![1 A Study on Approximate
Fixed Point Property in
Intuitionistic Fuzzy
n-Normed Linear Spaces
Pradip Debnath
Assam University
CONTENTS
1.1 Introduction .......................................................................................................1
1.2 Preliminaries......................................................................................................2
1.3 Approximate Fixed Points in IFnNLS...............................................................3
1.4 Intuitionistic n-Fuzzy Contraction and Nonexpansive Mappings .....................5
Bibliography ............................................................................................................12
1.1 INTRODUCTION
Fuzzy set theory [31] has widespread applications in different branches of mathemat-
ical science such as theory of functions [18,28], topological and metric spaces [10,14,
19], and approximation theory [1]. It has also been applied in control of chaos [13],
quantum physics [22], computer programming [16], population dynamics [3], and
nonlinear dynamical systems [17].
The initial notion of fuzzy norm was put forward by Katsaras [20]. It was further
improved and re-defined by various mathematicians considering particular areas of
application [2,12,19,22,29]. The notion of an intuitionistic fuzzy n-normed linear
space (IFnNLS) [23] generalizes an intuitionistic fuzzy normed space which was
introduced by Saadati and Park [24].
The concept of fixed point property of mappings plays a significant role in the
investigation of analytic properties of a normed linear space. In this chapter, we in-
troduce the concept of approximate fixed point property in an IFnNLS. We also es-
tablish the relation between asymptotic regularity and fixed point property. Further,
we introduce and investigate the properties of various types of intuitionistic n-fuzzy
contraction mappings and their connection with fixed point property.
DOI: 10.1201/9781003312017-1 1](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-18-320.jpg)
![2 Soft Computing
The definition of convergence of a sequence in an IFnNLS is crucial for the in-
vestigation of its analytic properties. A new and modified definition of convergence
was introduced in [25,26]. The results of this chapter are established on the basis of
this definition. For more relevant work, we refer to [4–7,11,15].
1.2 PRELIMINARIES
The same topology is induced by an intuitionistic fuzzy metric and a fuzzy met-
ric [15]. Hence, to generate original and new results in intuitionistic fuzzy setting, it
was necessary to re-define the notion of intuitionistic fuzzy norm [21,30]. To serve
this purpose, an improved definition of an IFnNLS was put forward by Debnath and
Sen [8,9] as given next.
Definition 1.1 The five-tuple (V,η,γ,∗,◦) is called an IFnNLS, where V is a vector
space of dimension d ≥ n over a field F, ∗ is a continuous t-norm, ◦ is a continuous
t-conorm, η,γ are fuzzy sets on Vn × (0,∞), η signifies the degree of membership,
and γ signifies the degree of non-membership of (u1,u n
2,...,un,t) ∈ V ×(0,1). The
following conditions are satisfied for every (u1,u n
2,...,un) ∈ V and s,t > 0:
(i) η(u1,u2,...,un,r) = 0 and γ(u1,u2,...,un,r) = 1 for all non-positive real num-
ber t,
(ii) η(u1,u2,...,un,r) = 1 and γ(u1,u2,...,un,r) = 0 for all positive r if and only
if u1,u2,...,un are linearly dependent,
(iii) η(u1,u2,...,un,r) and γ(u1,u2,...,un,r) are invariant under any permutation
of u1,u2,...,un,
(iv) η(u u ,cu t
1, 2,... n,r) = η(u1,u2,...,un, )
|c| and γ(u1,u2,...,cun,r) =
γ(u1,u r
2,...,un, )
| if c = 0, ∈
c| c F,
(v) η(u1,u2,...,u
′
n,s)∗η(u1,u2,...,u ≤
n,r) η(u1,u2,...,u
′
n +un,s+r),
(vi) η u u2 u s γ u
′ ′
( 1, ,..., n, )◦ (u1, 2,...,un,r) ≥ γ(u1,u2,...,un +un,s+r),
(vii) η(u1,u2,...,un,r) : (0,∞) → [0,1] and γ(u1,u2,...,un,r) : (0,∞) → [0,1] are
continuous in r,
(viii) limr→∞ η(u1,u2,...,un,r) = 1 and limr→0 η(u1,u2,...,un,r) = 0,
(ix) limr→∞ γ(u1,u2,...,un,r) = 0 and limr→0 γ(u1,u2,...,un,r) = 1.
Definition 1.2 [25,26] Let (V,η,γ,∗,◦) be an IFnNLS. A sequence v = {vk} in V
is called convergent to ς ∈ V with respect to the intuitionistic fuzzy n-norm (IFnN)
(η n
,γ) if, for every ε ∈ (0,1), r > 0 and u1,u2,...,un−1 ∈V, there exists k0 ∈ N such
that η(u1,u2,...,un−1,vk −ς,r) > 1−ε and γ(u1,u2,...,un−1,vk −ς,r) < ε for all
η n
k k . We denote it by η n ( ,γ
γ limv ς or v
)
≥ 0 ( , ) − = k → ς as k → ∞.
proposition 1.1 [27] In an IFnNLS V, η γ n
( , ) − limv = ς if and only if for every
r > 0 and u1,u2,...,un−1 ∈V, η(u1,...,un−1,vk −ς,r) → 1 and γ(u1,...,un−1,vk −
ς,r) → 0 as k → ∞.
Definition 1.3 [25,26] Let (V,η,γ,∗,◦) be an IFnNLS. A sequence v = {vk} in V
is said to be Cauchy with respect to the IFnN (η,γ n
) if, for every ε ∈ (0,1), r > 0
̸](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-19-320.jpg)
![A Study on Approximate Fixed Point Property 3
and u1,u2,...,un−1 ∈V, there exists k0 ∈ N such that η(u1,u2,...,un−1,vk −vm,r) >
1−ε and γ(u1,u2,...,un−1,vk −vm,r) < ε for all k,m ≥ k0.
Definition 1.4 An IFnNLS V is complete with respect to the IFnN (η,γ n
) if every
Cauchy sequence in it is convergent.
proposition 1.2 [27] If every Cauchy sequence in an IFnNLS V has a convergent
subsequence, then V is complete.
1.3 APPROXIMATE FIXED POINTS IN IFNNLS
Now we are ready to present our main results. First we define the concept of an intu-
itionistic n-fuzzy approximate fixed point (InFAFP) in an IFnNLS as given below.
Definition 1.5 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. ϖ0 ∈ V
is said to be an intuitionistic n-fuzzy approximate fixed point (InFAFP) of Γ if for
every ε > 0 and ω1,ω2,...,ωn−1 ∈ V we have
µ(ω1,ω2,...,ωn−1,Γ(ϖ0)−ϖ0,s) > 1−ε
and ν(ω1,ω2,...,ωn−1,Γ(ϖ0)−ϖ0,s) < ε
for all s > 0. We denote the set of all InFAFP of Γ by AFPε (Γ).
Further, we say that the mapping Γ has the InFAFP property if the set AFPε (Γ)
is nonempty for every ε > 0 and ω1,ω2,...,ωn−1 ∈ V.
The example below illustrates Definition 1.5.
Example 1.1 Let V = (0,1 n
) ,ω n
i = (ωi1,ωi2,...,ωin) ∈ (0,1) for each i = 1,2,...,n
with ' '
' ω ω
' 11 ··· 1n '
'
' . . '
∥ω1,ω2,...,ω .
n∥ = abs' . . .
.
' . . ',
'
' ωn1 ··· ω '
nn
and let a ∗ b = ab, a ◦ b = min{a + b,1} for all a,b ∈ [0,1]. Now for all
w n
1,w r
2,...,wn ∈ (0,1) and r > 0, let us define η(w1,w2,...,wn,r) = r+∥w1,w2,...,wn∥
and γ w w w r ∥w1,w2,...,w
( , ,..., , ) = n∥
1 2 n . Then ((0,1 n
) , ,
r+∥w1,w η γ,∗,◦)
∥ an IFnNLS.
2,...,w is
n
Consider the mapping Γ : V → V defined by
Γ(θ1,θ 1 2 2
2,...,θn) = (θ1 ,θ2 ,...,θn )
for all (θ1,θ2,...,θn) ∈ V. Clearly, Γ has no fixed point in V. Hence, we try to
investigate its InFAFP.
We observe that for every ε > 0, w1,w2,...,w n
n−1 ∈ (0,1) and r > 0, there exists
u ∈ (0,1 n
) such that
r
η(w1,w2,...,wn−1,Γ(u)−u,r) = > 1−ε
r +∥w1,w2,...,wn−1,Γ(u)−u∥](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-20-320.jpg)
![2 Sequential Extended
Parametric and Sequential
Extended Fuzzy b-Metrics
with an Application in
Integral Equations
Marija V. Paunović
University of Kragujevac
Samira Hadi Bonab, Vahid Parvaneh,
and Farhan Golkarmanesh
Islamic Azad University
CONTENTS
2.1 Introduction .....................................................................................................15
2.2 Main Results....................................................................................................17
2.3 Some Fixed Point Theorems ...........................................................................19
2.4 Sequential Extended Fuzzy b-Metric Spaces ..................................................23
2.5 Application ......................................................................................................25
Bibliography ............................................................................................................27
2.1 INTRODUCTION
In recent years, metric spaces have been generalized by many authors, which are
expressed in different approaches. Many interesting spaces of the two-variable met-
ric type are: b−metric space [2,5], rectangular metric space [4], parametric metric
space [1,16,26], extended parametric b-metric space [29], sequential extended S-
metric space [20], JS-metric space [17], extended b-metric space [19,30], modular
metric space [25], multiplicative metric space [3], cone b- metric space [12], C∗-
algebra valued metric space [27,28], vector-valued metric space [13,14], etc.
The purpose of this chapter is to introduce a new type of generalized metric
spaces, called the sequential extended parametric b-metric space (SEPbMS), as a
DOI: 10.1201/9781003312017-2 15](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-32-320.jpg)
![16 Soft Computing
generalization of extended parametric b-metric space [29], using JS-contractive type
mappings. Also, we prove some fixed point theorems for JS-contractive type map-
pings in SEPbMSs, and we obtain some new fixed point results in triangular sequen-
tial extended fuzzy b-metric spaces (SEFbMSs) induced by this new structure. An
example and an application are given to confirm the results.
For further details, the readers are referred to the book [6] and the references
therein.
Definition 2.1 [29] In a nonempty set V, the mapping ρ : 2
V × (0,+∞) → [0,+∞)
is said to be a parametric metric on V provided that,
1. ρ(ℏ,ℏ′,ς) = 0 if and only if ℏ = ℏ′;
2. ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0;
3. ρ(ℏ,ℏ′,ς) ≤ ρ(ℏ,υ,ς)+ρ′ (υ,ℏ′,ς) for all ℏ,ℏ′, ∈
ℏ υ V and for all ς > 0.
Then (V,ρ) is called a parametric metric space.
Definition 2.2 [32] In a non-empty set V a mapping ρ : 2
V ×(0,+∞) → [0,+∞) is
said to be an extended parametric b-metric (EPbM) if for all ℏ,ℏ′,υ ∈ V and ς > 0:
(a) ρ(ℏ,ℏ′,ς) = 0 implies ℏ = ℏ′ for all ς > 0;
(b) ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0;
(c) ρ(ℏ,ℏ′
,ς) ≤ Ω(ρ(ℏ,υ,ς)+ρ(υ,ℏ′
,ς)),
where Ω : [0,∞) → [0,∞) is an unto strictly increasing continuous function with
Ω−1(ς) ≤ ς ≤ Ω(ς).
Then (V,ρ) is called an EPbMS with control function Ω.
Obviously, for Ω(ς) = sς, the EPbM reduces to parametric b-metric.
Let V be a non-empty set and ϑ 2
g : V → [0,∞] be a mapping. For any ℏ ∈ V, let
us define the set
C(ϑg,V,ℏ) = {{ℏn} ⊂ V : lim ϑg( n, ) = 0}.
n→∞
Definition 2.3 [17] Let ϑg : 2
V → [0,∞] be a mapping which satisfies:
1. ϑg(ℏ,ℏ′) = 0 implies ℏ = ℏ′;
2. for every ℏ,ℏ′ ∈ V, we have ϑg(ℏ,ℏ′) = ϑ ′
g(ℏ ,ℏ);
3. if (ℏ,ℏ′ 2
) ∈ V and {ℏn} ∈ C(ϑg,V,ℏ), then ϑ ′
g(ℏ,ℏ ) ≤ plimsupϑg(ℏn,ℏ′), for
n→∞
ℏ ℏ (2.1)
some p > 0.
The pair (V,ϑg) is called a JS-metric space.
Since the Banach fixed point theorem is very attractive and practical, many re-
searchers have tried to generalize it in recent years. These generalizations have either
been done in generalized metric spaces or constructing the new contractions in which
the Banach contraction being obtained as a special case of these new contractions.
Of course, the generalization of Banach’s fixed point theorem is not limited to met-
ric spaces. Rather, the Cartesian product of metric spaces and the Darbo’s theorem
in terms of measure of noncompactness are also generalizations of Banach’s fixed
point theorem. Each of these theorems, in turn, has been studied in various ways by
many authors.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-33-320.jpg)
![Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 17
It should be noted that by obtaining these beautiful results, the existence and
uniqueness of the solution for the functional integral equations, fractional integral
equations, differential equations, differential fractional equations, and matrix equa-
tions have been proved.
Unswerving with Ref. [18], consider the family of all functions θ : (0,∞) → (1,∞)
so that:
(θ1) θ is increasing;
(θ2) lim θ(ρn) = 1 iff lim ρn = 0 for each sequence {ρn} ⊆ (0,∞);
n→∞ n→∞
(θ θ
3) lim (ρ)−1
ρκ = λ for some κ ∈ (0,1) and for some λ ∈ (0,∞] is signified by J0.
ρ→0+
Theorem 2.1 [17,18] A self-mapping ϒ on a complete metric space (V,d) so that
ℏ,ℏ′
∈ V, d(ϒℏ,ϒℏ′
) = 0 ⇒ θ(d ,ℏ α
(ϒℏ,ϒℏ′
)) ≤ θ(d(ℏ ′
))
̸
where θ ∈ J0 and α ∈ (0,1) possesses a unique fixed point.
Reminder that the Banach contraction principle is a specific instance of Theorem
2.1.
We denote by J the family of functions θ : (0,∞) → (1,∞) so that:
(θ1) θ is continues and increasing;
(θ2) lim θ(ρn) = θ(ρ) iff lim ρn = ρ for each sequence {ρn} ⊆ (0,∞);
n→∞ n→∞
In this chapter, via combination of Kannan contractions, Chatterjea contractions,
and JS-contractions, we state and prove some fixed point results in two classes of
generalized metric spaces.
2.2 MAIN RESULTS
In this section, first we introduce a new type of EPbMSs. To expand such a concept,
we first define,
{ }
ð(ρ,V,ℏ) := {ℏn} ⊂ V : lim ρ(ℏn,ℏ,ς) = 0 for all ς > 0 ,
n→∞
where ρ : 2
V ×(0,+∞) → [0,+∞) is a given mapping.
Definition 2.4 In a non-empty set V, a mapping ρ : 2
V ×(0,+∞) → [0,+∞) is said
to be a SEPbM if for all ℏ,ℏ′ ∈ V and for all ς > 0:
(a) ρ(ℏ,ℏ′,ς) = 0 for all ς > 0 implies ℏ = ℏ′;
(b) ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0;
(c) ρ(ℏ,ℏ′
,ς) ≤ Ω limsupρ(ℏ ,ℏ′
n ,ς) , where {ℏn} ∈ ð(ρ,V,ℏ) and Ω :
n→∞
[0,∞] → [0,∞] is an unto strictly increasing continuous function with Ω−1(ς) ≤ ς ≤
Ω(ς) for all 0 ≤ ς < ∞.
The triplet (V,ρ,Ω) is called a SEPbMS. For simplicity, we denote it by (V,ρ).](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-34-320.jpg)
![18 Soft Computing
Definition 2.5 Let (V,ρ) be a SEPbMS, {ℏn} ⊆ V and ℏ ∈ V.
(i) {ℏn} is said to be convergent and converges to ℏ, if {ℏn} ∈ ð(ρ,V,ℏ).
(ii) {ℏn} is said to be Cauchy, if lim ρ(ℏn,
→
ℏm,ς) = 0 for all ς > 0.
n,m ∞
(iii) (V,ρ) is said to be complete, if every Cauchy sequence is a convergent se-
quence.
Example 2.1 Let the triplet (V,ρ,Ω) be an EPbMS. If we define ρ(ℏ,ℏ′,ς) =
eς ℏ ℏ′ p
( − ) (p ≥ 1), then the pair (V,ρ) will be a SEPbMS on V for Ω(ς) = e2pς
for all ς ≥ 0.
proposition 2.1 If (V,ρ) be a parametric b-metric space with parameter s, then ρ
is also a SEPbM on V.
Proof. If (V,ρ) be a parametric b-metric space, then ρ undoubtedly fulfills the first
two circumstances of Definition 2.4. We just indicate that ρ also gratifies the third
condition of Definition 2.4.
Since ρ is a parametric b-metric, then for all ℏ,ℏ′ ∈ V and for any se-
quence {ℏn} ∈ ð(ρ,V,ℏ), according to triangle inequality we have, ρ(ℏ,ℏ′
,ς) ≤
slimsupρ(ℏ ,ℏ′
n ,ς) for all ς > 0.
n→∞
Then if we choose Ω(ς) = sς for all ς ∈ [0,∞], then we have ρ(ℏ,ℏ′
,ς) ≤
Ω limsupρ(ℏn,ℏ′
,ς) for all ℏ,ℏ′ ∈ V, for all ς > 0 and for all {ℏn} ∈ ð(ρ,V,ℏ).
n→∞
Therefore, (V,ρ) is also a SEPbMS.
Definition 2.6 Let (V,ρ) and (Y,ρ∗) be two SEPbMSs. A mapping ϒ : V → Y is
called continuous at a point ℏ ∈ V if for any ε > 0 there is δε > 0 such that for any
υ ∈ V, ρ∗(ϒυ,ϒℏ,ς) < ε whenever ρ(υ,ℏ,ς) < δε , for all ς > 0. ϒ is said to be
continuous on V if ϒ is continuous at each point of V.
proposition 2.2 In a SEPbMS (V,ρ), if a sequence {ℏn} is convergent, then it con-
verges to a unique element in V.
Proof. Suppose ℏ,ℏ′ ∈ V and ς > 0 be arbitrary such that ℏn → ℏ and ℏn → ℏ′
as n → ∞. Then we have, ρ(ℏ,ℏ′
,ς) ≤ Ω limsupρ(ℏn,ℏ′
,ς) which implies that
n→∞
ρ(ℏ,ℏ′,ς) ≤ Ω(0) = 0, i.e., ℏ = ℏ′.
proposition 2.3 Let (V,ρ) be a SEPbMS and {ℏn} ⊂ V converges to some ℏ ∈ V.
Then ρ(ℏ,ℏ,ς) = 0.
Proof. Since {ℏn} converges to ℏ ∈ V, so lim ρ(ℏn,ℏ,ς) = 0. Therefore, we have
n→∞
ρ(ℏ,ℏ,ς) ≤ Ω limsupρ(ℏnℏ,ς) = Ω(0) = 0,
n→∞
which implies ρ(ℏ,ℏ,ς) = 0.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-35-320.jpg)
![Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 19
proposition 2.4 For a Cauchy sequence {ℏn} in a SEPbMS (V,ρ,Ω) such that
Ω−1 is continuous, if {ℏn} has a convergent sub-sequence {ℏn }
k
which converges to
ℏ ∈ V, then {ℏn} also converges to ℏ ∈ V.
Proof. From condition (c) of Definition 2.4 we have, ρ(ℏn,ℏ,ς) ≤ Ω
limsupρ(ℏn,ℏnk
,ς) which implies that Ω−1
(ρ(ℏn,ℏ,ς)) ≤ limsupρ(ℏn,ℏnk
,ς)
k→∞ k→∞
for all n ∈ N and ς > 0.
Because of the Cauchyness of the sequence {ℏn}, it follows that lim ρ(ℏn,ℏnk
,ς)
n,k→∞
= 0 and thus, Ω−1 (ρ(ℏn,ℏ,ς)) → 0 as n → ∞ which implies that ρ(ℏn,ℏ,ς) → 0 as
n ∞ 1
→ , since Ω− is continuous. Hence, {ℏn} converges to ℏ ∈ V.
proposition 2.5 In a SEPbMS (V,ρ), if a self-mapping ϒ is continuous at ℏ ∈ V,
then {ϒℏn} ∈ ð(ρ,V,ϒℏ) for any sequence {ℏn} ∈ ð(ρ,V,ℏ).
