are necessarily led to conceive of the Creator as possessing the intellectual powers described in previous pages. He perceives, conceives, imagines, judges and remembers just as we do.
- 71. So also all our varied susceptibilities to pleasure and pain exist in the Eternal Mind. The desire of good and the fear of evil which are the motive power in the human mind, exist also in the divine. Thus by the light of nature we settle the question that the existence of susceptibilities to pain and evil are not the results of the Creator's will, but are a part of the eternal nature of things which he did not originate or control. All the minds we ever knew or heard of are moved to action by desire to gain happiness and escape pain, and as we can conceive of no other kind of mind than our own, we must attribute to the Creator this foundation element of mental activity. Thus we are led to attribute to the Creator all those susceptibilities included in the moral sense, as described in previous pages. His mind, like ours, feels that whatever makes the most happiness with the least [pg 100] evil is right; that is to say, it is fitted to the eternal nature of things, of which his own mind is a part. So also the Creator possesses that sense of justice implanted in our own minds, which involves the desire of good to those who make happiness, and of evil to those who destroy happiness; and which also demands that such retributions be proportioned to the good and evil done, and to the power of the agent. So also we must conceive of the Creator as possessing the susceptibility of conscience, which includes in the very constitution of mind retributions for right and wrong action. Again, we are led to conceive of God as a rational free agent, with power to choose either that which excites the strongest desire or that which is perceived to be best on the whole for all concerned, even if it does not excite the strongest desire. Again, we are to conceive of the Creator as possessing a belief in those principles of reason which he has implanted in our minds, and
- 72. made our guide in all matters, both of temporal and religious concern. Again, our experience of the nature and history of mind, leads to the inference that no being has existed from all eternity in solitude, but that there is more than one eternal, uncreated mind, and that all their powers of enjoyment from giving and receiving happiness in social relations have been in exercise from eternal ages. This is the just and natural deduction of reason and experience, as truly as the deduction that there is at least one eternal First Cause. Again, all our experience of mind involves the idea of the mutual relation of minds. We perceive that minds are made to match to other minds, so that there [pg 101] can be no complete action of mind, according to its manifest design, except in relation to other beings. A mind can not love till there is another mind to call forth such emotion. A mind can not bring a tithe of its power into appropriate action except in a community of minds. The conception of a solitary being, with all the social powers and sympathies of the human mind infinitely enlarged, and yet without any sympathizing mind to match and meet them, involves the highest idea of unfitness and imperfection conceivable, while it is contrary to our uniform experience of the nature and history of mind. It has been argued that the unity of design in the works of nature proves that there is but one creating mind. This is not so, for in all our experience of the creations of finite beings no great design was ever formed without a combination of minds, both to plan and to execute. The majority of minds in all ages, both heathen and Christian, have always conceived of the Creator as in some way existing so as to involve the ideas of plurality and of the love and communion of one mind with another. And yet the unity and harmony of all created things as parts of one and the same design, teach a degree of unity in the authorship of the universe never known in the complex action of finite minds.
- 73. Thus a unity and plurality in the Creator of all things is educed by reason and experience from the works of nature. [pg 102]
- 74. Chapter XIX. The Nature of Mind Our Guide to the Moral Attributes of God. Having employed the principles of common sense to gain a knowledge of the natural attributes of God, we are next to employ the same principles to gain his moral character; or those attributes which are exhibited in willing. In other words, we are to seek the character of God as expressed in his works or deeds. In our experience of the moral character of minds in this world, we find that some of the highest grades as to intellect and susceptibilities, are lowest as to good-willing. How is it, then, with the highest mind of all? Does he so prefer evil to good, that he deliberately plans for the production of evil when he has power to produce happiness in its place? Or does he sometimes prefer evil and sometimes good, with the variable humors of the human race? Or does he always prefer good when it costs him no trouble or sacrifice, but never when it does? Or is he one who invariably chooses what is best for all, even when it involves painful sacrifices to himself? In seeking a reply to these momentous questions, we return once more to the principle of common sense before stated, i.e., the nature of any work or contrivance is proof of the character and design of the author. In examining the works of the Creator, we find that the material world impresses us as wisely adjusted and good in construction, only as it is fitted to give enjoyment to sentient beings. It is the intelligent, [pg 103] feeling, acting minds that give the value to
- 75. every other existence. If there were no minds, all perception of beauty, fitness and goodness would perish. It is minds, therefore, which are the chief works of the Creator's hand, and which give value to all others. If the nature of these minds is evil, then the author of them is proved to be evil by his works. If their nature is good and perfect, then their author is proved to be good and perfect. Here again we are driven back to our own minds to gain the only conceptions possible to us, not only of wisdom, but of goodness or benevolence. On examination, we shall find that we can form no idea of these qualities which does not involve a limitation of power. Our idea of power is that which we gain when we will to move our bodies or to make any other change, and this change ensues. Our only idea of a want of power is gained when the choice or willing of a change or event does not produce it. Whenever, therefore, it shall appear that the Creator wills or wishes a thing to exist or to be changed, and that change or existence does not follow his so willing, we can not help believing that he has not the power to produce it? Again; our idea of perfectness always has reference to power; for a thing is regarded as perfect in construction only when there is no power in God or man to make it better. When any arrangement is as good as it can be, so that neither God nor man has power to make it better, we regard it as perfect, even when there is some degree of evil involved. We are now prepared to define what is included in [pg 104] the terms perfect wisdom and perfect benevolence, when applied to the Creator or to any other being, thus: A perfectly wise being is one who invariably wills the best possible ends and the best possible means of accomplishing those ends.