Proof. Let ε > 0 be given. Since ϒ is continuous at ℏ, then there exists δε > 0 such
that ρ(υ,ℏ,ς) < δε implies ρ(ϒυ,ϒℏ,ς) < ε.
As {ℏn} converges to ℏ, so for δε > 0, there exists N ∈ N such that ρ(ℏn,ℏ,ς) < δε
for all n ≥ N. Therefore, for any n ≥ N, ρ(ϒℏn,ϒℏ,ς) < ε and thus ϒℏn → ϒℏ as
n → ∞, that is, {ϒℏn} ∈ ð(ρ,V,ϒℏ).
Let (V,ρ) be a SEPbMS with associate function Ω. Define,
{ }
B(ℏ,η) := ℏ′
∈ V : ρ(ℏ,ℏ′
,ς) < ρ(ℏ,ℏ,ς)+η for all ς > 0
and
B[ℏ,η] := ℏ′
∈ V : ρ(ℏ,ℏ′
,ς) ≤ ρ(ℏ,ℏ,ς)+η for all ς > 0
for all ℏ ∈ V and for all η > 0.
{ }
Remark 2.1 Evidently,
τρ := {0
/}∪{V(= 0
/) ⊂ V : for any ℏ ∈ V, there is η > 0 so that B(ℏ,η) ⊂ V}
forms a topology on V.
̸
2.3 SOME FIXED POINT THEOREMS
In this section, we suppose that θ ∈ J .
Theorem 2.2 Let (V,ρ) be a complete SEPbMS and ϒ : V → V be a mapping such
that:
(i) θ(ρ(ϒℏ,ϒℏ′,ς)) ≤ θ(ρ(ℏ,ℏ′ α
,ς)) for all ℏ,ℏ′ ∈ V and for some α ∈ (0,1),
(ii) there is ℏ0 ∈ V so that
{ ( ) }
δ(ρ,ϒ,ℏ0) := sup ρ ϒi
ℏ0,ϒ j
ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞.
Then ϒ has at least one fixed point in V. Moreover if ℏ and ℏ′ are two fixed points of
ϒ in V with ρ(ℏ,ℏ′,ς) < ∞ then ℏ = ℏ′.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-36-320.jpg)
![Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 21
(ii) there is ℏ0 ∈ V so that
∆(ρ,ϒ,ℏ0) := sup ρ ϒi
ℏ0,ϒ j
ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞.
{ ( ) }
Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ∈ N, converges to some
ℏ ∈ V. If ρ(ℏ,ϒℏ,ς) < ∞, then ℏ ∈ V is a fixed point of ϒ. Furthermore, if ℏ′ is a
fixed point of ϒ in V such that ρ(ℏ,ℏ′,ς) < ∞ and ρ(ℏ′,ℏ′,ς) < ∞, then ℏ = ℏ′.
Proof. For all p ≥ 1 and for all i, j = 1,2 p
,..., θ ρ(ϒ +iℏ p
,ϒ +j
0 ℏ0,ς)
ρ(ϒ p−1+i
θ Ω
ℏ0,ϒ p+i γ
ℏ0,
− ς
1 )+ρ(ϒ p−1+jℏ0,ϒ p+jℏ0,ς) p γ
≤ ≤ ( ( , , ))
2 θ ∆ ρ ϒ ℏ0 .
( )
This implies that
( )
θ ∆ ρ ϒ p+1
ℏ θ sup ρ ϒ p+i
ℏ ϒ p+j
( , , 0) = ( 0, ℏ0,ς p
) ≤ θ (∆(ρ,ϒ ,ℏ γ
0))
i,j≥1
for all p ≥ 1.
Then continuing in an analogous technique as in Theorem 2.2, it can be effort-
lessly exposed that {ℏn} is a Cauchy sequence in V and by the completeness of V
there is some ℏ ∈ V such that limℏn = ℏ.
n
Now,
θ(ρ(ℏn+1,ϒℏ,ς)) = θ(ρ(ϒℏn,ϒℏ,ς))
≤ θ(Ω−1
[ρ(ℏ γ
n,ϒℏn,ς)+ρ(ℏ,ϒℏ,ς)])
≤ θ(Ω−1
[ρ(ℏ γ
n,ℏn+1,ς)+ρ(ℏ,ϒℏ,ς)]) ,
for all n ≥ 0, which implies that
limsupθ(ρ(ℏn+1,ϒℏ,ς)) ≤ θ 1
(Ω−
[ρ(ℏ,ϒℏ,ς γ
)]) < ∞.
n→∞
On the other hand,
ρ(ℏ,ϒℏ,ς) ≤ Ω limsupρ(ℏn+1,ϒℏ,ς)
n→∞
(
≤ Ω θ−
)
1
(θ(Ω−1
[ρ(ℏ,ϒℏ,ς γ
)]) ) .
If ρ(ℏ,ϒℏ,ς) > 0 then,
Ω−1
(ρ(ℏ,ϒℏ,ς)) ≤ θ−1
θ(Ω−1
[ρ(ℏ γ
,ϒℏ,ς)]) < Ω−1
(ρ(ℏ,ϒℏ,ς)),
( )
a contradiction. Hence, ϒℏ = ℏ, i.e., ℏ ∈ V is a fixed point of ϒ.
Now, if ℏ′ is a fixed point of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞ and ρ(ℏ′,ℏ′,ς) < ∞, then
we have
]
ρ ′
[
θ ′
ς θ ρ ϒ ϒ ς θ ρ ϒ
γ
( (ℏ,ℏ , )) = ( ( ℏ, ℏ , )) ≤ (ℏ, ℏ,ς)+ρ(ℏ′
,ϒℏ′
,ς) = 0,
as ρ(ℏ′,ℏ′,ς) = 0, therefore ℏ = ℏ′.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-38-320.jpg)
![22 Soft Computing
Theorem 2.4 In a complete SEPbMS (V,ρ), if ϒ : V → V be a mapping so that:
[ρ ′ ℏ′ β
(ℏ,ϒℏ ,ς)+ρ( ,ϒℏ,ς)]
(i) θ(ρ(ϒℏ,ϒℏ′,ς)) ≤ θ for all ℏ,ℏ′ ∈ V, for all
2
ς > 0 and for some β ∈ (0,1),
(ii) there is ℏ0 ∈ V so that
∆(ρ,ϒ,ℏ0) := sup ρ ϒiℏ j
0,ϒ ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞,
then, the Picard iterating sequence {ℏ n
n}, ℏn =ϒ ℏ0 for all n ≥ 1, converges to some
ℏ ∈ V. If limsupρ(ℏn,ϒℏ,ς) < ∞, then ℏ ∈ V is a fixed point of ϒ, and if ℏ′ is a fixed
n→∞
point of ϒ in V such that ρ(ℏ,ℏ′,ς) < ∞, then ℏ = ℏ′.
{ ( ) }
Proof. By parallel argument as in Theorem 2.2, {ℏn} is a Cauchy sequence in V, and
by completeness of V it converges to an element ℏ ∈ V.
Now, for all n ∈ N∪{0}, we have
θ(ρ(ℏn+1,ϒℏ,ς)) = θ(ρ(ϒℏn,ϒℏ,ς))
≤ θ ([ρ(ℏn,ϒℏ,ς β
)+ρ(ℏ,ϒℏn,ς)])
≤ θ ([ρ(ℏn,ϒℏ,ς)+ρ(ℏn+1,ℏ,ς β
)]) ,
which implies that
θ limsupρ ℏ ϒℏ ς θ limsupρ ℏ ϒℏ ς β
( ( n+1, , )) ≤ ( ( n, , ))
n→∞ n→∞
and hence limsupρ(ℏn,ϒℏ,ς) = 0. Therefore,
n→∞
ρ(ℏ,ϒℏ,ς) ≤ Ω limsupρ(ℏn,ϒℏ,ς) = Ω(0) = 0
n→∞
and as a result, ℏ = ℏ.
ϒ
If ℏ′ be a fixed point of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞, then we have
β
′ [ρ(ℏ,ϒℏ′,ς)+ρ(ℏ′,ϒℏ,ς)]
θ(ρ(ℏ,ℏ ,ς)) = θ(ρ(ϒℏ,ϒℏ′
,ς)) ≤ θ
2
= θ(ρ(ℏ,ℏ′
,ς β
)) ,
which implies ρ(ℏ,ℏ′,ς) = 0, therefore, ℏ = ℏ′.
Example 2.2 Consider V = [0,1] and ρ(ℏ,ℏ′,ς) = (ς|ℏ − ℏ′|) + ln(1 + ς|ℏ − ℏ′|)
for all ℏ,ℏ′ ∈ V and for all ς > 0. Then ρ forms a SEPbM on V with the function
Ω(s) = s+ln(1+s) for all s ≥ 0.
Define ϒ : ℏ
V → V by ϒℏ = 4 for all ℏ ∈ V. Then ϒ gratifies all the circumstances of
Theorem 2.2 for α 1
= 2 , θ(ς) = eς and evidently, ϒ has a unique fixed point 0 ∈ V,
because
θ(ρ ϒℏ ϒℏ′
ς ς ℏ 4 ℏ′ 4 ln 1 ς ℏ 4 ℏ′ 4
( , , )) = e( | / − / |+ ( + | / − / |))
ς ℏ ℏ′ ln 1 ς ℏ ℏ′ 1 2
≤ [e( | − |+ ( + | − |))
] /
1 2
≤ θ(ρ(ℏ,ℏ′
,ς)) /
.
For more details, the readers are referred to the book [11].](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-39-320.jpg)
![Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 23
2.4 SEQUENTIAL EXTENDED FUZZY B-METRIC SPACES
In this section, stimulated by the work existing in Ref. [16], we present the percep-
tion of a SEFbMS. We generate an association between SEPbM and SEFbM and
present some new fixed point results in SEFbMS. For more details on fuzzy metric
and its generalization, the readers are referred to [7]– [10], [21–24] and the refer-
ences therein.
Definition 2.7 (Schweizer and Sklar [31]) A binary operation ⋆ : [0,1 2
] → [0,1] is
called a continuous t-norm if:
(T1) ⋆ is commutative and associative;
(T2) ⋆ is continuous;
(T3) ℏ⋆1 = ℏ for all ℏ ∈ [0,1];
(T4) ℏ⋆ℏ′ ≤ ℜ⋆ℜ′ when ℏ ≤ ℜ and ℏ′ ≤ ℜ′, with ℏ,ℏ′,ℜ,ℜ′ ∈ [0,1].
Definition 2.8 [29] A triplet (V,M,∗) is supposed to be a fuzzy metric space if V is
an unselective set, ∗ is a continuous t-norm (CTN) and M is a fuzzy set on 2
V ×(0,∞)
so that, for all ℏ,ℏ′,υ ∈ V and ς,s > 0,
(i) M(ℏ,ℏ′,ς) > 0;
(ii) M(ℏ,ℏ′,ς) = 1 for all ς > 0 if and only if ℏ = ℏ′;
′ ′
(iii) M(ℏ,ℏ ,ς) = M(ℏ ,ℏ,ς);
(iv) M(ℏ,ℏ′,ς)∗M(ℏ′,υ,s) ≤ M(ℏ,υ,ς +s);
(v) M(ℏ,ℏ′,·) : (0,∞) → [0,1] is continuous;
The function M(ℏ,ℏ′,ς) means the degree of closeness among ℏ and ℏ′
regarding t.
Definition 2.9 [16] A fuzzy b-metric space is an ordered triplet (V,B,⋆) such that
2
V is a nonempty set, ⋆ is a continuous ς-norm and B is a fuzzy set on V ×(0,∞) so
that for all ℏ,ℏ′,υ ∈ V and ς,s > 0,
(F1) B(ℏ,ℏ′,ς) > 0;
(F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′;
(F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς);
(F4) B(ℏ,ℏ′,ς)⋆B(ℏ′,υ,s) ≤ B(ℏ,υ,b(ς +s)) where b ≥ 1;
(F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left.
Definition 2.10 [29] An ordered quadruple (V,B,⋆,Ω) in which V is a nonempty
set, 2
⋆ is a CTN and B is a fuzzy set on V × (0,∞) so that for all ℏ,ℏ′,υ ∈ V and
ς,s > 0,
(F1) B(ℏ,ℏ′,ς) > 0;
(F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′;
(F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς);
(F4) B(ℏ,ℏ′,ς)⋆B(ℏ′,υ,s) ≤ B(ℏ,υ,Ω(ς +s));
(F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left,
is an extended fuzzy b-metric space.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-40-320.jpg)
![24 Soft Computing
Now, let be a non-empty set and B be a fuzzy set on 2
V V ×(0,∞). For any ℏ ∈ V,
let
ð(B,V,ℏ) = {{ℏn} ⊂ V : lim B(ℏn,ℏ,ς) = 1 for all ς > 0}. (2.2)
n→∞
Definition 2.11 A SEFbMS is an ordered quadruple (V,B,⋆,Ω) in which V is a
nonempty set, ⋆ is a CTN and B is a fuzzy set on 2
V ×(0,∞) so that for all ℏ,ℏ′,υ ∈ V
and ς,s > 0,
(F1) B(ℏ,ℏ′,ς) > 0;
(F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′;
(F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς);
(F4) Ω[limsupB(ℏ ′ ′
n,ℏ ,ς)] ≤ B(ℏ,ℏ ,ς) where {ℏn} ∈ ð(B,V,ℏ) and Ω : [0,∞) →
[0 1
,∞) is an unto strictly increasing continuous function with Ω− (ς) ≤ ς ≤ Ω(ς) for
all 0 ≤ ς < ∞;
(F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left.
Definition 2.12 Let (V,B,⋆,Ω) be a SEFbMS and {ℏn} be a sequence in V and
ℏ ∈ V.
(i) {ℏn} is supposed to be convergent and converges to ℏ if {ℏn} ∈ ð(B,V,ℏ).
(ii) {ℏn} is supposed to be Cauchy if lim B(
n,m→
ℏn,ℏm,ς) = 1 for all ς > 0.
∞
(iii) (V,B,⋆,Ω) is called complete if every Cauchy sequence is a convergent se-
quence.
Definition 2.13 The SEFbM (V,B,⋆,Ω) is called Ω-convertible whenever,
1 1
−1 ≤ Ω[ −
Ω[limsupB(ℏ ,ℏ′
n ,ς)] limsupB(ℏn,ℏ′
1].
,ς)
for all ℏ,ℏ′,υ ∈ V, ς > 0 and ℏn ∈ ð(B,V,ℏ).
′ 1
Remark 2.2 Notice that ρ(ℏ,ℏ ,ς) = −
′
1 is a SEPbM whenever B is a
B(ℏ,ℏ ,ς)
Ω-convertible SEFbM.
As an application of Remark 2.2 and the results recognized in Section 3, we can
deduce the subsequent results in SEFbMSs.
Theorem 2.5 Let (V,B,⋆,Ω) be an Ω-convertible complete SEFbMS andϒ : V → V
be a mapping so that:
α
1 1
(i) θ −1 ≤ θ −
B(ϒℏ,ϒℏ′
1 for all ℏ,ℏ′ ∈ V and for some
,ς) B(ℏ,ℏ′,ς)
α ∈ (0,1),
(ii) there is ℏ0 ∈ V so that
1
∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,..., ς > 0 <
B(ϒiℏ ,ϒ j
∞.
0 ℏ0,ς)
Then ϒ takes at least one fixed point in V. Furthermore, if ℏ and ℏ′ are two fixed
1
points of ϒ in V with −
B(ℏ,ℏ′
1 < ∞, then ℏ = ℏ′.
,ς)](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-41-320.jpg)
![Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 25
Theorem 2.6 Let (V,B,⋆,Ω) be a Ω-convertible complete SEFbMS and ϒ : V → V
such that:
(i)
1 1 γ
−1+
1 −
B(ℏ,ϒℏ,ς) (ℏ′,ϒℏ′
1
B ,ς)
θ( −
′
1 ≤ θ 1
) Ω−
B(ϒℏ,ϒℏ ,ς) ,
2
for all ℏ,ℏ′ ∈ V and for some γ ∈ (0,1),
(ii) there is ℏ0 ∈ V such that
1
∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,..., <
i
ς > 0
B(ϒ ℏ ,ϒ j
∞.
0 ℏ0,ς)
Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ∈ N, converges to some
1
ℏ ∈ V. If − 1 < ∞, then ℏ ∈ V is a fixed point of ϒ. Moreover, if ℏ′ is
B(ℏ,ϒℏ,ς)
1 1
a fixed point of ϒ in V such that − < − <
B(ℏ,ℏ′
1 ∞ and 1 ∞, then
,ς) B(ℏ′,ℏ′,ς)
ℏ = ℏ′.
Theorem 2.7 Let (V,B,⋆,Ω) be a Ω-convertible complete SEFbMS and ϒ : V → V
be a mapping so that:
(i)
β
1 1
1
1 −1+ −
B
(ℏ,ϒℏ′,ς) B(ℏ′,ϒℏ,ς)
θ( −1) ≤
B(ϒℏ,ϒ ′
θ
,ς) ,
ℏ 2
for all ℏ,ℏ′ ∈ V, ς > 0 and for some β ∈ (0,1),
(ii) there is ℏ0 ∈ V so that
1
∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,...,
B(ϒ ℏ j
ς > 0 <
i
∞.
0,ϒ ℏ0,ς)
Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ≥ 1, converges to some
1
ℏ ∈ V. If limsup −1 < ∞, then ℏ ∈ V is a fixed point of ϒ. Also, if ℏ′ is
n→∞ B(ℏn,ϒℏ,ς)
1
a fixed point of ϒ in V such that −
B(ℏ,ℏ′
1 < ∞, then ℏ = ℏ′.
,ς)
2.5 APPLICATION
Let V = C[0,T] be the set of real continuous functions defined on [0,T] and ρ :
2
V ×(0,∞) → [0,∞) be defined by:
ρ(ℏ,ℏ′
,α) = sup ℏ(ς p
(e−ας
| )−ℏ′
(ς)| ) for all ℏ,ℏ′
∈ Λ and all ς > 0, p ≥ 1.
0≤ς≤T](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-42-320.jpg)
![26 Soft Computing
Then (V,ρ) is a complete SEPbM space with Ω(ς) = 2p−1ς for all ς ≥ 0. Now,
let us study the integral equation:
T
ℏ(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ(s))ds,
0
(2.3)
where h : [0,T] → R, F : [0,T 2 2
] → [0,∞) and K : [0,T] × R → R are continuous
functions.
Theorem 2.8 Assume that the subsequent suppositions are fulfilled:
(i) for all ς,s ∈ [0,T] we have
p α
|K(ς,s,ℏ(s))−K(ς,s,ℏ′
(s s p
))| ≤ A(e−
max |ℏ(s)−ℏ′
(s)| ), p ≥ 1,0 ≤ A < 1,
0≤s≤ϒ
f
(ii) sup T 1
t∈[0,T] [ |
0 F(t,s q
)| ds]q ≤ 1.
Then, the integral equation (2.3) has a unique solution u ∈ Λ.
(iii) there is ℏ0 ∈ C[0,T] such that
sup sup − j
(e ας
ℏ p
|ϒi
ℏ0 −ϒ 0| ) : i, j = 1,2,..., ς > 0 < ∞,
0≤ς≤T
where
T
ϒ(ℏ0)(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ0(s))ds, ℏ ∈ V, ς,s ∈ [0,T].
0
Proof. Let us define ϒ : V → V by
T
ϒ(ℏ)(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ(s))ds, ℏ ∈ V, ς,s ∈ [0,T].
0
Then by conditions (i)- (iii), for all ϕ,ψ ∈ V we get
ρ ϒ ϕ ας p
( ( ),ϒ(ψ),ς) = sup(e− |ϒ(ϕ)(ς)−ϒ(ψ)(ς)| )
f
= sup −ας T
ς∈ | {K
[0,T] e 0 F(ς,s) (ς s p
, ,ϕ(s))−K(ς,s,ψ(s))}ds|
f 1
T f
[ ς
1 p
≤ sup ∈ e−ας q p
|F( ,s T
)| ds]q [ |K −
ς [0,T] 0 0 (ς,s,ϕ(s)) K(ς,s,ψ(s))| ds] p
( )
A e−αs
≤ sup ≤ ≤ {|ϕ(s)−ψ(s p
)| }
0 s T
≤ Aρ(ϕ,ψ,ς) for all ς ∈ [0,T], for A ∈ (0,1) and for all ϕ,ψ ∈ V.
Hereafter, the circumstances of Theorem 2.2 (with θ(ς) = eς ) are fulfilled, and
thus ϒ has a unique fixed point in V, namely, the nonlinear integral equation (2.3)
has a unique solution in C[0,T].
For more examples of applications, the readers are referred to the book [11] and
the references therein.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-43-320.jpg)
![30 Soft Computing
3.1.1 BACKGROUND
In order to model the situations where data or elements are imprecise or vague and
to represent a mathematical structure for such types of situations, an extended con-
cept of set theory called fuzzy set theory was established by Zadeh [32] in 1965.
Simultaneously, Kaleva and Seikkala [21] initiated the idea of fuzzy metric space.
Simultaneously, several mathematicians like Kramosil and Michalek [26], George
and Veeramani [14], etc. modified the notion of fuzzy metric space (FMS). The con-
cept of weakly compatible maps was established by Jungck and Rhoads [20] for
metric spaces. The development of metric space in multiple ways is an exciting area
of research for the mathematicians. By considering a weaker condition, in place of
triangular inequality, Bakhtin [4] and Czerwik [7] introduced the notion of b metric
space.
Heilpern [18] initiated the study fixed point theory and developed an extended
version of the Banach’s contraction principle in fuzzy metric spaces. The con-
cept of contraction-type fixed point results in FMS was established by Gregori and
Sapena [17]. Some more generalized and extended work in the settings of fuzziness
may be found in Refs. [1,3,5,6,8–13,16,22–25,27–31].
3.1.2 MAIN GOAL
The predominant aim of this chapter is to define the notion of rational-type fuzzy
contraction in FbMS and establish some new fixed point results. After that the exis-
tence and uniqueness of fixed point for rational-type fuzzy contraction in G-complete
FbMS is established. We also provide an application in support of the results.
3.2 BASIC DEFINITIONS
Below we discuss a few preliminary definitions which are essential for our main
results.
Definition 3.1 Consider a binary operation ∗ : [0,1]×[0,1] → [0,1]. Then ∗ is known
as a continuous t −norm if it satisfies the condition:
(i) ∗ is associative and commutative,
(ii) ∗ is continuous,
(iii) α ∗1 = α for all α ∈ [0,1],
(iv) α ∗b ≤ β ∗d whenever α ≤ β and b ≤ d and α,β,c,d ∈ [0,1].
Definition 3.2 Consider a function d : S × S −→ R, where S = φ. Then for all
s1,s2,s3 ∈ S, (S,d) is called a metric space if it satisfies the following conditions:
̸
(i) d(s1,s2) ≥ 0 and d(s1,s2) = 0 iff s1 = s2.
(ii) d(s1,s2) = d(s2,s1).
(iii) d(s1,s3) ≤ d(s1,s2)+d(s2,s3).](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-47-320.jpg)
![Analytical Sequel of Rational-Type Fuzzy Contraction 31
Definition 3.3 Suppose X is a classical set, called the universe and A ∈ X. The
membership of A is considered as a characteristic function µA from X to {0,1} such
that
1 iff x ∈ A
µA(x) =
0 iff x ∈
/ A.
{0,1} is called a valuation set. If {0,1} is allowed to be [0,1], A is said to be a fuzzy
set.
Kramosil and Michalek [26] defined fuzzy metric space as follows:
Definition 3.4 [26] Consider a set X = φ and a continuous t-norm ∗. Suppose M
is a fuzzy set on X2 × R. Then for all a1,a2,a3 ∈ X and t,s ∈ R, (X,M,∗) is called
fuzzy metric space if it satisfies the following axioms:
̸
(i) M(a1,a2,t) = 0 ∀ t ≤ 0.
(ii) M(a1,a2,t) = 1 ∀ t > 0 iff a1 = a2.
(iii) M(a1,a2,t) = M(a2,a1,t).
(iv) M(a1,a2,t)∗M(a2,a3,s) ≤ M(a1,a3,t +s).
(v) M(a1,a2,t) : (0,∞) → [0,1] is left continuous.
(vi) limt→∞ M(a1,a2,t) = 1.
George and Veeramani [14,15] made an appealing modification of fuzzy metric
spaces in the following way:
Definition 3.5 [14] Consider a set X = φ and a continuous t-norm ∗. Suppose M is
a fuzzy set on X2 ×(0,∞). Then for all a1,a2,a3 ∈ X and t,s ∈ R, (X,M,∗) is called
fuzzy metric space if it satisfies the following axioms:
̸
(i) M(a1,a2,t) > 0.
(ii) M(a1,a2,t) = 1 ∀ t > 0 if and only if a1 = a2.
(iii) M(a1,a2,t) = M(a2,a1,t).
(iv) M(a1,a2,t)∗M(a2,a3,s) ≤ M(a1,a3,t +s).
(v) M(a1,a2,t) : (0,∞) → [0,1] is continuous.
Definition 3.6 [19] Consider a non-empty set S and a continuous t-norm ∗. Suppose
P is a fuzzy set on S ×S ×(0,∞) such that for all u,v,w ∈ S and α,β > 0 following
conditions are holds:
(i) P(h̄1,h̄2,α) > 0,
(ii) P(h̄1,h̄2,α) = 1 ⇐⇒ h̄1 = h̄2,
(iii) P(h̄1,h̄2,α) = P(h̄2,h̄1,α),
(iv) P(h̄1,h̄2,·) : (0,∞) → (0,1] is continuous,
β
(v) P(h̄1,h̄ α
3,α +β) ≥ ∗(P(h̄1,h̄2, ),P(h̄2,h̄3, ))
b b
Then (S,P,∗) is called a FbMS.](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-48-320.jpg)
![32 Soft Computing
Definition 3.7 [2] Consider a metric space (Y,d) and a function T : Y → Y. Then
T is called a contraction mapping or contraction if there exists a constant α (called
constant of contraction), with 0 ≤ α < 1, such that
d(T(y1),T(y2)) ≤ αd(y1,y2),∀y1,y2 ∈ Y.
Definition 3.8 [17] Let (U,Mr,∗) be a FbM-space, v1 ∈ U and a sequence (µj) in
U is fuzzy-contractive if there exists α ∈ (0,1) such that
1
− ) > ≥
Mr( , for t , j
µ 1 ≤ α 1
( −1 0 1
j,µj+1,t) Mr(µj−1,µj,t)
Definition 3.9 [17] Let (U,Mr,∗) be a FbM-space. A sequence (µj) in U is said to
be G-Cauchy if
limj Mr(µj,µj+p,t) = 1, for t > 0 and p > 0.
And (U,Mr,∗) is called G-complete if every G-Cauchy sequence is convergent.
Definition 3.10 Let (U,Mr,∗) be a FbM-space. Then Mr is said to be triangular if
it satisfied the following property
1
− − )
M ( , ∗,t) 1 1 1
≤ ( −1)+( 1 ,
r µ1 µ Mr(µ µ t t
1, , ) M
b r(µ,µ∗, )
b
for all µ,µ1,µ∗ ∈ U, t > 0.
Definition 3.11 [17] Suppose (U,Mr,∗) is a FbM-space. Construct a mapping f :
U → U. If for all µ1,µ∗ ∈ U, t > 0, there exists α ∈ (0,1) such that
1
−
M (f(µ ),f(µ∗),t) 1 1
≤ α( −
r 1 Mr(µ1,µ∗,t) 1),
Then f is called fuzzy-contractive.
Next we elaborate the results of the chapter.
3.3 MAIN RESULTS OF THE CHAPTER
In this section, we put forward the definition of rational-type fuzzy contraction in
Fb-MS. After defining the main concept, we provide some propositions and related
theorems.
3.3.1 DEFINITION OF RATIONAL-TYPE FUZZY CONTRACTION IN FB-MS
Definition 3.12 Consider a FbMS (U,Mr,∗) and a function f : U → U. Then f
is said to be rational-type fuzzy contraction(RTF-contraction) if there exists α,β ∈
[0,1) such that for all µ1,µ∗ ∈ U and t > 0,
1 1
−1 ≤α( −1)
Mr(f(µ1), f(µ∗),t) Mr(µ1,µ∗,t)
Mr(µ ,
β
1 µ∗,t)
+ ( −1)
Mr(µ1, f t 2t
(µ1), )∗Mr(µ∗, f( 1), )
b µ b](https://image.slidesharecdn.com/24127982-250512045615-534b98ef/85/Soft-Computing-Recent-Advances-And-Applications-In-Engineering-And-Mathematical-Sciences-1st-Edition-Pradip-Debnath-49-320.jpg)
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Title: Sermons of the Rev. Francis A. Baker, Priest of the
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Author: Francis A. Baker
Contributor: A. F. Hewit
Release date: February 3, 2019 [eBook #58812]
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*** START OF THE PROJECT GUTENBERG EBOOK SERMONS OF
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Sermons Of The
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Soft Computing Recent Advances And Applications In Engineering And Mathematical Sciences 1st Edition Pradip Debnath
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- 6. Soft Computing This book explores soft computing techniques in a systematic manner starting from their initial stage to recent developments in this area. The book presents a survey of the existing knowledge and the current state-of-the-art development through cutting- edge original new contributions from the researchers. Soft Computing: Recent Ad- vances and Applications in Engineering and Mathematical Sciences presents a sur- vey of the existing knowledge and the current state-of-the-art development through cutting-edge original new contributions from the researchers. As suggested by the title, this book particularly focuses on the recent advances and applications of soft computing techniques in engineering and mathematical sci- ences. Chapter 1 describes the contribution of soft computing techniques towards a new paradigm shift. The subsequent chapters present a systematic application of fuzzy logic in mathematical sciences and decision-making. New research directions are also provided at the end of each chapter. The application of soft computing in health sciences and in the modeling of epidemics including the effects of vaccination are also examined. Sustainability of green product development, optimum design of 3D steel frame, digitalization investment analysis in the maritime industry, forecast- ing return rates of individual pension funds are among some of the topics where engineering and industrial applications of soft computing have been studied in the book. The readers of this book will require minimum prerequisites of undergraduate studies in computation and mathematics. This book is meant for graduate students, faculty, and researchers who are apply- ing soft computing in engineering and mathematics. New research directions are also provided at the end of each chapter.
- 7. Edge AI in Future Computing Series Editors: Arun Kumar Sangaiah SCOPE, VIT University, Tamil Nadu Mamta Mittal G. B. Pant Government Engineering College, Okhla, New Delhi Soft Computing Techniques in Engineering, Health, Mathematical and Social Sciences Pradip Debnath and S. A. Mohiuddine Machine Learning for Edge Computing: Frameworks, Patterns and Best Practices Amitoj Singh, Vinay Kukreja, and Taghi Javdani Gandomani Internet of Things: Frameworks for Enabling and Emerging Technologies Bharat Bhushan, Sudhir Kumar Sharma, Bhuvan Unhelkar, Muhammad Fazal Ijaz, and Lamia Karim Soft Computing: Recent Advances and Applications in Engineering and Mathematical Sciences Pradip Debnath, Oscar Castillo, and Poom Kumam For more information about this series, please visit: https://www.routledge.com/Edge-AI-in-Future-Computing/book-series/EAIFC
- 8. Soft Computing Recent Advances and Applications in Engineering and Mathematical Sciences Edited by Pradip Debnath Department of Applied Science and Humanities, Assam University Silchar, India Oscar Castillo Tijuana Institute of Technology, Mexico Poom Kumam Department of Mathematics, King Mongkut’s University of Technology Thonburi, Thailand
- 9. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, trans- mitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. ISBN: 978-1-032-31831-8 (hbk) ISBN: 978-1-032-31917-9 (pbk) ISBN: 978-1-003-31201-7 (ebk) DOI: 10.1201/9781003312017 Typeset in Times by codeMantra
- 10. Contents Preface......................................................................................................................vii Editors.......................................................................................................................ix Contributor................................................................................................................xi Chapter 1 A Study on Approximate Fixed Point Property in Intuitionistic Fuzzy n-Normed Linear Spaces.................................1 Pradip Debnath Chapter 2 Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics with an Application in Integral Equations..........15 Marija V. Paunović, Samira Hadi Bonab, Vahid Parvaneh, and Farhan Golkarmanesh Chapter 3 Analytical Sequel of Rational-Type Fuzzy Contraction in Fuzzy b-Metric Spaces ..................................................................29 Nabanita Konwar Chapter 4 Weak-Wardowski Contractions in Generalized Triple-Controlled Modular Metric Spaces and Generalized Triple-Controlled Fuzzy Metric Spaces ........................................45 Marija V. Paunović, Samira Hadi Bonab, and Vahid Parvaneh Chapter 5 Some First-Order-Like Methods for Solving Systems of Nonlinear Equations ......................................................................67 Sani Aji, Poom Kumam, and Wiyada Kumam Chapter 6 Cubic Inverse Soft Set ...................................................................87 Srinivasan Vijayabalaji and Kaliyaperumal Punniyamoorthy v
- 11. vi Contents Chapter 7 Inverse Soft-Rough Matrices.........................................................97 Srinivasan Vijayabalaji Chapter 8 New Observations on Lacunary I-Invariant Convergence for Sequences in Fuzzy Cone Normed Spaces............................107 Omer ¨ Kisi, Mehmet Gur ¨ dal, and Erhan Guler ¨ Chapter 9 Some Convergent Sequence Spaces of Fuzzy Star-Shaped Numbers ......................................................................................125 Erhan Guler ¨ and Omer ¨ Kisi Chapter 10 Digitalization Investment Analysis in Maritime Industry with Interval−Valued Pythagorean Fuzzy Present Worth Analysis ............................................................................141 Eda Bolturk ¨ Chapter 11 Composite Mapping on Hesitant Fuzzy Soft Classes..................153 Manash Jyoti Borah and Bipan Hazarika Chapter 12 Ulam Stability of Mixed Type Functional Equation in Non-Archimedean IFN-Space.................................................167 K. Tamilvanan, S. A. Mohiuddine, and N. Revathi Chapter 13 Optimum Design of 3D Steel Frames with Composite Slabs Using Adaptive Harmony Search Method.........................179 Mehmet Polat Saka, Ibrahim Aydogdu, Refik Burak Taymus, and Zong Woo Geem Chapter 14 Fostering Sustainability in Open Innovation, to Select the Right Partner on Green Product Development ......................211 Ricardo Santos, Polinho Katina, José Soares, Anouar Hallioui, Joao Matias, and Fernanda Mendes Index.......................................................................................................................233
- 12. Preface This book collects chapters from eminent contemporary researchers across the coun- tries working on the theory and applications of soft computing techniques. The book presents a survey of the existing knowledge and also current state of the art devel- opment through cutting-edge original new contributions from the researchers. As suggested by the title, this book particularly focuses on the recent advances and ap- plications of soft computing techniques in engineering and mathematical sciences. The first Chapter presents a study on approximate fixed point property in a gen- eralized fuzzy normed space. Chapters 2–4 consist of new fixed point results and thier applications in different types of metric spaces such as fuzzy b-metric spaces and controlled fuzzy metric spaces. Chapter 5 describes new first order-like meth- ods for solving nonlinear equations. Cubic inverse soft sets have been studied in Chapter 6, whereas inverse soft-rough matrices are introduced in Chapter 7. Some new convergence results concerning fuzzy normed spaces and fuzzy numbers have been described in Chapters 8 and 9, respectively. Digitalization investment analysis in maritime industry with picture fuzzy sets are studied in Chapter 10. Chapter 11 contains a study on composite mapping on hesitant fuzzy soft classes. In Chapter 12 we have Ulam stability of mixed type functional equation in fuzzy normed space. An interesting investigation on optimum design of 3D steel frames with composite slabs has been presented in Chapter 13. Finally, in Chapter 14, we discuss fostering sustainability in open innovation to select the right partner on green product devel- opment. This book is meant for graduate students, faculties and researchers willing to learn and apply fuzziness and soft computing in engineering and mathematics. New re- search directions have been presented within the chapters to enable the researchers to further advance their research. The readers of this book will require minimum pre-requisites of undergraduate studies in computation and mathematics. vii
- 14. Editors Pradip Debnath is an Assistant Professor (in Mathematics) at the Department of Applied Science and Humanities, Assam University, Silchar (a central university), India. He received his Ph.D. in Mathematics from the National Institute of Technol- ogy Silchar, India. His research interests include fixed point theory, nonlinear func- tional analysis, soft computing and mathematical statistics. He has published more than 60 papers in various journals of international repute and is reviewer for more than 40 international journals. Dr. Debnath is also a reviewer for “Mathematical Re- views” published by the American Mathematical Society. He is the Lead Editor of the books “Metric Fixed Point Theory - Applications in Science, Engineering and Behavioural Sciences” (2021, Springer Nature), “Soft Computing Techniques in En- gineering, Health, Mathematical and Social Sciences” (2021, CRC Press) and “Fixed Point Theory and Fractional Calculus: Recent Advances and Applications” (2022, Springer Nature). He has successfully guided Ph.D. students in the areas of fuzzy logic, soft computing and fixed point theory. He has recently completed a Basic Sci- ence Research Project on fixed point theory funded by the UGC, the Government of India. Having been an academic gold medalist during his post-graduation stud- ies from Assam University, Silchar, Dr. Debnath has qualified several national-level examinations in mathematics in India. Oscar Castillo holds the Doctor in Science degree (Doctor Habilitatus) in Com- puter Science from the Polish Academy of Sciences (with the Dissertation “Soft Computing and Fractal Theory for Intelligent Manufacturing”). He is a Professor of Computer Science in the Graduate Division, Tijuana Institute of Technology, Ti- juana, Mexico. In addition, he is serving as Research Director of Computer Science and head of the research group on Hybrid Fuzzy Intelligent Systems. Currently, he is President of HAFSA (Hispanic American Fuzzy Systems Association) and Past Pres- ident of IFSA (International Fuzzy Systems Association). Prof. Castillo is also Chair of the Mexican Chapter of the Computational Intelligence Society (IEEE). He also belongs to the Technical Committee on Fuzzy Systems of IEEE and to the Task Force on “Extensions to Type-1 Fuzzy Systems”. He is also a member of NAFIPS, IFSA and IEEE. He belongs to the Mexican Research System (SNI Level 3). His research interests are in Type-2 Fuzzy Logic, Fuzzy Control, Neuro-Fuzzy and Genetic-Fuzzy hybrid approaches. He has published over 300 journal papers, 10 authored books, 40 edited books, 200 papers in conference proceedings, and more than 300 chapters in edited books, in total 910 publications according to Scopus (H index=63, and more than 1050 publications according to Research Gate (H index=74 in Google Scholar). He has been Guest Editor of several successful Special Issues in the past, like in the following journals: Applied Soft Computing, Intelligent Systems, Information Sciences, Non-Linear Studies, Fuzzy Sets and Systems, JAMRIS and Engineering Letters. He is currently Associate Editor of the Information Sciences Journal, Ap- plied Soft Computing Journal, Engineering Applications of Artificial Intelligence, ix
- 15. x Editors Granular Computing Journal and the International Journal on Fuzzy Systems. Fi- nally, he has been elected IFSA Fellow in 2015 and MICAI Fellow member in 2017. He has been recognized as Highly Cited Researcher in 2017 and 2018 by Clarivate Analytics because of having multiple highly cited papers in Web of Science. Poom Kumam received the B.S., M.Sc., and Ph.D. degrees in mathematics from Burapha University (BUU), Chiang Mai University (CMU), and Naresuan Univer- sity (NU), respectively. In 2008, he received a grant from Franco-Thai Cooperation for short-term visited at the Laboratoire de Mathematiques, Universite de Bretagne Occidentale, France. He was also a Visiting Professor for a short-term research with Professor Anthony To-Ming Lau at the University of Alberta, AB, Canada. He is currently a Full Professor with the Department of Mathematics, King Mongkut’s University of Technology Thonburi (KMUTT), where he is also the Head of the KMUTT Fixed Point Theory and Applications Research Group since 2007 and also leading of the Theoretical and Computational Science Center (TaCS-Center) in 2014 (now, became to TaCS-Center of Excellence in 2021). He has successfully advised 5 master’s, and 44 Ph.D. graduates. He had won of the most important awards for mathematicians. The first one is the TRF-CHE-Scopus Young Researcher Award in 2010 that is the award given by the corporation from three organizations: Thailand Research Fund (TRF), the Commission of Higher Education (CHE), and Elsevier Publisher (Scopus). The second award was in 2012 when he received the TWAS Prize for Young Scientist in Thailand, which is given by the Academy of Sciences for the Developing World TWAS (UNESCO) together with the National Research Council of Thailand. In 2014, the third award is the Fellowship Award for Outstand- ing Contribution to Mathematics from International Academy of Physical Science, Allahabad, India. In 2015 Dr. Poom Kumam has been awarded Thailand Frontier Au- thor Award 2015, Award for outstanding researcher who has published works and has often been used as a reference or evaluation criteria of the database Web of Science. Moreover, In 2016 Dr. Poom Kumam has been awarded 2016 Thailand Frontier Re- searcher Awards on Innovation Forum: Discovery, Protection, Commercialization By Intellectual Property & Science, and Thomson Reuters. Dr. Poom Kumam has been Highly Cited Researcher (HCR 2015, 2016, 2017). Moreover, he has been received KMUTT-HALL OF FAME 2017, In Honour of the Recipients of Academic Awards, KMUTT Young Researcher Awards, Excellence in Teaching Awards for 2016. In 2019 he received 2019 CMMSE Prize Winner: The CMMSE prize is given to com- putational researchers for important contributions in the developments of Numerical Methods for Physics, Chemistry, Engineering and Economics, from CMMSE Con- ference June 30 to July 6, 2019, Rota, Cadiz - Spain. He has also been listed and ranked in the 197th Place in General Mathematics among the Top 2% Scientists in the World 2021 (Published by Stanford University in USA). He served on the editorial boards of various international journals and has also published more than 800 papers in Scopus and Web of Science (WoS) database and also delivers many invited talks on different international conferences every year all around the world. Furthermore, his research interest focuses on Fixed Point The- ory, fractional differential equations and Optimization with related with optimization problems in both pure science and applied science.