- 76. An imperfectly wise being is one who does not invariably do this. A perfectly benevolent being is one who invariably wills the most good and the least evil in his power. An imperfectly benevolent being is one who does not invariably will thus. The degree in which a being is ranked as wise and good is estimated by the extent to which his willing good or evil corresponds with his power. Thus it appears that, in a system where evil exists, the very idea of perfect benevolence and wisdom involves the supposition of a limitation of power. To return, then, to the question as proposed at the commencement of the chapter—Is the Creator a being who prefers good to evil invariably, or is he one who only sometimes prefers evil to good, and at other times prefers good to evil, with the varying humors of man; or does he invariably choose what is best for all, even in cases where it may cost personal sacrifice and suffering to himself? It will be the object of what follows to prove that the last supposition is the true one. In attempting this, we again take the principle of common sense, that “the nature of any contrivance proves the design and character of the author.” Then we proceed to a review of the nature, first of mind, and next of the material world, to prove that the design or chief end of the Creator is, not to make happiness [pg 105] irrespective of the amount, but to produce the greatest possible happiness with the least possible evil. In other words, we are to seek for proof that God has done all things for the best, so that he has no power to do better. In still another form, we are to seek for evidence, in the nature of God's works, that he has ever done the best he could, so that the amount of evil that ever was or ever will exist, is not caused by his
- 77. willing it, but by his want of power to prevent it; so that any change would be an increase of evil and a lessening of good to the universe as a whole. In pursuing this attempt, it will be needful to reproduce two or three chapters of a work by the author, already before the public, entitled, The Bible and the People; or, Common Sense applied to Religion. In this work the nature of mind is presented very much more in detail, for the same purpose as that here indicated. What will now follow is a brief review of previous chapters in that work, as a summary of the evidence there presented that the chief end of God in all his works is to produce the greatest possible happiness with the least possible evil. Whenever we find any contrivances all combining to secure a certain good result, which, at the same time, involve some degree of inevitable evil, and then discover that there are contrivances to diminish and avoid this evil, we properly infer that the author intended to secure as much of the good with as little of the evil as possible. For example, a traveler finds a deserted mine, and all around he discovers contrivances for obtaining gold, and, at the same time, other contrivances for getting rid of the earth mixed with it. The [pg 106] inevitable inference would be that the author of these contrivances designed to secure as much gold with as little earth as possible; and should any one say that he could have had more gold and less earth if he chose to, the answer would be that there is no evidence of this assertion, but direct evidence against it. Again: should we discover a piece of machinery in which every contrivance tended to secure a speed in movement, produced by the friction of wheels against a rough surface, and at the same time other contrivances were found for diminishing all friction that was useless, we should infer that the author designed to secure the greatest possible speed with the least possible friction.
- 78. In like manner, if we can show that mind is a contrivance that acts by the influence of fear of evil, and that pain seems as indispensable to the action of a free agent as friction is to motion; if we can show that there is no contrivance in mind or matter which is designed to secure suffering as its primary end; if we can, on the contrary, show that the direct end of all the organizations of mind and matter is to produce happiness; if we can show that it is only the wrong action of mind that involves most of the pain yet known, so that right action, in its place, would secure only happiness; if we can show contrivances for diminishing pain, and also contrivances for increasing happiness by means of the inevitable pain involved in the system of things, then the just conclusion will be gained that the Author of the system of mind and matter designed “to produce the greatest possible happiness with the least possible evil.” [pg 107] In the pages which follow, we shall present evidence exhibiting all these particulars. The only way in which we learn the nature of a thing is to observe its qualities and actions. This is true of mind as much as it is of matter. Experience and observation teach that the nature of mind is such, that the fear of suffering is indispensable to secure a large portion of the enjoyment within reach of its faculties, and that the highest modes of enjoyment can not be secured except by sacrifice, and thus by more or less suffering. This appears to be an inevitable combination, as much so as friction is inevitable in machinery. We have the evidence of our own consciousness that it is fear of evil to ourselves or to others that is the strongest motive power to the mind. If we should find that no pain resulted from burning up our own bodies, or from drowning, or from any other cause; if every one perceived that no care, trouble, or pain resulted from losing all kinds of enjoyment, the effort to seek it would be greatly diminished.