- 16. Contributors Sani Aji Department of Mathematics Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT) Bangkok, Thailand and Department of Mathematics Faculty of Science Gombe State University Gombe, Nigeria Ibrahim Aydogdu Department of Civil Engineering Akdeniz University Antalya, Turkey Eda Boltürk Istanbul Settlement and Custody Bank Inc. Istanbul, Turkiye Samira Hadi Bonab Department of Mathematics, Ardabil Branch Islamic Azad University Ardabil, Iran Manash Jyoti Borah Department of Mathematics Bahona College Jorhat, India Zong Woo Geem College of IT Convergence Gachon University Seongnam, Korea Farhan Golkarmanesh Department of Mathematics, Sanandaj Branch Islamic Azad University Sanandaj, Iran Erhan Güler Department of Mathematics Faculty of Science Bartin University Bartin, Turkey Mehmet Gürdal Department of Mathematics Faculty of Arts and Sciences Suleyman Demirel University Isparta, Turkey Bipan Hazarika Department of Mathematics Gauhati University Guwahati, India Polinho Katina Department of Informatics University of South Carolina Upstate Spartanburg, South Carolina Ömer Kisi Department of Mathematics Faculty of Science Bartin University Nabanita Konwar Department of Mathematics Birjhora Mahavidyalaya Bongaigaon, Assam xi
- 17. xii Contributors Wiyada Kumam Applied Mathematics for Science and Engineering Research Unit (AMSERU) Department of Mathematics and Computer Science Faculty of Science and Technology Rajamangala University of Technology Thanyaburi (RMUTT) Pathum Thani, Thailand Joao Matias Department of Economics, Industrial Engineering and Tourism – GOVCOPP University of Aveiro Aveiro, Portugal Fernanda Mendes ESAI Lisbon, Portugal S. A. Mohiuddine Department of General Required Courses, Mathematics The Applied College King Abdulaziz University Jeddah, Saudi Arabia and Operator Theory and Applications Research Group Department of Mathematics, Faculty of Science King Abdulaziz University Jeddah, Saudi Arabia Vahid Parvaneh Department of Mathematics, Gilan-E-Gharb Branch Islamic Azad University Gilan-E-Gharb, Iran Marija V. Paunović Faculty of Hotel Management and Tourism University of Kragujevac Kragujevac, Serbia Kaliyaperumal Punniyamoorthy Department of Mathematics Rajalakshmi Engineering College (Autonomous) Chennai, India N. Revathi Department of Computer Science Periyar University PG Extension Centre Dharmapuri, India Mehmet Polat Saka Department of Engineering Sciences Middle East Technical University Ankara, Turkey Ricardo Santos GOVCOPP University of Aveiro Aveiro, Portugal José Soares Department of Management – ADVANCE University of Lisbon Lisbon, Portugal Anouar Hallioui Department of Industrial Engineering Sidi Mohamed Ben Abdellah University Fez, Morocco K. Tamilvanan Department of Mathematics, Faculty of Science & Humanities R.M.K. Engineering College Tamil Nadu, India Refik Burak Taymus Department of Civil Engineering Yuzuncu yil University Van, Turkey Srinivasan Vijayabalaji Department of Mathematics (S&H) University College of Engineering Panruti (A Constituent College of Anna University) Panruti, India
- 18. 1 A Study on Approximate Fixed Point Property in Intuitionistic Fuzzy n-Normed Linear Spaces Pradip Debnath Assam University CONTENTS 1.1 Introduction .......................................................................................................1 1.2 Preliminaries......................................................................................................2 1.3 Approximate Fixed Points in IFnNLS...............................................................3 1.4 Intuitionistic n-Fuzzy Contraction and Nonexpansive Mappings .....................5 Bibliography ............................................................................................................12 1.1 INTRODUCTION Fuzzy set theory [31] has widespread applications in different branches of mathemat- ical science such as theory of functions [18,28], topological and metric spaces [10,14, 19], and approximation theory [1]. It has also been applied in control of chaos [13], quantum physics [22], computer programming [16], population dynamics [3], and nonlinear dynamical systems [17]. The initial notion of fuzzy norm was put forward by Katsaras [20]. It was further improved and re-defined by various mathematicians considering particular areas of application [2,12,19,22,29]. The notion of an intuitionistic fuzzy n-normed linear space (IFnNLS) [23] generalizes an intuitionistic fuzzy normed space which was introduced by Saadati and Park [24]. The concept of fixed point property of mappings plays a significant role in the investigation of analytic properties of a normed linear space. In this chapter, we in- troduce the concept of approximate fixed point property in an IFnNLS. We also es- tablish the relation between asymptotic regularity and fixed point property. Further, we introduce and investigate the properties of various types of intuitionistic n-fuzzy contraction mappings and their connection with fixed point property. DOI: 10.1201/9781003312017-1 1
- 19. 2 Soft Computing The definition of convergence of a sequence in an IFnNLS is crucial for the in- vestigation of its analytic properties. A new and modified definition of convergence was introduced in [25,26]. The results of this chapter are established on the basis of this definition. For more relevant work, we refer to [4–7,11,15]. 1.2 PRELIMINARIES The same topology is induced by an intuitionistic fuzzy metric and a fuzzy met- ric [15]. Hence, to generate original and new results in intuitionistic fuzzy setting, it was necessary to re-define the notion of intuitionistic fuzzy norm [21,30]. To serve this purpose, an improved definition of an IFnNLS was put forward by Debnath and Sen [8,9] as given next. Definition 1.1 The five-tuple (V,η,γ,∗,◦) is called an IFnNLS, where V is a vector space of dimension d ≥ n over a field F, ∗ is a continuous t-norm, ◦ is a continuous t-conorm, η,γ are fuzzy sets on Vn × (0,∞), η signifies the degree of membership, and γ signifies the degree of non-membership of (u1,u n 2,...,un,t) ∈ V ×(0,1). The following conditions are satisfied for every (u1,u n 2,...,un) ∈ V and s,t > 0: (i) η(u1,u2,...,un,r) = 0 and γ(u1,u2,...,un,r) = 1 for all non-positive real num- ber t, (ii) η(u1,u2,...,un,r) = 1 and γ(u1,u2,...,un,r) = 0 for all positive r if and only if u1,u2,...,un are linearly dependent, (iii) η(u1,u2,...,un,r) and γ(u1,u2,...,un,r) are invariant under any permutation of u1,u2,...,un, (iv) η(u u ,cu t 1, 2,... n,r) = η(u1,u2,...,un, ) |c| and γ(u1,u2,...,cun,r) = γ(u1,u r 2,...,un, ) | if c = 0, ∈ c| c F, (v) η(u1,u2,...,u ′ n,s)∗η(u1,u2,...,u ≤ n,r) η(u1,u2,...,u ′ n +un,s+r), (vi) η u u2 u s γ u ′ ′ ( 1, ,..., n, )◦ (u1, 2,...,un,r) ≥ γ(u1,u2,...,un +un,s+r), (vii) η(u1,u2,...,un,r) : (0,∞) → [0,1] and γ(u1,u2,...,un,r) : (0,∞) → [0,1] are continuous in r, (viii) limr→∞ η(u1,u2,...,un,r) = 1 and limr→0 η(u1,u2,...,un,r) = 0, (ix) limr→∞ γ(u1,u2,...,un,r) = 0 and limr→0 γ(u1,u2,...,un,r) = 1. Definition 1.2 [25,26] Let (V,η,γ,∗,◦) be an IFnNLS. A sequence v = {vk} in V is called convergent to ς ∈ V with respect to the intuitionistic fuzzy n-norm (IFnN) (η n ,γ) if, for every ε ∈ (0,1), r > 0 and u1,u2,...,un−1 ∈V, there exists k0 ∈ N such that η(u1,u2,...,un−1,vk −ς,r) > 1−ε and γ(u1,u2,...,un−1,vk −ς,r) < ε for all η n k k . We denote it by η n ( ,γ γ limv ς or v ) ≥ 0 ( , ) − = k → ς as k → ∞. proposition 1.1 [27] In an IFnNLS V, η γ n ( , ) − limv = ς if and only if for every r > 0 and u1,u2,...,un−1 ∈V, η(u1,...,un−1,vk −ς,r) → 1 and γ(u1,...,un−1,vk − ς,r) → 0 as k → ∞. Definition 1.3 [25,26] Let (V,η,γ,∗,◦) be an IFnNLS. A sequence v = {vk} in V is said to be Cauchy with respect to the IFnN (η,γ n ) if, for every ε ∈ (0,1), r > 0 ̸
- 20. A Study on Approximate Fixed Point Property 3 and u1,u2,...,un−1 ∈V, there exists k0 ∈ N such that η(u1,u2,...,un−1,vk −vm,r) > 1−ε and γ(u1,u2,...,un−1,vk −vm,r) < ε for all k,m ≥ k0. Definition 1.4 An IFnNLS V is complete with respect to the IFnN (η,γ n ) if every Cauchy sequence in it is convergent. proposition 1.2 [27] If every Cauchy sequence in an IFnNLS V has a convergent subsequence, then V is complete. 1.3 APPROXIMATE FIXED POINTS IN IFNNLS Now we are ready to present our main results. First we define the concept of an intu- itionistic n-fuzzy approximate fixed point (InFAFP) in an IFnNLS as given below. Definition 1.5 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. ϖ0 ∈ V is said to be an intuitionistic n-fuzzy approximate fixed point (InFAFP) of Γ if for every ε > 0 and ω1,ω2,...,ωn−1 ∈ V we have µ(ω1,ω2,...,ωn−1,Γ(ϖ0)−ϖ0,s) > 1−ε and ν(ω1,ω2,...,ωn−1,Γ(ϖ0)−ϖ0,s) < ε for all s > 0. We denote the set of all InFAFP of Γ by AFPε (Γ). Further, we say that the mapping Γ has the InFAFP property if the set AFPε (Γ) is nonempty for every ε > 0 and ω1,ω2,...,ωn−1 ∈ V. The example below illustrates Definition 1.5. Example 1.1 Let V = (0,1 n ) ,ω n i = (ωi1,ωi2,...,ωin) ∈ (0,1) for each i = 1,2,...,n with ' ' ' ω ω ' 11 ··· 1n ' ' ' . . ' ∥ω1,ω2,...,ω . n∥ = abs' . . . . ' . . ', ' ' ωn1 ··· ω ' nn and let a ∗ b = ab, a ◦ b = min{a + b,1} for all a,b ∈ [0,1]. Now for all w n 1,w r 2,...,wn ∈ (0,1) and r > 0, let us define η(w1,w2,...,wn,r) = r+∥w1,w2,...,wn∥ and γ w w w r ∥w1,w2,...,w ( , ,..., , ) = n∥ 1 2 n . Then ((0,1 n ) , , r+∥w1,w η γ,∗,◦) ∥ an IFnNLS. 2,...,w is n Consider the mapping Γ : V → V defined by Γ(θ1,θ 1 2 2 2,...,θn) = (θ1 ,θ2 ,...,θn ) for all (θ1,θ2,...,θn) ∈ V. Clearly, Γ has no fixed point in V. Hence, we try to investigate its InFAFP. We observe that for every ε > 0, w1,w2,...,w n n−1 ∈ (0,1) and r > 0, there exists u ∈ (0,1 n ) such that r η(w1,w2,...,wn−1,Γ(u)−u,r) = > 1−ε r +∥w1,w2,...,wn−1,Γ(u)−u∥
- 21. 4 Soft Computing and ∥w ,w2,...,w γ w w 1 n−1,Γ(u)−u∥ ( 1, 2,...,wn−1,Γ(u)−u,r) = < ε. r +∥w1,w2,...,wn−1,Γ(u)−u∥ Hence, we conclude that Γ has InFAFP property. Definition 1.6 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. Γ is said to be intuitionistic n-fuzzy asymptotic regular (InFAR) of Γ if for every ϖ ∈ V, s > 0 and ω1,ω2,...,ωn−1 ∈ V we have lim µ k 1 k (ω1,ω2,...,ωn−1,Γ + (ϖ)−Γ (ϖ),s) = 1 k→∞ and lim ν ω ω ω k+1 k ( 1, 2,..., n−1,Γ (ϖ)−Γ (ϖ),s) = 0. k→∞ Theorem 1.1 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. If Γ is InFAR, then it has InFAFP property. Proof. Let ϖ0 ∈V. Since Γ is InFAR, we have for every s > 0 and ω1,ω2,...,ωn−1 ∈ V that lim µ(ω ,ω k ,ω ,... ,Γ +1 1 2 n−1 (ϖ)−Γk (ϖ),s) = 1 k→∞ and lim ν ω 1 ( 1,ω2,...,ωn−1,Γk+ (ϖ0)−Γk (ϖ0),s) = 0. k→∞ In this case, for every ε > 0, there exists k0 ∈ N such that µ(ω ,ω k ,...,ω ,Γ +1 k 1 2 n−1 (ϖ0)−Γ (ϖ0),s) > 1−ε and ν(ω1,ω2,...,ω k+1 k n−1,Γ (ϖ0)−Γ (ϖ0),s) < ε for every k k ≥ k0. If we denote Γ (ϖ0) by ϑ0, we have µ k 1 k (ω ω2,...,ω + 1, n−1,Γ (ϖ0)−Γ (ϖ0),s) = µ(ω1,ω2,...,ω k k n−1,Γ(Γ (ϖ0))−Γ (ϖ0),s) = µ(ω1,ω2,...,ωn−1,Γ(ϑ0)−ϑ0,s) > 1−ε and ν(ω1,ω2,...,ω + n Γk 1 −1, (ϖ0)−Γk (ϖ0),s) k = ν(ω k 1,ω2,...,ωn−1,Γ(Γ (ϖ0))−Γ (ϖ0),s) = ν(ω1,ω2,...,ωn−1,Γ(ϑ0)−ϑ0,s) < ε. This proves that ϑ0 is an InFAFP of Γ.
- 22. A Study on Approximate Fixed Point Property 5 1.4 INTUITIONISTIC N-FUZZY CONTRACTION AND NONEXPANSIVE MAPPINGS In this section, we introduce the concepts of intuitionistic n-fuzzy contraction and nonexpansive mappings in an IFnNLS and investigate their properties. Definition 1.7 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. Γ is said to be intuitionistic n-fuzzy contraction (InFC) if there exists p ∈ (0,1) such that for any s > 0 and ω1,ω2,...,ωn−1 ∈ V we have µ(ω1,ω2,...,ωn−1,Γ(θ)−Γ(φ), ps) ≥ µ(ω1,ω2,...,ωn−1,θ −φ,s) and ν(ω1,ω2,...,ωn−1,Γ(θ)−Γ(φ), ps) ≤ ν(ω1,ω2,...,ωn−1,θ −φ,s). for all θ,φ ∈ V. Theorem 1.2 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be an InFC. Then AFPε (Γ) is nonempty for every ε ∈ (0,1). Proof. Fix ω1,ω2,...,ωn−1 ∈ V and s > 0. Also let ϖ ∈ V. Then µ k (ω1,ω 1 2,...,ωn−1 Γ (ϖ k , )−Γ + (ϖ),s) = µ(ω1,ω k 1 k 2,...,ωn ,Γ(Γ − −1 (ϖ))−Γ(Γ (ϖ)),s) s ≥ µ(ω1,ω2,...,ωn−1,Γk−1 k (ϖ)−Γ (ϖ), ) p ≥ µ(ω k ,ω k 2 1 s 1 2,...,ω , − − n−1 Γ (ϖ)−Γ (ϖ), ) p2 ≥ ... s ≥ µ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ) pk and ν(ω k k 1 1,ω2,...,ωn−1,Γ (ϖ)−Γ + (ϖ),s) = ν(ω1,ω2,...,ω 1 Γ k−1 k n− , (Γ (ϖ))−Γ(Γ (ϖ)),s) k s ≤ ν(ω1,ω2,...,ω −1 k n−1,Γ (ϖ)−Γ (ϖ), ) p ≤ ν(ω ω k ,ω 2 (ϖ k ,..., ,Γ − )−Γ −1 s 1 2 n−1 (ϖ), ) p2 ≤ ... s ≤ ν(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ). k p For p ∈ (0,1 t ), as k → ∞, we have pk → ∞. Thus, using the properties of the intuitionistic fuzzy norm, we have that µ(ω1,ω2,...,ω k k 1 n−1,Γ (ϖ)−Γ + (ϖ),s) → 1
- 23. 6 Soft Computing and ν(ω1,ω2,...,ωn−1,Γk (ϖ)−Γk+1 (ϖ),s) → 0 as k → ∞. Therefore, we conclude that AFPε (Γ) is nonempty for every ε ∈ (0,1). Example 1.2 Consider the IFnNLS (V = (0,1 n ) ,η,γ,∗,◦) as in Example 1.1. Also, consider the mapping Γ : V → V defined by θ Γ(θ1,θ2,...,θ 1 θ2 θn n) = ( , ,..., ) 2 2 2 for all (θ1,θ2,...,θn) ∈ V. Clearly, Γ has no fixed point in V. We prove that Γ is an InFC mapping. For every w1,w2,...,wn−1 ∈ (0,1 n n ) and s > 0, and for all u,v ∈ (0,1) we have that s s η(w1,w2,...,wn−1,Γ 2 (u)−v, ) = 2 s +∥w1,w2,..., 2 wn−1,Γ(u)−v∥ = η(w1,w2,...,wn−1,u−v,s) and s ∥w ,w ,...,w ,Γ(u)−v∥ γ n 1,Γ 1 (w1,w , w 2 −1 2 ..., n− (u)−v, ) = 2 s +∥ 2 w1,w2,...,wn−1,Γ(u)−v∥ = γ(w1,w2,...,wn−1,u−v,s). Now, for any ε ∈ (0,1), we have u η(w1,w2,...,wn−1,u−Γ(u),s) = η(w1,w2,...,wn−1,u− ,s) 2 u = η(w1,w2,...,wn−1, ,s) 2 s = s+∥w1,w2,...,wn−1, u ∥ 2 > 1−ε. Similarly, we can prove that γ(w1,w2,...,wn−1,u−Γ(u),s) < ε. Hence, Γ has InFAFP property. Definition 1.8 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be a mapping. If there exists κ ∈ (0,1) and L > 0 such that for any fixed ω1,ω2,...,ωn−1 ∈ V, we have s η(ω1,ω2,...,ωn−1,Γ(θ)−Γ(φ),s) ≥ η(ω1,ω2,...,ωn−1,θ −φ, ) κ s ∗η(ω1,ω2,...,ωn−1,φ −Γ(θ), ) L
- 24. A Study on Approximate Fixed Point Property 7 and s γ(ω1,ω2,...,ωn−1,Γ(θ)−Γ(φ),s) ≤ γ(ω1,ω2,...,ωn−1,θ −φ, ) κ s ◦γ(ω1,ω2,...,ωn−1,φ −Γ(θ), ) L for all s > 0 and θ,φ ∈ V, then Γ is called an intuitionistic n-fuzzy weak contraction operator (InFWCO). Theorem 1.3 Let (V,η,γ,∗,◦) be an IFnNLS and Γ : V → V be an InFWCO. Then AFPε (Γ) is nonempty for every ε ∈ (0,1). Proof. Fix ω1,ω2,...,ωn−1 ∈ V. Also let ϖ ∈ V, ε ∈ (0,1) and s > 0. Then µ(ω1,ω k 2,...,ωn−1,Γ k 1 (ϖ)−Γ + (ϖ),s) = µ(ω1,ω k 2,...,ωn−1,Γ(Γ −1 (ϖ k ))−Γ(Γ (ϖ)),s) ≥ µ k 1 k s (ω 2,...,ωn−1,Γ − 1,ω (ϖ)−Γ (ϖ), ) κ ∗ µ(ω1,ω k k s 2,...,ωn−1,Γ (ϖ)−Γ (ϖ), ) L s µ(ω k−1 k = 1,ω2,...,ωn−1,Γ (ϖ)−Γ (ϖ), )∗1 κ s = µ(ω1,ω k 1 k 2,...,ω − n−1,Γ (ϖ)−Γ (ϖ), ) κ s ≥ µ(ω ,ω k−2 k−1 1 2,...,ωn−1,Γ (ϖ)−Γ (ϖ), ) κ2 ∗ µ(ω 1 ,ω k k 1 s 1 2,...,ωn−1,Γ − (ϖ)−Γ − (ϖ), ) L s = µ(ω ,ω ,...,ω ,Γk−2 1 2 n 1 (ϖ)−Γk−1 − (ϖ), )∗1 κ2 = µ(ω1,ω2,...,ω k 2 k 1 s n−1,Γ − (ϖ)−Γ − (ϖ), ) κ2 ≥ ... k−(k−1) s = µ(ω1,ω2,...,ωn−1,Γ (ϖ)−Γk−(k−2) (ϖ), ) κ(k−1) s = µ(ω1,ω ,...,ω 2 2 n−1,Γ(ϖ)−Γ (ϖ), ) κ(k−1) s ≥ µ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ) κk s ∗ µ(ω1,ω2,...,ωn−1,Γ(ϖ)−Γ(ϖ), ) L s ≥ µ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), )∗1 κk s = µ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ). κk
- 25. 8 Soft Computing Similarly, for the non-membership function γ, we have γ(ω1,ω k 2,...,ωn−1,Γ k 1 (ϖ)−Γ + (ϖ),s) = γ(ω1,ω k 2,...,ωn−1,Γ(Γ −1 (ϖ k ))−Γ(Γ (ϖ)),s) ≤ γ k 1 k s (ω 2,...,ωn−1,Γ − 1,ω (ϖ)−Γ (ϖ), ) κ ◦γ(ω1,ω k k s 2,...,ωn−1,Γ (ϖ)−Γ (ϖ), ) L s γ(ω k−1 k = 1,ω2,...,ωn−1,Γ (ϖ)−Γ (ϖ), )◦0 κ s = γ(ω1,ω k 1 k 2,...,ω − n−1,Γ (ϖ)−Γ (ϖ), ) κ s ≤ γ(ω ,ω k−2 k−1 1 2,...,ωn−1,Γ (ϖ)−Γ (ϖ), ) κ2 ◦γ(ω 1 ,ω k k 1 s 1 2,...,ωn−1,Γ − (ϖ)−Γ − (ϖ), ) L s = γ(ω ,ω ,...,ω ,Γk−2 1 2 n 1 (ϖ)−Γk−1 − (ϖ), )◦0 κ2 = γ(ω1,ω2,...,ω k 2 k 1 s n−1,Γ − (ϖ)−Γ − (ϖ), ) κ2 ≤ ... k−(k−1) s = γ(ω1,ω2,...,ωn−1,Γ (ϖ)−Γk−(k−2) (ϖ), ) κ(k−1) s = γ(ω1,ω ,...,ω 2 2 n−1,Γ(ϖ)−Γ (ϖ), ) κ(k−1) s ≤ γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ) κk s ◦γ(ω1,ω2,...,ωn−1,Γ(ϖ)−Γ(ϖ), ) L s ≤ γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), )◦0 κk s = γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ). κk Since s κk → ∞, as k → ∞, using the properties of the intuitionistic fuzzy n-norm, we conclude that an InFWCO has approximate fixed point property. Finally, we introduce and study the properties of an intuitionistic n-fuzzy nonex- pansive mapping and establish its connection with approximate fixed point property. Definition 1.9 Let (V,η,γ,∗,◦) be an IFnNLS. The mapping Γ : V → V is said to be intuitionistic n-fuzzy nonexpansive if for every ω1,ω2,...,ωn−1 ∈ V and s > 0, we have η(ω1,ω2,...,ωn−1,Γ(ϖ)−Γ(θ),s) ≥ η(ω1,ω2,...,ωn−1,ϖ −θ,s) and γ(ω1,ω2,...,ωn−1,Γ(ϖ)−Γ(θ),s) ≤ γ(ω1,ω2,...,ωn−1,ϖ −θ,s) for all ϖ,θ ∈ V.
- 26. A Study on Approximate Fixed Point Property 9 Definition 1.10 Let (V,η,γ,∗,◦) be an IFnNLS and J ⊂ V. Then J is said to have intuitionistic n-fuzzy approximate fixed point (InFAFP) property if every intuition- istic n-fuzzy nonexpansive mapping Γ : V → V satisfies the property that for fixed ω1,ω2,...,ωn−1 ∈ V and s > 0, we have sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),s) : ϖ ∈ V} = 1 and inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),s) : ϖ ∈ V} = 0. Theorem 1.4 Let (V,η,γ,∗,◦) be an IFnNLS having InFAFP property and J be a dense subset of V. Then J has InFAFP property. Proof. Let Γ : V → V be an intuitionistic n-fuzzy nonexpansive mapping. First we prove that for fixed ω1,ω2,...,ωn−1 ∈ V, t > 0 and s > 0, sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = sup{η(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J} and inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = inf{γ(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J}. Since J ⊂ V, we have that sup{η(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J} ≥ sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} and inf{γ(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J} ≤ inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J}. (η,γ n Let θ ∈ V. For J is dense, there exists a sequence {θ ) k} in J such that θk → θ. We know that for each k ∈ N and t,s > 0, sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} ≥ η(ω1,ω2,...,ωn−1,θk −Γ(θk),t) ≥ η(ω1,ω2,...,ωn−1,θk −θ +θ −Γ(θ)+Γ(θ)−Γ(θk),t) t t ≥ η(ω1,ω2,...,ωn−1,θk −θ, )∗η(ω1,ω2,...,ωn−1,θ −Γ(θ), ) 3 3 t ∗η(ω1,ω2,...,ωn−1,θk −Γ(θk), ) 3
- 27. 10 Soft Computing and inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} ≤ γ(ω1,ω2,...,ωn−1,θk −Γ(θk),t) ≤ γ(ω1,ω2,...,ωn−1,θk −θ +θ −Γ(θ)+Γ(θ)−Γ(θk),t) t t ≤ γ(ω1,ω2,...,ωn−1,θk −θ, )∗γ(ω1,ω2,...,ωn−1,θ −Γ(θ), ) 3 3 t ∗γ(ω1,ω2,...,ωn−1,θk −Γ(θk), ). 3 Since Γ is intuitionistic n-fuzzy nonexpansive, it is clearly intuitionistic n-fuzzy continuous. η n If θ ( ,γ) k → θ, we have η(ω1,ω2,...,ωn−1,Γ(θk)−Γ(θ),t) ≥ η(ω1,ω2,...,ωn−1,θk −θ,t) → 1 and γ(ω1,ω2,...,ωn−1,Γ(θk)−Γ(θ),t) ≤ γ(ω1,ω2,...,ωn−1,θk −θ,t) → 0 as k → ∞. η Thus, we have Γ(θ ( ,γ n ) η γ n k) → Γ(θ) when θ ( , ) k → θ. Hence, from the last inequality, we have sup{η(ω1,ω2,...,ωn−1,θ −Γ(θ),t) : θ ∈ J} t ≥ η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ) 3 and inf{γ(ω1,ω2,...,ωn−1,θ −Γ(θ),t) : θ ∈ J} t ≤ γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ), ) 3 for all ϖ ∈ V and t > 0 with fixed ω1,ω2,...,ωn−1 ∈ V. Thus if we assume t = ′ 3 t , then sup{η(ω1,ω2,...,ωn−1,θ −Γ(θ),t) : θ ∈ J} ≥ sup{η(ω1,ω2,...,ω ′ n−1,ϖ −Γ(ϖ),t ) : ϖ ∈ J} and inf{γ(ω1,ω2,...,ωn−1,θ −Γ(θ),t) : θ ∈ J} ≤ inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t′ ) : ϖ ∈ J}. Thus, our first claim is proved.
- 28. A Study on Approximate Fixed Point Property 11 Now consider any intuitionistic n-fuzzy nonexpansive mapping ΓJ : J → J. Since γ J is dense, there exists a sequence {θ (η n , ) k} in J such that θk → θ for any θ ∈ J. Since an intuitionistic n-fuzzy nonexpansive mapping is continuous, ΓJ is intu- itionistic n-fuzzy continuous and it can be extended on V by defining Γ(θ) = (η,γ n ) − lim ΓJ(θk). k→∞ Hence, Γ may be considered as an intuitionistic n-fuzzy nonexpansive mapping on V. Thus, we have η(ω1,ω2,...,ωn−1,Γ(θ)−Γ(ϖ),t) = limsupη(ω1,ω2,...,ωn−1,Γ(θk)−Γ(ϖk),t) k→∞ ≥ limsupη(ω1,ω2,...,ωn−1,θk −ϖk,t) k→∞ and γ(ω1,ω2,...,ωn−1,Γ(θ)−Γ(ϖ),t) = limsupγ(ω1,ω2,...,ωn−1,Γ(θk)−Γ(ϖk),t) k→∞ ≤ limsupγ(ω1,ω2,...,ωn−1,θk −ϖk,t) k→∞ for all θ,ϖ ∈ V and t > 0 with fixed ω1,ω2,...,ωn−1 ∈ V. Since V has InFAFP property, we have sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = sup{η(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J} = 1 and inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = inf{γ(ω1,ω2,...,ωn−1,θ −Γ(θ),s) : θ ∈ J} = 0. Thus, given any intuitionistic n-fuzzy nonexpansive mapping Γ on J, we have sup{η(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = 1 and inf{γ(ω1,ω2,...,ωn−1,ϖ −Γ(ϖ),t) : ϖ ∈ J} = 0 and J has InFAFP property. This completes the proof.
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- 32. 2 Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics with an Application in Integral Equations Marija V. Paunović University of Kragujevac Samira Hadi Bonab, Vahid Parvaneh, and Farhan Golkarmanesh Islamic Azad University CONTENTS 2.1 Introduction .....................................................................................................15 2.2 Main Results....................................................................................................17 2.3 Some Fixed Point Theorems ...........................................................................19 2.4 Sequential Extended Fuzzy b-Metric Spaces ..................................................23 2.5 Application ......................................................................................................25 Bibliography ............................................................................................................27 2.1 INTRODUCTION In recent years, metric spaces have been generalized by many authors, which are expressed in different approaches. Many interesting spaces of the two-variable met- ric type are: b−metric space [2,5], rectangular metric space [4], parametric metric space [1,16,26], extended parametric b-metric space [29], sequential extended S- metric space [20], JS-metric space [17], extended b-metric space [19,30], modular metric space [25], multiplicative metric space [3], cone b- metric space [12], C∗- algebra valued metric space [27,28], vector-valued metric space [13,14], etc. The purpose of this chapter is to introduce a new type of generalized metric spaces, called the sequential extended parametric b-metric space (SEPbMS), as a DOI: 10.1201/9781003312017-2 15
- 33. 16 Soft Computing generalization of extended parametric b-metric space [29], using JS-contractive type mappings. Also, we prove some fixed point theorems for JS-contractive type map- pings in SEPbMSs, and we obtain some new fixed point results in triangular sequen- tial extended fuzzy b-metric spaces (SEFbMSs) induced by this new structure. An example and an application are given to confirm the results. For further details, the readers are referred to the book [6] and the references therein. Definition 2.1 [29] In a nonempty set V, the mapping ρ : 2 V × (0,+∞) → [0,+∞) is said to be a parametric metric on V provided that, 1. ρ(ℏ,ℏ′,ς) = 0 if and only if ℏ = ℏ′; 2. ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0; 3. ρ(ℏ,ℏ′,ς) ≤ ρ(ℏ,υ,ς)+ρ′ (υ,ℏ′,ς) for all ℏ,ℏ′, ∈ ℏ υ V and for all ς > 0. Then (V,ρ) is called a parametric metric space. Definition 2.2 [32] In a non-empty set V a mapping ρ : 2 V ×(0,+∞) → [0,+∞) is said to be an extended parametric b-metric (EPbM) if for all ℏ,ℏ′,υ ∈ V and ς > 0: (a) ρ(ℏ,ℏ′,ς) = 0 implies ℏ = ℏ′ for all ς > 0; (b) ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0; (c) ρ(ℏ,ℏ′ ,ς) ≤ Ω(ρ(ℏ,υ,ς)+ρ(υ,ℏ′ ,ς)), where Ω : [0,∞) → [0,∞) is an unto strictly increasing continuous function with Ω−1(ς) ≤ ς ≤ Ω(ς). Then (V,ρ) is called an EPbMS with control function Ω. Obviously, for Ω(ς) = sς, the EPbM reduces to parametric b-metric. Let V be a non-empty set and ϑ 2 g : V → [0,∞] be a mapping. For any ℏ ∈ V, let us define the set C(ϑg,V,ℏ) = {{ℏn} ⊂ V : lim ϑg( n, ) = 0}. n→∞ Definition 2.3 [17] Let ϑg : 2 V → [0,∞] be a mapping which satisfies: 1. ϑg(ℏ,ℏ′) = 0 implies ℏ = ℏ′; 2. for every ℏ,ℏ′ ∈ V, we have ϑg(ℏ,ℏ′) = ϑ ′ g(ℏ ,ℏ); 3. if (ℏ,ℏ′ 2 ) ∈ V and {ℏn} ∈ C(ϑg,V,ℏ), then ϑ ′ g(ℏ,ℏ ) ≤ plimsupϑg(ℏn,ℏ′), for n→∞ ℏ ℏ (2.1) some p > 0. The pair (V,ϑg) is called a JS-metric space. Since the Banach fixed point theorem is very attractive and practical, many re- searchers have tried to generalize it in recent years. These generalizations have either been done in generalized metric spaces or constructing the new contractions in which the Banach contraction being obtained as a special case of these new contractions. Of course, the generalization of Banach’s fixed point theorem is not limited to met- ric spaces. Rather, the Cartesian product of metric spaces and the Darbo’s theorem in terms of measure of noncompactness are also generalizations of Banach’s fixed point theorem. Each of these theorems, in turn, has been studied in various ways by many authors.
- 34. Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 17 It should be noted that by obtaining these beautiful results, the existence and uniqueness of the solution for the functional integral equations, fractional integral equations, differential equations, differential fractional equations, and matrix equa- tions have been proved. Unswerving with Ref. [18], consider the family of all functions θ : (0,∞) → (1,∞) so that: (θ1) θ is increasing; (θ2) lim θ(ρn) = 1 iff lim ρn = 0 for each sequence {ρn} ⊆ (0,∞); n→∞ n→∞ (θ θ 3) lim (ρ)−1 ρκ = λ for some κ ∈ (0,1) and for some λ ∈ (0,∞] is signified by J0. ρ→0+ Theorem 2.1 [17,18] A self-mapping ϒ on a complete metric space (V,d) so that ℏ,ℏ′ ∈ V, d(ϒℏ,ϒℏ′ ) = 0 ⇒ θ(d ,ℏ α (ϒℏ,ϒℏ′ )) ≤ θ(d(ℏ ′ )) ̸ where θ ∈ J0 and α ∈ (0,1) possesses a unique fixed point. Reminder that the Banach contraction principle is a specific instance of Theorem 2.1. We denote by J the family of functions θ : (0,∞) → (1,∞) so that: (θ1) θ is continues and increasing; (θ2) lim θ(ρn) = θ(ρ) iff lim ρn = ρ for each sequence {ρn} ⊆ (0,∞); n→∞ n→∞ In this chapter, via combination of Kannan contractions, Chatterjea contractions, and JS-contractions, we state and prove some fixed point results in two classes of generalized metric spaces. 2.2 MAIN RESULTS In this section, first we introduce a new type of EPbMSs. To expand such a concept, we first define, { } ð(ρ,V,ℏ) := {ℏn} ⊂ V : lim ρ(ℏn,ℏ,ς) = 0 for all ς > 0 , n→∞ where ρ : 2 V ×(0,+∞) → [0,+∞) is a given mapping. Definition 2.4 In a non-empty set V, a mapping ρ : 2 V ×(0,+∞) → [0,+∞) is said to be a SEPbM if for all ℏ,ℏ′ ∈ V and for all ς > 0: (a) ρ(ℏ,ℏ′,ς) = 0 for all ς > 0 implies ℏ = ℏ′; (b) ρ(ℏ,ℏ′,ς) = ρ(ℏ′,ℏ,ς) for all ς > 0; (c) ρ(ℏ,ℏ′ ,ς) ≤ Ω limsupρ(ℏ ,ℏ′ n ,ς) , where {ℏn} ∈ ð(ρ,V,ℏ) and Ω : n→∞ [0,∞] → [0,∞] is an unto strictly increasing continuous function with Ω−1(ς) ≤ ς ≤ Ω(ς) for all 0 ≤ ς < ∞. The triplet (V,ρ,Ω) is called a SEPbMS. For simplicity, we denote it by (V,ρ).
- 35. 18 Soft Computing Definition 2.5 Let (V,ρ) be a SEPbMS, {ℏn} ⊆ V and ℏ ∈ V. (i) {ℏn} is said to be convergent and converges to ℏ, if {ℏn} ∈ ð(ρ,V,ℏ). (ii) {ℏn} is said to be Cauchy, if lim ρ(ℏn, → ℏm,ς) = 0 for all ς > 0. n,m ∞ (iii) (V,ρ) is said to be complete, if every Cauchy sequence is a convergent se- quence. Example 2.1 Let the triplet (V,ρ,Ω) be an EPbMS. If we define ρ(ℏ,ℏ′,ς) = eς ℏ ℏ′ p ( − ) (p ≥ 1), then the pair (V,ρ) will be a SEPbMS on V for Ω(ς) = e2pς for all ς ≥ 0. proposition 2.1 If (V,ρ) be a parametric b-metric space with parameter s, then ρ is also a SEPbM on V. Proof. If (V,ρ) be a parametric b-metric space, then ρ undoubtedly fulfills the first two circumstances of Definition 2.4. We just indicate that ρ also gratifies the third condition of Definition 2.4. Since ρ is a parametric b-metric, then for all ℏ,ℏ′ ∈ V and for any se- quence {ℏn} ∈ ð(ρ,V,ℏ), according to triangle inequality we have, ρ(ℏ,ℏ′ ,ς) ≤ slimsupρ(ℏ ,ℏ′ n ,ς) for all ς > 0. n→∞ Then if we choose Ω(ς) = sς for all ς ∈ [0,∞], then we have ρ(ℏ,ℏ′ ,ς) ≤ Ω limsupρ(ℏn,ℏ′ ,ς) for all ℏ,ℏ′ ∈ V, for all ς > 0 and for all {ℏn} ∈ ð(ρ,V,ℏ). n→∞ Therefore, (V,ρ) is also a SEPbMS. Definition 2.6 Let (V,ρ) and (Y,ρ∗) be two SEPbMSs. A mapping ϒ : V → Y is called continuous at a point ℏ ∈ V if for any ε > 0 there is δε > 0 such that for any υ ∈ V, ρ∗(ϒυ,ϒℏ,ς) < ε whenever ρ(υ,ℏ,ς) < δε , for all ς > 0. ϒ is said to be continuous on V if ϒ is continuous at each point of V. proposition 2.2 In a SEPbMS (V,ρ), if a sequence {ℏn} is convergent, then it con- verges to a unique element in V. Proof. Suppose ℏ,ℏ′ ∈ V and ς > 0 be arbitrary such that ℏn → ℏ and ℏn → ℏ′ as n → ∞. Then we have, ρ(ℏ,ℏ′ ,ς) ≤ Ω limsupρ(ℏn,ℏ′ ,ς) which implies that n→∞ ρ(ℏ,ℏ′,ς) ≤ Ω(0) = 0, i.e., ℏ = ℏ′. proposition 2.3 Let (V,ρ) be a SEPbMS and {ℏn} ⊂ V converges to some ℏ ∈ V. Then ρ(ℏ,ℏ,ς) = 0. Proof. Since {ℏn} converges to ℏ ∈ V, so lim ρ(ℏn,ℏ,ς) = 0. Therefore, we have n→∞ ρ(ℏ,ℏ,ς) ≤ Ω limsupρ(ℏnℏ,ς) = Ω(0) = 0, n→∞ which implies ρ(ℏ,ℏ,ς) = 0.