- 79. If we could desire good enough to exert ourselves to seek it, and yet should feel no discomfort in failing; if we could lose every thing, and feel no sense of pain or care, the stimulus to action which experience has shown to be most powerful and beneficent would be lost. We find that abundance of ease and prosperity enervates mental power, and that mind increases in all that is grand and noble, and also in the most elevating happiness, by means of danger, care and pain. We may properly infer, then, that evil is a necessary part of the experience of a perfectly-acting mind. [pg 108] So strong is the conviction that painful penalties are indispensable, that the kindest parents and the most benevolent rulers are the most sure to increase rather than diminish those that are already involved in the existing nature of things. Again: without a revelation we have no knowledge of any kind of mind but by inference from our experience in this state of being. All we know of the Eternal First Cause is by a process of reasoning, inferring that his nature must be like the only minds of which we have any knowledge. We assume, then, that he is a free agent, regulated by desire for happiness and fear of evil. We thus come to the conclusion that this organization of mind is a part of the fixed and eternal nature of things, and does not result from the will of the Creator. His own is the eternal pattern of an all- perfect mind, and our own are formed on this perfect model, with susceptibilities to pain as an indispensable motive power in gaining happiness. We will now recapitulate some of the particulars in the laws and constitution of mind which tend to establish the position that its Creator's grand design is “to produce the greatest possible happiness with the least possible evil.”
- 80. Intellectual Powers. First, then, in reference to the earliest exercise of mind in sensation. The eye might have been so made that light would inflict pain, and the ear so that sound would cause only discomfort. And so of all the other senses. But the condition of a well-formed, healthy infant [pg 109] is a most striking illustration of the adaptation of the senses to receive enjoyment. Who could gaze on the countenance of such a little one, as its various senses are called into exercise without such a conviction? The delight manifested as the light attracts the eye, or as pleasant sounds charm the ear, or as the limpid nourishment gratifies its taste, or as gentle motion and soft fondlings soothe the nerves of touch, all testify to the benevolent design of its Maker. Next come the pleasures of perception as the infant gradually observes the qualities of the various objects around, and slowly learns to distinguish its mother and its playthings from the confused mass of forms and colors. Then comes the gentle curiosity as it watches the movement of its own limbs, and finally discovers that its own volitions move its tiny fingers, while the grand idea that it is itself a cause is gradually introduced. Next come the varied intellectual pleasures as the several powers are exercised in connection with the animate and material world around, in acquiring the meaning of words, and in imitating the sounds and use of language. The adult, in toiling over the dry lexicon, little realizes the pleasure with which the little one is daily acquiring the philosophy, grammar, and vocabulary of its mother tongue.
- 81. A child who can not understand a single complete sentence, or speak an intelligible phrase, will sit and listen with long-continued delight to the simple enunciation of words, each one of which presents a picture to his mind of a dog, a cat, a cow, a horse, a whip, a ride, and many other objects and scenes that have given pleasure in the past; while the single words, [pg 110] without any sentences, bring back, not only vivid conceptions of these objects, but a part of the enjoyment with which they have been connected. Then, as years pass by, the intellect more and more administers pleasure, while the reasoning powers are developed, the taste cultivated, the imagination exercised, the judgment employed, and the memory stored with treasures for future enjoyment. In the proper and temperate use of the intellectual powers, there is a constant experience of placid satisfaction, or of agreeable and often of delightful emotions, while no one of these faculties is productive of pain, except in violating the laws of the mental constitution.
- 82. The Susceptibilities. In regard to the second general class of mental powers—the susceptibilities—the first particular to be noticed is the ceaseless and all-pervading desire to gain happiness and escape pain. This is the mainspring of all voluntary activity; for no act of volition will take place till some good is presented to gain, or some evil to shun. At the same time, as has been shown, the desire to escape evil is more potent and effective than the desire for good. Thousands of minds that rest in passive listlessness, when there is nothing to stimulate but hope of enjoyment, will exert every physical and mental power to escape impending evil. The seasons of long-continued prosperity in nations always tend to a deterioration of intellect and manhood. It is in seasons of danger alone that fear wakes up the highest energies, and draws forth the heroes of the race. Mind, then, is an existence having the power of that [pg 111] self- originating action of choice which constitutes free agency, while this power can only be exercised when desires are excited to gain happiness or to escape pain. This surely is the highest possible evidence that its Author intended mind should thus act. But a mind may act to secure happiness and avoid pain to itself, and yet may gain only very low grades of enjoyment, while much higher are within reach of its faculties. So, also, it may act to gain happiness for itself as the chief end in such ways as to prevent or destroy the higher happiness of others around. In reference to this, we find those susceptibilities which raise man to the dignity of a rational and moral being.