- 36. Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 19 proposition 2.4 For a Cauchy sequence {ℏn} in a SEPbMS (V,ρ,Ω) such that Ω−1 is continuous, if {ℏn} has a convergent sub-sequence {ℏn } k which converges to ℏ ∈ V, then {ℏn} also converges to ℏ ∈ V. Proof. From condition (c) of Definition 2.4 we have, ρ(ℏn,ℏ,ς) ≤ Ω limsupρ(ℏn,ℏnk ,ς) which implies that Ω−1 (ρ(ℏn,ℏ,ς)) ≤ limsupρ(ℏn,ℏnk ,ς) k→∞ k→∞ for all n ∈ N and ς > 0. Because of the Cauchyness of the sequence {ℏn}, it follows that lim ρ(ℏn,ℏnk ,ς) n,k→∞ = 0 and thus, Ω−1 (ρ(ℏn,ℏ,ς)) → 0 as n → ∞ which implies that ρ(ℏn,ℏ,ς) → 0 as n ∞ 1 → , since Ω− is continuous. Hence, {ℏn} converges to ℏ ∈ V. proposition 2.5 In a SEPbMS (V,ρ), if a self-mapping ϒ is continuous at ℏ ∈ V, then {ϒℏn} ∈ ð(ρ,V,ϒℏ) for any sequence {ℏn} ∈ ð(ρ,V,ℏ). Proof. Let ε > 0 be given. Since ϒ is continuous at ℏ, then there exists δε > 0 such that ρ(υ,ℏ,ς) < δε implies ρ(ϒυ,ϒℏ,ς) < ε. As {ℏn} converges to ℏ, so for δε > 0, there exists N ∈ N such that ρ(ℏn,ℏ,ς) < δε for all n ≥ N. Therefore, for any n ≥ N, ρ(ϒℏn,ϒℏ,ς) < ε and thus ϒℏn → ϒℏ as n → ∞, that is, {ϒℏn} ∈ ð(ρ,V,ϒℏ). Let (V,ρ) be a SEPbMS with associate function Ω. Define, { } B(ℏ,η) := ℏ′ ∈ V : ρ(ℏ,ℏ′ ,ς) < ρ(ℏ,ℏ,ς)+η for all ς > 0 and B[ℏ,η] := ℏ′ ∈ V : ρ(ℏ,ℏ′ ,ς) ≤ ρ(ℏ,ℏ,ς)+η for all ς > 0 for all ℏ ∈ V and for all η > 0. { } Remark 2.1 Evidently, τρ := {0 /}∪{V(= 0 /) ⊂ V : for any ℏ ∈ V, there is η > 0 so that B(ℏ,η) ⊂ V} forms a topology on V. ̸ 2.3 SOME FIXED POINT THEOREMS In this section, we suppose that θ ∈ J . Theorem 2.2 Let (V,ρ) be a complete SEPbMS and ϒ : V → V be a mapping such that: (i) θ(ρ(ϒℏ,ϒℏ′,ς)) ≤ θ(ρ(ℏ,ℏ′ α ,ς)) for all ℏ,ℏ′ ∈ V and for some α ∈ (0,1), (ii) there is ℏ0 ∈ V so that { ( ) } δ(ρ,ϒ,ℏ0) := sup ρ ϒi ℏ0,ϒ j ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞. Then ϒ has at least one fixed point in V. Moreover if ℏ and ℏ′ are two fixed points of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞ then ℏ = ℏ′.
- 37. 20 Soft Computing Proof. We define, { ( ∆ ρ ϒ p+1 ℏ : sup ρ ϒ p+i ℏ ϒ p+ j ( , , 0) = 0, ℏ0,ς : i, j = 1,2,..., ς > 0 , for all p ≥ 1. Clearly, ∆(ρ,ϒ p+1,ℏ0) ≤ ∆(ρ,ϒ,ℏ0) < ∞ for all p ≥ 1. ) } Then for all p ≥ 1 and for all i, j = 1,2,..., θ ρ ϒ p+i ℏ ϒ p+j ℏ ς θ ρ ϒ p−1+i ℏ ϒ p−1+j α ( 0, 0, ) ≤ ( 0, ℏ0,ς) ≤ θ(∆(ρ,ϒ p ,ℏ α 0)) , ( ) ( ) which implies, for all p ≥ 1, ( ) θ ∆(ρ p 1 p i p j ,ϒ + ,ℏ0) = θ sup ρ(ϒ + ℏ + 0,ϒ ℏ0,ς) i,j≥1 ≤ θ (∆(ρ p ,ϒ ,ℏ α 0)) ( ≤ θ ∆(ρ,ϒ p− ) 1 α2 ,ℏ0) . . . ≤ θ (∆(ρ,ϒ,ℏ αp 0)) . Let ℏi =ϒℏi−1 =ϒiℏ0 for all i ∈ N. For all m > n ≥ 1 we have, θ ρ ℏ ℏ ς θ ρ ϒn ℏ ϒm ( ( n, m, )) = ( ( 0, ℏ0,ς)) θ ρ ϒn−1+1 ℏ ϒn−1+(m = −n 1 ( + ) 0, ℏ0,ς) ≤ θ (∆(ρ,ϒn ,ℏ0)) ≤ θ (∆(ρ,ϒ,ℏ αn−1 0)) −→ 1 as n → ∞. Therefore, {ℏn} is a Cauchy sequence in V. From the completeness of V, {ℏn} is convergent. Let limℏ α n = ℏ ∈ V. Now, θ(ρ(ϒℏ,ϒℏn,ς)) ≤ θ(ρ(ℏ,ℏn,ς)) → 1 as n n → ∞. Therefore, ℏn+1 →ϒℏ as n → ∞. Hence, by Proposition 2.2, we haveϒℏ = ℏ, i.e., ℏ ∈ V is a fixed point of ϒ. Now, if ℏ and ℏ′ be two fixed points of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞, then θ(ρ(ℏ,ℏ′ ,ς)) = θ(ρ(ϒℏ,ϒℏ′ ,ς)) ≤ θ(ρ(ℏ,ℏ′ ,ς α )) , which gives ρ(ℏ,ℏ′,ς) = 0, hence ℏ = ℏ′. Theorem 2.3 Let (V,ρ) be a complete SEPbMS and ϒ : V → V so that: (i) ( )+ρ(ℏ′ ′ ) −1 ρ(ℏ,ϒℏ,ς ,ϒℏ′,ς γ ) θ ρ(ϒℏ,ϒℏ ,ς) ≤ θ Ω , 2 for all ℏ,ℏ′ ∈ V and for some γ ∈ (0,1),
- 38. Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 21 (ii) there is ℏ0 ∈ V so that ∆(ρ,ϒ,ℏ0) := sup ρ ϒi ℏ0,ϒ j ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞. { ( ) } Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ∈ N, converges to some ℏ ∈ V. If ρ(ℏ,ϒℏ,ς) < ∞, then ℏ ∈ V is a fixed point of ϒ. Furthermore, if ℏ′ is a fixed point of ϒ in V such that ρ(ℏ,ℏ′,ς) < ∞ and ρ(ℏ′,ℏ′,ς) < ∞, then ℏ = ℏ′. Proof. For all p ≥ 1 and for all i, j = 1,2 p ,..., θ ρ(ϒ +iℏ p ,ϒ +j 0 ℏ0,ς) ρ(ϒ p−1+i θ Ω ℏ0,ϒ p+i γ ℏ0, − ς 1 )+ρ(ϒ p−1+jℏ0,ϒ p+jℏ0,ς) p γ ≤ ≤ ( ( , , )) 2 θ ∆ ρ ϒ ℏ0 . ( ) This implies that ( ) θ ∆ ρ ϒ p+1 ℏ θ sup ρ ϒ p+i ℏ ϒ p+j ( , , 0) = ( 0, ℏ0,ς p ) ≤ θ (∆(ρ,ϒ ,ℏ γ 0)) i,j≥1 for all p ≥ 1. Then continuing in an analogous technique as in Theorem 2.2, it can be effort- lessly exposed that {ℏn} is a Cauchy sequence in V and by the completeness of V there is some ℏ ∈ V such that limℏn = ℏ. n Now, θ(ρ(ℏn+1,ϒℏ,ς)) = θ(ρ(ϒℏn,ϒℏ,ς)) ≤ θ(Ω−1 [ρ(ℏ γ n,ϒℏn,ς)+ρ(ℏ,ϒℏ,ς)]) ≤ θ(Ω−1 [ρ(ℏ γ n,ℏn+1,ς)+ρ(ℏ,ϒℏ,ς)]) , for all n ≥ 0, which implies that limsupθ(ρ(ℏn+1,ϒℏ,ς)) ≤ θ 1 (Ω− [ρ(ℏ,ϒℏ,ς γ )]) < ∞. n→∞ On the other hand, ρ(ℏ,ϒℏ,ς) ≤ Ω limsupρ(ℏn+1,ϒℏ,ς) n→∞ ( ≤ Ω θ− ) 1 (θ(Ω−1 [ρ(ℏ,ϒℏ,ς γ )]) ) . If ρ(ℏ,ϒℏ,ς) > 0 then, Ω−1 (ρ(ℏ,ϒℏ,ς)) ≤ θ−1 θ(Ω−1 [ρ(ℏ γ ,ϒℏ,ς)]) < Ω−1 (ρ(ℏ,ϒℏ,ς)), ( ) a contradiction. Hence, ϒℏ = ℏ, i.e., ℏ ∈ V is a fixed point of ϒ. Now, if ℏ′ is a fixed point of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞ and ρ(ℏ′,ℏ′,ς) < ∞, then we have ] ρ ′ [ θ ′ ς θ ρ ϒ ϒ ς θ ρ ϒ γ ( (ℏ,ℏ , )) = ( ( ℏ, ℏ , )) ≤ (ℏ, ℏ,ς)+ρ(ℏ′ ,ϒℏ′ ,ς) = 0, as ρ(ℏ′,ℏ′,ς) = 0, therefore ℏ = ℏ′.
- 39. 22 Soft Computing Theorem 2.4 In a complete SEPbMS (V,ρ), if ϒ : V → V be a mapping so that: [ρ ′ ℏ′ β (ℏ,ϒℏ ,ς)+ρ( ,ϒℏ,ς)] (i) θ(ρ(ϒℏ,ϒℏ′,ς)) ≤ θ for all ℏ,ℏ′ ∈ V, for all 2 ς > 0 and for some β ∈ (0,1), (ii) there is ℏ0 ∈ V so that ∆(ρ,ϒ,ℏ0) := sup ρ ϒiℏ j 0,ϒ ℏ0,ς : i, j = 1,2,..., ς > 0 < ∞, then, the Picard iterating sequence {ℏ n n}, ℏn =ϒ ℏ0 for all n ≥ 1, converges to some ℏ ∈ V. If limsupρ(ℏn,ϒℏ,ς) < ∞, then ℏ ∈ V is a fixed point of ϒ, and if ℏ′ is a fixed n→∞ point of ϒ in V such that ρ(ℏ,ℏ′,ς) < ∞, then ℏ = ℏ′. { ( ) } Proof. By parallel argument as in Theorem 2.2, {ℏn} is a Cauchy sequence in V, and by completeness of V it converges to an element ℏ ∈ V. Now, for all n ∈ N∪{0}, we have θ(ρ(ℏn+1,ϒℏ,ς)) = θ(ρ(ϒℏn,ϒℏ,ς)) ≤ θ ([ρ(ℏn,ϒℏ,ς β )+ρ(ℏ,ϒℏn,ς)]) ≤ θ ([ρ(ℏn,ϒℏ,ς)+ρ(ℏn+1,ℏ,ς β )]) , which implies that θ limsupρ ℏ ϒℏ ς θ limsupρ ℏ ϒℏ ς β ( ( n+1, , )) ≤ ( ( n, , )) n→∞ n→∞ and hence limsupρ(ℏn,ϒℏ,ς) = 0. Therefore, n→∞ ρ(ℏ,ϒℏ,ς) ≤ Ω limsupρ(ℏn,ϒℏ,ς) = Ω(0) = 0 n→∞ and as a result, ℏ = ℏ. ϒ If ℏ′ be a fixed point of ϒ in V with ρ(ℏ,ℏ′,ς) < ∞, then we have β ′ [ρ(ℏ,ϒℏ′,ς)+ρ(ℏ′,ϒℏ,ς)] θ(ρ(ℏ,ℏ ,ς)) = θ(ρ(ϒℏ,ϒℏ′ ,ς)) ≤ θ 2 = θ(ρ(ℏ,ℏ′ ,ς β )) , which implies ρ(ℏ,ℏ′,ς) = 0, therefore, ℏ = ℏ′. Example 2.2 Consider V = [0,1] and ρ(ℏ,ℏ′,ς) = (ς|ℏ − ℏ′|) + ln(1 + ς|ℏ − ℏ′|) for all ℏ,ℏ′ ∈ V and for all ς > 0. Then ρ forms a SEPbM on V with the function Ω(s) = s+ln(1+s) for all s ≥ 0. Define ϒ : ℏ V → V by ϒℏ = 4 for all ℏ ∈ V. Then ϒ gratifies all the circumstances of Theorem 2.2 for α 1 = 2 , θ(ς) = eς and evidently, ϒ has a unique fixed point 0 ∈ V, because θ(ρ ϒℏ ϒℏ′ ς ς ℏ 4 ℏ′ 4 ln 1 ς ℏ 4 ℏ′ 4 ( , , )) = e( | / − / |+ ( + | / − / |)) ς ℏ ℏ′ ln 1 ς ℏ ℏ′ 1 2 ≤ [e( | − |+ ( + | − |)) ] / 1 2 ≤ θ(ρ(ℏ,ℏ′ ,ς)) / . For more details, the readers are referred to the book [11].
- 40. Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 23 2.4 SEQUENTIAL EXTENDED FUZZY B-METRIC SPACES In this section, stimulated by the work existing in Ref. [16], we present the percep- tion of a SEFbMS. We generate an association between SEPbM and SEFbM and present some new fixed point results in SEFbMS. For more details on fuzzy metric and its generalization, the readers are referred to [7]– [10], [21–24] and the refer- ences therein. Definition 2.7 (Schweizer and Sklar [31]) A binary operation ⋆ : [0,1 2 ] → [0,1] is called a continuous t-norm if: (T1) ⋆ is commutative and associative; (T2) ⋆ is continuous; (T3) ℏ⋆1 = ℏ for all ℏ ∈ [0,1]; (T4) ℏ⋆ℏ′ ≤ ℜ⋆ℜ′ when ℏ ≤ ℜ and ℏ′ ≤ ℜ′, with ℏ,ℏ′,ℜ,ℜ′ ∈ [0,1]. Definition 2.8 [29] A triplet (V,M,∗) is supposed to be a fuzzy metric space if V is an unselective set, ∗ is a continuous t-norm (CTN) and M is a fuzzy set on 2 V ×(0,∞) so that, for all ℏ,ℏ′,υ ∈ V and ς,s > 0, (i) M(ℏ,ℏ′,ς) > 0; (ii) M(ℏ,ℏ′,ς) = 1 for all ς > 0 if and only if ℏ = ℏ′; ′ ′ (iii) M(ℏ,ℏ ,ς) = M(ℏ ,ℏ,ς); (iv) M(ℏ,ℏ′,ς)∗M(ℏ′,υ,s) ≤ M(ℏ,υ,ς +s); (v) M(ℏ,ℏ′,·) : (0,∞) → [0,1] is continuous; The function M(ℏ,ℏ′,ς) means the degree of closeness among ℏ and ℏ′ regarding t. Definition 2.9 [16] A fuzzy b-metric space is an ordered triplet (V,B,⋆) such that 2 V is a nonempty set, ⋆ is a continuous ς-norm and B is a fuzzy set on V ×(0,∞) so that for all ℏ,ℏ′,υ ∈ V and ς,s > 0, (F1) B(ℏ,ℏ′,ς) > 0; (F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′; (F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς); (F4) B(ℏ,ℏ′,ς)⋆B(ℏ′,υ,s) ≤ B(ℏ,υ,b(ς +s)) where b ≥ 1; (F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left. Definition 2.10 [29] An ordered quadruple (V,B,⋆,Ω) in which V is a nonempty set, 2 ⋆ is a CTN and B is a fuzzy set on V × (0,∞) so that for all ℏ,ℏ′,υ ∈ V and ς,s > 0, (F1) B(ℏ,ℏ′,ς) > 0; (F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′; (F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς); (F4) B(ℏ,ℏ′,ς)⋆B(ℏ′,υ,s) ≤ B(ℏ,υ,Ω(ς +s)); (F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left, is an extended fuzzy b-metric space.
- 41. 24 Soft Computing Now, let be a non-empty set and B be a fuzzy set on 2 V V ×(0,∞). For any ℏ ∈ V, let ð(B,V,ℏ) = {{ℏn} ⊂ V : lim B(ℏn,ℏ,ς) = 1 for all ς > 0}. (2.2) n→∞ Definition 2.11 A SEFbMS is an ordered quadruple (V,B,⋆,Ω) in which V is a nonempty set, ⋆ is a CTN and B is a fuzzy set on 2 V ×(0,∞) so that for all ℏ,ℏ′,υ ∈ V and ς,s > 0, (F1) B(ℏ,ℏ′,ς) > 0; (F2) B(ℏ,ℏ′,ς) = 1 if and only if ℏ = ℏ′; (F3) B(ℏ,ℏ′,ς) = B(ℏ′,ℏ,ς); (F4) Ω[limsupB(ℏ ′ ′ n,ℏ ,ς)] ≤ B(ℏ,ℏ ,ς) where {ℏn} ∈ ð(B,V,ℏ) and Ω : [0,∞) → [0 1 ,∞) is an unto strictly increasing continuous function with Ω− (ς) ≤ ς ≤ Ω(ς) for all 0 ≤ ς < ∞; (F5) B(ℏ,ℏ′,·) : (0,+∞) → (0,1] is continuous from the left. Definition 2.12 Let (V,B,⋆,Ω) be a SEFbMS and {ℏn} be a sequence in V and ℏ ∈ V. (i) {ℏn} is supposed to be convergent and converges to ℏ if {ℏn} ∈ ð(B,V,ℏ). (ii) {ℏn} is supposed to be Cauchy if lim B( n,m→ ℏn,ℏm,ς) = 1 for all ς > 0. ∞ (iii) (V,B,⋆,Ω) is called complete if every Cauchy sequence is a convergent se- quence. Definition 2.13 The SEFbM (V,B,⋆,Ω) is called Ω-convertible whenever, 1 1 −1 ≤ Ω[ − Ω[limsupB(ℏ ,ℏ′ n ,ς)] limsupB(ℏn,ℏ′ 1]. ,ς) for all ℏ,ℏ′,υ ∈ V, ς > 0 and ℏn ∈ ð(B,V,ℏ). ′ 1 Remark 2.2 Notice that ρ(ℏ,ℏ ,ς) = − ′ 1 is a SEPbM whenever B is a B(ℏ,ℏ ,ς) Ω-convertible SEFbM. As an application of Remark 2.2 and the results recognized in Section 3, we can deduce the subsequent results in SEFbMSs. Theorem 2.5 Let (V,B,⋆,Ω) be an Ω-convertible complete SEFbMS andϒ : V → V be a mapping so that: α 1 1 (i) θ −1 ≤ θ − B(ϒℏ,ϒℏ′ 1 for all ℏ,ℏ′ ∈ V and for some ,ς) B(ℏ,ℏ′,ς) α ∈ (0,1), (ii) there is ℏ0 ∈ V so that 1 ∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,..., ς > 0 < B(ϒiℏ ,ϒ j ∞. 0 ℏ0,ς) Then ϒ takes at least one fixed point in V. Furthermore, if ℏ and ℏ′ are two fixed 1 points of ϒ in V with − B(ℏ,ℏ′ 1 < ∞, then ℏ = ℏ′. ,ς)
- 42. Sequential Extended Parametric and Sequential Extended Fuzzy b-Metrics 25 Theorem 2.6 Let (V,B,⋆,Ω) be a Ω-convertible complete SEFbMS and ϒ : V → V such that: (i) 1 1 γ −1+ 1 − B(ℏ,ϒℏ,ς) (ℏ′,ϒℏ′ 1 B ,ς) θ( − ′ 1 ≤ θ 1 ) Ω− B(ϒℏ,ϒℏ ,ς) , 2 for all ℏ,ℏ′ ∈ V and for some γ ∈ (0,1), (ii) there is ℏ0 ∈ V such that 1 ∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,..., < i ς > 0 B(ϒ ℏ ,ϒ j ∞. 0 ℏ0,ς) Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ∈ N, converges to some 1 ℏ ∈ V. If − 1 < ∞, then ℏ ∈ V is a fixed point of ϒ. Moreover, if ℏ′ is B(ℏ,ϒℏ,ς) 1 1 a fixed point of ϒ in V such that − < − < B(ℏ,ℏ′ 1 ∞ and 1 ∞, then ,ς) B(ℏ′,ℏ′,ς) ℏ = ℏ′. Theorem 2.7 Let (V,B,⋆,Ω) be a Ω-convertible complete SEFbMS and ϒ : V → V be a mapping so that: (i) β 1 1 1 1 −1+ − B (ℏ,ϒℏ′,ς) B(ℏ′,ϒℏ,ς) θ( −1) ≤ B(ϒℏ,ϒ ′ θ ,ς) , ℏ 2 for all ℏ,ℏ′ ∈ V, ς > 0 and for some β ∈ (0,1), (ii) there is ℏ0 ∈ V so that 1 ∆(B,ϒ,ℏ0) := sup −1 : i, j = 1,2,..., B(ϒ ℏ j ς > 0 < i ∞. 0,ϒ ℏ0,ς) Then the Picard iterating sequence {ℏn}, ℏn =ϒnℏ0 for all n ≥ 1, converges to some 1 ℏ ∈ V. If limsup −1 < ∞, then ℏ ∈ V is a fixed point of ϒ. Also, if ℏ′ is n→∞ B(ℏn,ϒℏ,ς) 1 a fixed point of ϒ in V such that − B(ℏ,ℏ′ 1 < ∞, then ℏ = ℏ′. ,ς) 2.5 APPLICATION Let V = C[0,T] be the set of real continuous functions defined on [0,T] and ρ : 2 V ×(0,∞) → [0,∞) be defined by: ρ(ℏ,ℏ′ ,α) = sup ℏ(ς p (e−ας | )−ℏ′ (ς)| ) for all ℏ,ℏ′ ∈ Λ and all ς > 0, p ≥ 1. 0≤ς≤T
- 43. 26 Soft Computing Then (V,ρ) is a complete SEPbM space with Ω(ς) = 2p−1ς for all ς ≥ 0. Now, let us study the integral equation: T ℏ(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ(s))ds, 0 (2.3) where h : [0,T] → R, F : [0,T 2 2 ] → [0,∞) and K : [0,T] × R → R are continuous functions. Theorem 2.8 Assume that the subsequent suppositions are fulfilled: (i) for all ς,s ∈ [0,T] we have p α |K(ς,s,ℏ(s))−K(ς,s,ℏ′ (s s p ))| ≤ A(e− max |ℏ(s)−ℏ′ (s)| ), p ≥ 1,0 ≤ A < 1, 0≤s≤ϒ f (ii) sup T 1 t∈[0,T] [ | 0 F(t,s q )| ds]q ≤ 1. Then, the integral equation (2.3) has a unique solution u ∈ Λ. (iii) there is ℏ0 ∈ C[0,T] such that sup sup − j (e ας ℏ p |ϒi ℏ0 −ϒ 0| ) : i, j = 1,2,..., ς > 0 < ∞, 0≤ς≤T where T ϒ(ℏ0)(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ0(s))ds, ℏ ∈ V, ς,s ∈ [0,T]. 0 Proof. Let us define ϒ : V → V by T ϒ(ℏ)(ς) = h(ς)+ F(ς,s)K(ς,s,ℏ(s))ds, ℏ ∈ V, ς,s ∈ [0,T]. 0 Then by conditions (i)- (iii), for all ϕ,ψ ∈ V we get ρ ϒ ϕ ας p ( ( ),ϒ(ψ),ς) = sup(e− |ϒ(ϕ)(ς)−ϒ(ψ)(ς)| ) f = sup −ας T ς∈ | {K [0,T] e 0 F(ς,s) (ς s p , ,ϕ(s))−K(ς,s,ψ(s))}ds| f 1 T f [ ς 1 p ≤ sup ∈ e−ας q p |F( ,s T )| ds]q [ |K − ς [0,T] 0 0 (ς,s,ϕ(s)) K(ς,s,ψ(s))| ds] p ( ) A e−αs ≤ sup ≤ ≤ {|ϕ(s)−ψ(s p )| } 0 s T ≤ Aρ(ϕ,ψ,ς) for all ς ∈ [0,T], for A ∈ (0,1) and for all ϕ,ψ ∈ V. Hereafter, the circumstances of Theorem 2.2 (with θ(ς) = eς ) are fulfilled, and thus ϒ has a unique fixed point in V, namely, the nonlinear integral equation (2.3) has a unique solution in C[0,T]. For more examples of applications, the readers are referred to the book [11] and the references therein.
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- 46. 3 Analytical Sequel of Rational-Type Fuzzy Contraction in Fuzzy b-Metric Spaces Nabanita Konwar Birjhora Mahavidyalaya CONTENTS 3.1 Introduction .....................................................................................................29 3.1.1 Background.........................................................................................30 3.1.2 Main Goal...........................................................................................30 3.2 Basic Definitions .............................................................................................30 3.3 Main Results of the Chapter............................................................................32 3.3.1 Definition of Rational-Type Fuzzy Contraction in Fb-MS.................32 3.3.2 Related Theorems of Rational-Type Fuzzy Contraction ....................33 3.3.3 Corollaries...........................................................................................39 3.4 Application of Rational-Type Fuzzy Contraction ...........................................40 3.5 Conclusion.......................................................................................................42 Bibliography ............................................................................................................42 3.1 INTRODUCTION In mathematical analysis, the study of the existence of fixed point of a function plays a significant role. With the help of fixed point of a function, one can verify the exis- tence of a solution of the function within a metric space. The notion of generalized b-metric spaces has recently contributed significantly to the study of fixed point the- ory. Such type of generalization can modulate complex situations more effectively for higher order sets and scale down the complexity of modeling systems. It also creates an effective platform for mathematical modeling and designing. DOI: 10.1201/9781003312017-3 29
- 47. 30 Soft Computing 3.1.1 BACKGROUND In order to model the situations where data or elements are imprecise or vague and to represent a mathematical structure for such types of situations, an extended con- cept of set theory called fuzzy set theory was established by Zadeh [32] in 1965. Simultaneously, Kaleva and Seikkala [21] initiated the idea of fuzzy metric space. Simultaneously, several mathematicians like Kramosil and Michalek [26], George and Veeramani [14], etc. modified the notion of fuzzy metric space (FMS). The con- cept of weakly compatible maps was established by Jungck and Rhoads [20] for metric spaces. The development of metric space in multiple ways is an exciting area of research for the mathematicians. By considering a weaker condition, in place of triangular inequality, Bakhtin [4] and Czerwik [7] introduced the notion of b metric space. Heilpern [18] initiated the study fixed point theory and developed an extended version of the Banach’s contraction principle in fuzzy metric spaces. The con- cept of contraction-type fixed point results in FMS was established by Gregori and Sapena [17]. Some more generalized and extended work in the settings of fuzziness may be found in Refs. [1,3,5,6,8–13,16,22–25,27–31]. 3.1.2 MAIN GOAL The predominant aim of this chapter is to define the notion of rational-type fuzzy contraction in FbMS and establish some new fixed point results. After that the exis- tence and uniqueness of fixed point for rational-type fuzzy contraction in G-complete FbMS is established. We also provide an application in support of the results. 3.2 BASIC DEFINITIONS Below we discuss a few preliminary definitions which are essential for our main results. Definition 3.1 Consider a binary operation ∗ : [0,1]×[0,1] → [0,1]. Then ∗ is known as a continuous t −norm if it satisfies the condition: (i) ∗ is associative and commutative, (ii) ∗ is continuous, (iii) α ∗1 = α for all α ∈ [0,1], (iv) α ∗b ≤ β ∗d whenever α ≤ β and b ≤ d and α,β,c,d ∈ [0,1]. Definition 3.2 Consider a function d : S × S −→ R, where S = φ. Then for all s1,s2,s3 ∈ S, (S,d) is called a metric space if it satisfies the following conditions: ̸ (i) d(s1,s2) ≥ 0 and d(s1,s2) = 0 iff s1 = s2. (ii) d(s1,s2) = d(s2,s1). (iii) d(s1,s3) ≤ d(s1,s2)+d(s2,s3).
- 48. Analytical Sequel of Rational-Type Fuzzy Contraction 31 Definition 3.3 Suppose X is a classical set, called the universe and A ∈ X. The membership of A is considered as a characteristic function µA from X to {0,1} such that 1 iff x ∈ A µA(x) = 0 iff x ∈ / A. {0,1} is called a valuation set. If {0,1} is allowed to be [0,1], A is said to be a fuzzy set. Kramosil and Michalek [26] defined fuzzy metric space as follows: Definition 3.4 [26] Consider a set X = φ and a continuous t-norm ∗. Suppose M is a fuzzy set on X2 × R. Then for all a1,a2,a3 ∈ X and t,s ∈ R, (X,M,∗) is called fuzzy metric space if it satisfies the following axioms: ̸ (i) M(a1,a2,t) = 0 ∀ t ≤ 0. (ii) M(a1,a2,t) = 1 ∀ t > 0 iff a1 = a2. (iii) M(a1,a2,t) = M(a2,a1,t). (iv) M(a1,a2,t)∗M(a2,a3,s) ≤ M(a1,a3,t +s). (v) M(a1,a2,t) : (0,∞) → [0,1] is left continuous. (vi) limt→∞ M(a1,a2,t) = 1. George and Veeramani [14,15] made an appealing modification of fuzzy metric spaces in the following way: Definition 3.5 [14] Consider a set X = φ and a continuous t-norm ∗. Suppose M is a fuzzy set on X2 ×(0,∞). Then for all a1,a2,a3 ∈ X and t,s ∈ R, (X,M,∗) is called fuzzy metric space if it satisfies the following axioms: ̸ (i) M(a1,a2,t) > 0. (ii) M(a1,a2,t) = 1 ∀ t > 0 if and only if a1 = a2. (iii) M(a1,a2,t) = M(a2,a1,t). (iv) M(a1,a2,t)∗M(a2,a3,s) ≤ M(a1,a3,t +s). (v) M(a1,a2,t) : (0,∞) → [0,1] is continuous. Definition 3.6 [19] Consider a non-empty set S and a continuous t-norm ∗. Suppose P is a fuzzy set on S ×S ×(0,∞) such that for all u,v,w ∈ S and α,β > 0 following conditions are holds: (i) P(h̄1,h̄2,α) > 0, (ii) P(h̄1,h̄2,α) = 1 ⇐⇒ h̄1 = h̄2, (iii) P(h̄1,h̄2,α) = P(h̄2,h̄1,α), (iv) P(h̄1,h̄2,·) : (0,∞) → (0,1] is continuous, β (v) P(h̄1,h̄ α 3,α +β) ≥ ∗(P(h̄1,h̄2, ),P(h̄2,h̄3, )) b b Then (S,P,∗) is called a FbMS.
- 49. 32 Soft Computing Definition 3.7 [2] Consider a metric space (Y,d) and a function T : Y → Y. Then T is called a contraction mapping or contraction if there exists a constant α (called constant of contraction), with 0 ≤ α < 1, such that d(T(y1),T(y2)) ≤ αd(y1,y2),∀y1,y2 ∈ Y. Definition 3.8 [17] Let (U,Mr,∗) be a FbM-space, v1 ∈ U and a sequence (µj) in U is fuzzy-contractive if there exists α ∈ (0,1) such that 1 − ) > ≥ Mr( , for t , j µ 1 ≤ α 1 ( −1 0 1 j,µj+1,t) Mr(µj−1,µj,t) Definition 3.9 [17] Let (U,Mr,∗) be a FbM-space. A sequence (µj) in U is said to be G-Cauchy if limj Mr(µj,µj+p,t) = 1, for t > 0 and p > 0. And (U,Mr,∗) is called G-complete if every G-Cauchy sequence is convergent. Definition 3.10 Let (U,Mr,∗) be a FbM-space. Then Mr is said to be triangular if it satisfied the following property 1 − − ) M ( , ∗,t) 1 1 1 ≤ ( −1)+( 1 , r µ1 µ Mr(µ µ t t 1, , ) M b r(µ,µ∗, ) b for all µ,µ1,µ∗ ∈ U, t > 0. Definition 3.11 [17] Suppose (U,Mr,∗) is a FbM-space. Construct a mapping f : U → U. If for all µ1,µ∗ ∈ U, t > 0, there exists α ∈ (0,1) such that 1 − M (f(µ ),f(µ∗),t) 1 1 ≤ α( − r 1 Mr(µ1,µ∗,t) 1), Then f is called fuzzy-contractive. Next we elaborate the results of the chapter. 3.3 MAIN RESULTS OF THE CHAPTER In this section, we put forward the definition of rational-type fuzzy contraction in Fb-MS. After defining the main concept, we provide some propositions and related theorems. 3.3.1 DEFINITION OF RATIONAL-TYPE FUZZY CONTRACTION IN FB-MS Definition 3.12 Consider a FbMS (U,Mr,∗) and a function f : U → U. Then f is said to be rational-type fuzzy contraction(RTF-contraction) if there exists α,β ∈ [0,1) such that for all µ1,µ∗ ∈ U and t > 0, 1 1 −1 ≤α( −1) Mr(f(µ1), f(µ∗),t) Mr(µ1,µ∗,t) Mr(µ , β 1 µ∗,t) + ( −1) Mr(µ1, f t 2t (µ1), )∗Mr(µ∗, f( 1), ) b µ b
- 50. Analytical Sequel of Rational-Type Fuzzy Contraction 33 3.3.2 RELATED THEOREMS OF RATIONAL-TYPE FUZZY CONTRACTION Theorem 3.1 Consider a G-complete FbMS (U,Mr,∗) and a rational-type fuzzy contraction mapping f : U → U with α + β = 1. Then in U, f has a unique fixed point. Proof. Suppose that µ0 ∈ U is fixed and µj+1 = f(µj), j ≥ 0. Then for t > 0, j ≥ 1, 1 1 −1 = −1 Mr(µj,µj+1,t) Mr(f(µj−1), f(µj),t) 1 ≤ α( −1) Mr(µj−1,µj,t) Mr(µj−1,µj,t) +β( −1) Mr(µ t j−1, f t 2 (µj−1), )∗Mr( b µj, f(µj−1), ) b 1 = α( −1) Mr(µj−1,µj,t) M β r(µj−1,µj,t) + ( −1) (3.1) M t 2t r(µj−1,µj, )∗ b Mr(µj,µj, ) b Therefore, for t > 0 1 1 −1 ≤ α( −1) (3.2) Mr(µj,µj+1,t) Mr(µj−1,µj,t) In a similar way we have, for t > 0 1 1 −1 ≤ α( −1) Mr(µj−1,µj,t) Mr(µj−2,µj−1,t) Therefore, from Eqs. (3.2) and (3.3), we have for t > 0, (3.3) 1 1 −1 ≤ α( −1) Mr(µj,µj+1,t) Mr(µj−1,µj,t) ≤ α2 1 ( −1) Mr(µj−2,µj−1,t) j 1 ≤ ··· ≤ α ( −1) Mr(µ0,µ1,t) −→ 0, as j −→ ∞. Therefore, (µj) is a fuzzy-contractive sequence in U. Hence for t > 0, limj→∞ Mr(µj,µj+1,t) = 1 Next we have to show that (µj) is a G-Cauchy sequence.
- 51. 34 Soft Computing Consider a fixed q ∈ N and let j ∈ N such that 1 1 1 Mr(µj,µj+q,t) = Mr(µj,µj+q,( + +···+ )t) q q q q−times t t ≥ Mr(µj,µj+1, )∗M qb r(µj+1,µj+2, ) qb t ∗···∗Mr(µj+q−1,µj+q, ) qb −→ 1∗1∗···∗1 1 = , as j −→ ∞. q−times Hence (µj) is a G-Cauchy sequence. Since (U,Mr,∗) is G-complete, there exists v1 ∈ U such that for t > 0, µj → v1, as j −→ ∞, limj→∞ Mr(µj,v1,t) = 1 As Mr is triangular, for t > 0, we have 1 1 1 −1 ≤ ( − Mr(v t 1)+( − t 1) 1, f(v1),t) Mr(v1,µj+1, ) Mr(f(µj), ) b f(v1), b 1 1 ≤ ( − t 1)+α( −1) Mr(v1,µj+1, ) b Mr(µj,v1,t) M +β r(µj,v1,t) ( −1) Mr(µj, f(µ t M v f µ 2t j), )∗ b r( 1, ( j), ) b 1 1 = ( − t 1)+α( −1) Mr(v1,µj+1, ) Mr(µj,v1,t) b M +β r(µj,v1,t) ( −1) M t 2t r(µj,µj+1, )∗ b Mr(v1,µj+1, ) b −→ 0, as j −→ ∞. Hence for t > 0, Mr(v1, f(v1),t) = 1 implies f(v1) = v1 Finally, we have to prove the uniqueness. Consider that ∃ z1 ∈ U such that f(z1) = z1 and f(v1) = v1, then we have 1 1 −1 = −1 Mr(v1,z1,t) Mr(f(v1), f(z1),t) 1 ≤ α( −1) Mr(v1,z1,t) M +β r(v1,z1,t) ( −1) M v f v t 2 r( , 1), )∗ b Mr(z1, f(v t 1 ( 1), ) b
- 52. Analytical Sequel of Rational-Type Fuzzy Contraction 35 1 ≤ α( −1) Mr(v1,z1,t) M (v ,z ,t) +β r 1 1 ( −1) M v v t 2 r 1, )∗ b Mr(z1,v t ( 1, 1, ) b 1 = α( −1) Mr(v1,z1,t) 1 = α( −1) Mr(f(v1), f(z1),t) ≤ α2 1 ( −1) Mr(v1,z1,t) ≤ ··· ≤ α j 1 ( −1) Mr(v1,z1,t) −→ 0, as j −→ ∞. Therefore, Mr(v1,z1,t) = 1 implies v1 = z1. Hence f has a unique fixed point. Theorem 3.2 Consider a G-complete FbMS (U,Mr,∗) where Mr satisfy the trian- gular inequality and a mapping f : U → U with α + β + 2γ + 2δ < 1 such that for all µ1,µ∗ ∈ U, t > 0, α,β,γ,δ ≥ 0 f satisfies the following property: 1 1 −1 ≤α( − ) Mr(f(µ ∗ ∗ 1 1), f(µ ),t) Mr(µ1,µ ,t) M (µ ,µ∗ t t , )∗M (µ∗, f(µ∗), ) +β r 1 b r b ( − ) Mr(µ1, f(µ t t 1 1), )∗ ( b M µ1 f(µ∗ r , ), ) b M t ∗ r(µ , ∗ γ 1, f(µ1) ) M 1 r(µ , f(µ ),t) + ( − + −1) Mr(µ1, f(µ∗ 2t ), ) b Mr(µ1, f(µ 2t 1), ) b 1 1 +δ( −1+ −1) M µ t r( 1, f µ1), ) b Mr(µ∗, f( ∗ t ( µ ), ) b (3.4) Then f has a unique fixed point in U. Proof. Consider a fixed µ0 ∈ U and µj+1 = f(µj), j ≥ 0. Now for t > 0, j ≥ 1 1 1 −1 = −1 Mr(µj,µj+1,t) Mr(f(µj−1), f(µj),t) 1 ≤ α( −1) Mr(µj−1,µj,t)
- 53. 36 Soft Computing M ( µ β r µj−1,µ t j, )∗ b M t r( j, f(µj), ) b + ( −1) M (µ t 2t r j−1, f(µj−1), )∗Mr(µj−1, f( j ) b µ ), b M (µ γ r j−1, f(µj−1),t) Mr(µj, f(µj),t) + ( −1+ −1) Mr(µj−1, f(µ 2t 2t j), ) ( , ( ), ) b Mr µj−1 f µj b 1 1 +δ( − − t 1+ t 1) Mr(µj−1, f(µj−1), ) b Mr(µj, f(µj), ) b 1 =α( −1) Mr(µj−1,µj,t) Mr(µj−1,µ t j, )∗ b M t r(µ +β j,µj+1, ) b ( −1) M t 2t r(µj−1,µj, )∗ b Mr(µj−1,µj+1, ) b Mr(µj−1,µj,t) Mr(µj,µj+1,t) +γ( −1+ −1) Mr(µj−1,µ 2t M µ µ 2t j+1, ) r( j−1, j+1, ) b b 1 1 +δ( −1+ −1) Mr(µ t 1 µ t j− , j, ) b Mr(µj,µj+1, ) b Since for t > 0 Mr(µj−1,µj+1,2t) ≥ Mr(µj−1,µj,t)∗Mr(µj,µj+1,t) we have 1 1 −1 ≤ λ( −1), Mr(µj,µj+1,t) Mr(µj−1,µj,t) (3.5) where λ α+β+γ+δ = < 1−γ−δ 1 Similarly, for t > 0 1 1 −1 ≤ λ( −1), Mr(µj−1,µj,t) Mr(µj−2,µj−1,t) (3.6) where λ α+β+γ+δ = < 1−γ−δ 1 Hence from Eqs. ( 3.5) and ( 3.6) we have for t > 0 1 1 −1 ≤ λ( −1) Mr(µj,µj+1,t) Mr(µj−1,µj,t) ≤ λ2 1 ( −1) Mr(µj−2,µj−1,t) ≤ ... ≤ λ j 1 ( −1) Mr(µ0,µ1,t) −→ 0, as j −→ ∞ Therefore, (µj) is a rational-type fuzzy-contractive sequence in U. Hence for t > 0, limj→∞ Mr(µj,µj+1,t) = 1 Next we have to show that (µj) is a G-Cauchy sequence.
- 54. Analytical Sequel of Rational-Type Fuzzy Contraction 37 Consider a fixed q ∈ N and let j ∈ N such that 1 1 1 Mr(µj,µj+q,t) = Mr(µj,µj+q,( + +···+ )t) q q q q−times t t ≥ Mr(µj,µj+1, )∗M qb r(µj+1,µj+2, ) qb t ∗···∗Mr(µj+q−1,µj+q, ) qb −→ 1∗1∗···∗1 1 = , as j −→ ∞. q−times Hence (µj) is a G-Cauchy sequence. Since (U,Mr,∗) is G-complete, ∃ v1 ∈ U such that for t > 0, µj → v1, as j −→ ∞, limj→∞ Mr(µj,v1,t) = 1 As Mr is triangular, for t > 0, we have 1 1 1 −1 ≤ ( −1)+( −1) Mr(v1, f(v1),t) Mr(v t t 1,µj+1, ) Mr(µj+1, f(v1), ) b b Hence we have, 1 1 −1 = −1 Mr(µj+1, f(v1),t) Mr(f(µj), f(v1),t) 1 ≤ α( −1) Mr(µj,v1,t) M t t β r(µj,v1, )∗ b Mr(v1, f(v1), ) b + ( −1) Mr(µj, f t (µj), )∗Mr(µj, f(v 2t 1), ) b b M (µ γ r(µj, f j),t) M (v , f(v ),t) + ( −1 r 1 1 + −1) M µ f v 2t M µ f v 2t r( j, ( 1), ) b r( j, ( 1), ) b 1 1 +δ( − − t 1+ Mr(µj, f µ t 1) ( j), ) Mr(v1, f(v1), ) b b 1 = α( −1) Mr(µj,v1,t) M t t r(µj,v1, )∗ b Mr(v1, f(v1), ) +β b ( −1) Mr(µ µ t j j+1, )∗ b Mr(µj, f(v 2t , 1), ) b M , M γ r(µj,µj+1 t) 1 r(v1, f(v1),t) + ( − + −1) Mr(µj, f(v 2t 1), ) b Mr(µj, f(v 2t 1), ) b 1 1 +δ( −1+ −1) Mr µ t ( j,µj+1, ) b Mr(v1, f t (v1), ) b
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- 59. The Project Gutenberg eBook of Sermons of the Rev. Francis A. Baker, Priest of the Congregation of St. Paul
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Sermons of the Rev. Francis A. Baker, Priest of the Congregation of St. Paul Author: Francis A. Baker Contributor: A. F. Hewit Release date: February 3, 2019 [eBook #58812] Language: English Credits: Produced by Don Kostuch *** START OF THE PROJECT GUTENBERG EBOOK SERMONS OF THE REV. FRANCIS A. BAKER, PRIEST OF THE CONGREGATION OF ST. PAUL ***
- 61. [Transcriber's Notes: This production was derived from https://archive.org/details/lifeofrevfrancis00hewi/page/n9] Sermons Of The Rev. Francis A. Baker,
- 62. Priest Of The Congregation Of St. Paul. With A Memoir Of His Life
- 63. BY Rev. A. F. Hewit. Fourth Edition. New York: Lawrence Kehoe, 145 Nassau Street. 1867. Entered according to Act of Congress, In the year 1865 By A. F. Hewit, In the Clerk's Office of the District Court of the United States for the Southern District of New York. PREFACE. In offering the Memoir and Sermons of this volume to the friends of F. Baker, and to the public, propriety requires of me a few words of explanation. The number of those who have been more or less interested in the events touched upon in the sketch of his life and labors is very great, and composed of many different classes of persons in various places, and of more than one religious
- 64. communion. I cannot suppose that all of them will read these pages, but it is likely that many will; and therefore a word is due to those who are more particularly interested, as well as to the general class of readers. I have to ask the indulgence of all my readers for having interwoven so much of my own history and my own reflections on the topics and events of the period included within the limits of the narrative. They have woven themselves in spontaneously, without any intention on my part, and on account of the close connexion between myself and the one whose career I have been describing; and I have been unable to unravel them from the texture of the narrative without breaking its threads. I have simply transferred to paper that picture of the past, long forgotten amid the occupations of an active life, which came up again, unbidden and with great vividness, before the eye of memory, during the hours while the remains of my brother and dearest friend lay robed in violet, waiting for the last solemn rites of the requiem to be fulfilled. If I have succeeded, I cannot but think that the picture will have something of the same interest for others that it has for myself. Those who knew and loved the original, will, I hope, prize it for his sake; and their own recollections will diffuse the coloring and animation of life over that which in itself is but a pale and indistinct sketch. For their sakes chiefly I have prepared it, so far as the mere personal motive of perpetuating the memory of a revered and beloved individual is concerned. But I have had a higher motive as my chief reason for undertaking the task: a desire to promote the glory of God, by preserving and extending the memory of the graces and virtues with which He adorned one of His most faithful children. I have wished to place before the world the example of one of the most signal conversions to the Catholic faith which has taken place in our country, as a lesson to all to imitate the pure and disinterested devotion to truth and conscience which it presents to them. Let me not be misunderstood. I do not present the example of his conversion, or that of the great number of persons of similar
- 65. character who have embraced the Catholic religion, as a proof sufficient by itself of the truth of that religion. I propose it as a specimen of many instances in which the power of the Catholic religion to draw intelligent minds and upright hearts to itself, and to inspire them with a pure and noble spirit of self-sacrifice in the cause of God and humanity, is exhibited. This is surely a sufficient motive for examining carefully the reasons and evidences on which their submission to the Church was grounded; and an incentive to seek for the truth, with an equally sincere intention to embrace it, at whatever cost or struggle it may demand. It may appear to the casual reader that I have drawn in this narrative an ideal portrait which exaggerates the reality. I do not think I have done so; and I believe the most competent judges will attest my strict fidelity to the truth of nature. If I have represented my subject as a most perfect and beautiful character, the model of a man, a Christian, and a priest of God, I have not exceeded the sober judgment of the most impartial witnesses. A Protestant Episcopal clergyman, of remarkable honesty and generosity of nature, said of him to a Catholic friend: "You have one perfect man among your converts." Another, a Catholic clergyman, whose coolness of judgment and reticence of praise are remarkable traits in his character, said, on hearing of his decease: "The best priest in New York is dead." I have no doubt that more than one would have been willing to give their own lives in place of his, if he could have been saved by the sacrifice. In narrating events connected with F. Baker's varied career, I have simply related those things of which I have had either personal knowledge, or the evidence furnished by his own correspondence with a very dear friend, aided by the information which that friend has furnished me. I have to thank this very kind and valued friend, the Rev. Dwight E. Lyman, for the aid he has given me in this way, which has increased so much the completeness and interest of the Memoir. I am also indebted to another, still dearer to the departed, for information concerning his early history and family.
- 66. I trust that those readers who are not members of the Catholic communion, especially such as have been the friends of the subject and the author of this memoir, will find nothing here to jar unnecessarily upon their sentiments and feelings. Fidelity to the deceased has required me not to conceal his conviction of the exclusive truth and authority of the doctrine and communion of the holy, Catholic, Apostolic, Roman Church. The same fidelity would prevent me, if my own principles did not do so, from mixing up with religious questions any thing savoring of personal arrogance, or directed to the vindication of private feelings, and retaliation upon individuals with whom religious conflicts have brought us into collision. I wish those who still retain their friendship for the dead, and whose minds will recur with interest to scenes of this narrative, in which they were concerned with him, to be assured of that lasting sentiment of regard which he carried with him to the grave, and which survives in the heart of the writer of these lines. In the history of F. Baker's missionary career, I have endeavored to select from the materials on hand such portions of the details of particular missions as would make the nature of the work in which he was engaged intelligible to all classes of readers, without making the narrative too tedious and monotonous. I have wished to present all the diverse aspects and all the salient points of his missionary life, and to give as varied and miscellaneous a collection of specimens from its records as possible. From the necessity of the case, only a small number of missions could be particularly noticed. Those which have been passed by have not been slighted, however, as less worthy of notice than the others, but omitted from the necessity of selecting those most convenient for illustration of the theme in hand. The statistics given, in regard to numbers, etc., in the history of our missions, have all been taken from records carefully made at the time, and based on an exact enumeration of the communions given. I trust this volume will renew and keep alive in the minds of those who took part in these holy scenes, and who hung on the lips of the eloquent preacher of God's word whose life and doctrine are contained in it, the memory of the holy
- 67. lessons of teaching and example by which he sought to lead them to heaven. Of the sermons contained in this volume, seventeen have been reprinted from the four volumes of "Sermons by the Paulists, 1861- 64;" and twelve published from MSS. Four of these are mission sermons, selected from the complete series, as the most suitable specimens of this species of discourse. The others are parochial sermons, preached in the parish church of St. Paul the Apostle, New York. There still remain a considerable number of sermons, more or less complete; but the confused and illegible state in which F. Baker left his MSS. has made the task of reading and copying them very laborious, and prevented any larger number from being prepared for publication at the present time. I leave these Sermons, with the Memoir of their author, to find their own way to those minds and hearts which are prepared to receive them, and to do the good for which they are destined by the providence of God. May we all have the grace to imitate that high standard of Christian virtue which they set before us, as true disciples of Jesus Christ our Lord! A. F. H. St. Paul's Church, Fifty-ninth Street, Advent, 1865. CONTENTS Page Memoir 13 Sermon
- 68. I. The Necessity of Salvation (Mission Sermon) 209 II. Mortal Sin (Mission Sermon) 226 III. The Particular Judgement (Mission Sermon) 239 IV. Heaven (Mission Sermon) 252 V. The Duty of Growing in Christian Knowledge (First Sunday in Advent) 263 VI. The Mission of St. John the Baptist (Second Sunday in Advent) 271 VII. God's Desire to be Loved (Christmas Day) 282 VIII. The Failure and Success of the Gospel (Sexagesima) 292 IX. The Work of Life (Septuagesima) 303 X. The Church's Admonition to the Individual Soul (Ash-Wednesday) 312 XI. The Negligent Christian (Third Sunday in Lent) 320 XII. The Cross, the Measure of Sin (Passion Sunday) 329 XIII. Divine Calls and Warnings (Lent) 340 XIV. The Tomb of Christ, the School of Comfort (Easter Sunday) 352 XV. St. Mary Magdalene at the Sepulchre (Easter Sunday) 360
- 69. XVI. The Preacher, the Organ of the Holy Ghost (Fourth Sunday after Easter) 370 XVII. The Two Wills in Man (Fourth Sunday after Easter) 380 XVIII. The Intercession of the Blessed Virgin the Highest Power of Prayer (Sunday within the Octave of the Ascension) 391 XIX. Mysteries in Religion (Trinity Sunday) 399 XX> The Worth of the Soul (Third Sunday after Pentecost) 408 XXI. The Catholic's Certitude concerning the Way of Salvation (Fifth Sunday after Pentecost) 418 XXII. The Presence of God (Fifth Sunday after Pentecost) 429 XXIII. Keeping the Law not Impossible (Ninth Sunday after Pentecost) 437 XXIV. The Spirit of Sacrifice (Feast of St. Laurence) 447 XXV. Mary's Destiny a Type of Ours (Assumption) 456 XXVI. Care for the Dead (Fifteenth Sunday after Pentecost) 465 XXVII. Success the Reward of Merit (Fifteenth Sunday after Pentecost) 475 XXVIII. The Mass the Highest Worship (Twenty-first Sunday after Pentecost) 484
- 70. XXIX. The Lessons of Autumn (Last Sunday after Pentecost) 493 MEMOIR. Memoir. Francis A. Baker was born in Baltimore, March 30, 1820. The name given him in baptism was Francis Asbury, after the Methodist bishop of that name; but when he became a Catholic he changed it to Francis Aloysius, in honor of St. Francis de Sales and St. Aloysius, to both of whom he had a special devotion, and both of whom he resembled in many striking points of character. He was of mixed German and English descent, and combined the characteristics of both races in his temperament of mind and body. He had also some of the Irish and older American blood in his veins. His paternal grandfather, William Baker, emigrated from Germany at an early age to Baltimore, where he married a young lady of Irish origin, and became a wealthy merchant. His maternal grandfather, the Rev. John Dickens, was an Englishman, a Methodist preacher, who resided chiefly in Philadelphia. His grandmother was a native of Georgia. During the great yellow-fever epidemic in Philadelphia, Mr. Dickens remained at his post, and his wife fell a victim to the disease, with her eldest daughter. His father was Dr. Samuel Baker, of Baltimore, and his mother, Miss Sarah
- 71. Dickens. Dr. Baker was an eminent physician and medical lecturer, holding the honorable positions of Professor of Materia Medica in the University of Maryland, and President of the Baltimore Medico- Chirurgical Society. There was a striking similarity in the character of Dr. Baker and his son Francis. The writer of an obituary notice of the father, in the Baltimore Athenæum, tells us that his early preceptors admired "the balance of the faculties of his mind," and that "his classmates were attached to him for his integrity and affectionate manners." In another passage, the same writer would seem to be describing Francis Baker, to those who knew him alone, and have never seen the original of the sketch. "The style of conversation with which Dr. Baker interested his friends, his patients, or the stranger, was marked with an unaffected simplicity. Even when he was most fluent and communicative, no one could suspect him of an ambition to shine. He spoke to give utterance to pleasing and useful thoughts on science, religion, and general topics, as if his chief enjoyment was to diffuse the charms of his own tranquillity. In social intercourse, his dignity was the natural attitude of his virtue. On the part of the trifling it required but little discernment to perceive the tacit warning that vulgar familiarity would find nothing congenial in him. He never engrossed conversation, and seemed always desirous of obtaining information by eliciting it from others. Whether he listened or spoke, his countenance, receiving impressions readily from his mind, was an expressive index of the tone of his various emotions and thoughts. The conduct of Dr. Baker as a physician, a Christian, and a citizen, was a mirror, reflecting the beautiful image of goodness in so distinct a form as to leave none to hesitate about the sincerity and purity of his feelings. It therefore constantly reminded many of 'the wisdom that is from above, which is first pure, then peaceable, gentle, easy to be entreated, full of mercy and good fruits, without partiality, and without hypocrisy.' The friendly sympathy and anxiety which he evinced in the presence of human suffering attached all classes of his patients to him, and he was very happy in his benevolent tact at winning the affection of children, even in their sickness." Dr. Baker was a member of the Methodist Church, and
- 72. an intimate friend of the celebrated and eloquent preacher Summerfield. He was not one, however, of the enthusiastic sort, but sober, quiet, and reserved. He never went through any period of religious excitement himself, or endeavored to practise on the susceptibilities of his children. He said of himself, as one of his intimate friends testifies, "that he did not know the period when he became religious, so gradually was his life regulated by the spiritual truths which enlightened his mind from childhood." He had no hostile feelings toward the Catholic Church, and was a great admirer and warm friend of the Sisters of Charity, many of whom I have heard frequently speak of him in terms of the most affectionate respect. His benevolence toward the poor was unbounded, and he was in fact endeared to all classes of the community, without exception, in Baltimore. Francis Baker had a very great respect for his father, and was very fond of talking of him to me, during the first period of our acquaintance, when his early recollections were fresh and recent in his mind. Of his mother he had but a faint remembrance, having been deprived of her at the age of seven years. It is easy to judge of her character, however, from that of her children, and of her sister, who was a mother to her orphans from the time of her death until her own life was ended among them. Mrs. Baker's brother, the Hon. Asbury Dickens, is well known as having been for nearly half a century the Secretary of the Senate of the United States, which position he held until his death, which occurred at an advanced age a few years since. Dr. Baker had four sons and two daughters. Only one of them, Dr. William George Baker, ever married, and he died without children: so that Dr. Samuel Baker left not a single grandchild after him to perpetuate his name or family—and of his children, one daughter only survives. Three of his sons were physicians of great promise, which they did not live to fulfil. Francis was his third son, and the one who most resembled him in character. Of his boyhood I know little, except that his companions at school who grew up to manhood, and preserved their acquaintance with him, were
- 73. extremely attached to him. One of them passed an evening and night in our house, as the guest of F. Baker, but a few months before his death, with great pleasure to both. I have also heard some of the good Sisters of Charity speak of having known the little Frank Baker as a boy, and mention the fact that he was very fond of visiting them. I am sure that his childhood was an extremely happy one until the period of his father's death. This event took place in October, 1835, when Francis was in his sixteenth year, and in the fiftieth year of Dr. Baker's life. It was very sudden and unexpected, and threw a shadow of grief and sadness over the future of his children, which was deepened by the subsequent untimely decease of the two eldest sons, Samuel and William. Francis was entered at Princeton College soon after his father's death, and graduated there with the class of 1839. I am not aware that his college life had any remarkable incidents. He was not ambitious of distinguishing himself, or inclined to apply himself to very severe study. I believe, however, that his standing was respectable, and his conduct regular and exemplary. He was not decidedly religious in his early youth. Methodism had no attraction for him, and the Calvinistic preaching at Princeton was repugnant to his reason and feelings. Whatever religious impressions he had in childhood were chiefly those produced by the Catholic Church, whose services he was fond of attending; but these were not deep or lasting. The early death of his father, and the consequent responsibility and care thrown upon him as the male head of the family, first caused him to reflect deeply, and to seek for some decided religious rule of his own life and conduct, and finally led him to join the Protestant Episcopal communion, and to resolve to prepare himself for the ministry. All the members of his family joined the same communion, and were baptized with him, in St. Paul's Church, by the rector of the parish, Dr. Wyatt. This event took place in 1841, or '42. Soon afterward, Mr. Baker formed an acquaintance with a young man, a candidate for orders and an inmate of the family of Dr. Whittingham, the Bishop of Maryland,
- 74. which was destined to ripen into a most endearing and life-long friendship, and to have a most important influence on his subsequent history. This gentleman was Dwight Edwards Lyman, a son of the Rev. Dr. Lyman a respectable Presbyterian minister, of the same age with Francis Baker, and an ardent disciple of the school of John Henry Newman. At the time of his baptism, Mr. Baker was only acquainted with church principles as they were taught by Dr. Wyatt, who was an old-fashioned High Churchman. The intercourse which he had with Mr. Lyman was the principal occasion of introducing him to an acquaintance with the Oxford movement, into which he very soon entered with his whole mind and heart. In 1842, Mr. Lyman was sent to St. James's College, near Hagerstown, where he remained several years, receiving orders in the interval. During this time, Mr. Baker kept up a frequent and most confidential correspondence with him, which is full of liveliness and humor in its earlier stages, but becomes more grave and serious as both advanced nearer to the time of their ordination. It continued during the entire period of their ministry in the Episcopal Church, and during the whole subsequent life of Mr. Baker, closing with a very playful letter written by the latter, a few days before his last illness. In one of these letters, he acknowledges his obligations to Mr. Lyman as the principal instrument of making him acquainted with Catholic principles, in these warm and affectionate words: "I do not know whether you are aware of the advantage I derived from you in the earlier part of our acquaintance, by reason of your greater familiarity with the Catholic system as exhibited in the Anglican Church. The influence you exerted was of a kind of which I can hardly suppose you to have been conscious; yet I am sure you will be gratified to think it was effectual, as I believe, to fix me more firmly in the system for which I had long entertained so profound a reverence and affection. These are benefits which I cannot forget, and which (if there were not other reasons of which I need not speak) must always keep a place for you in the heart of your unworthy friend."
- 75. The nature of the later correspondence between these two friends, and their mutual influence on each other, will appear later in this narrative. There are friendships which are formed in heaven, and in looking back upon that which grew up between these two young men of congenial spirit, and in which I was also a sharer in a subordinate degree, I cannot but admire the benignant ways of Divine Providence, by which those strands which afterward bound our existence together so closely were first interwoven. I had myself met Mr. Lyman, some years before this, and felt the charm of his glowing and enthusiastic advocacy of principles which were just beginning to germinate in my own mind. Soon after Lyman's removal to Hagerstown, I made the acquaintance of Mr. Baker, a circumstance which the latter mentions in his next letter to his friend in these words, which I trust I may be pardoned for quoting —— "The Bishop's family have a young man staying with them (Mr. H.), a convert to the Church, and one, I believe, of great promise. He was a Congregationalist minister, and Rev. Mr. B. read me a letter from him, dated about a month ago, before his coming into the Church, the tone of which was far more Catholic than that of many (alas!) of those who have been partakers of the holy treasures to be found only in her bosom. Mr. B. tells me that Church principles are silently spreading in the North, among the sects. In this place, I believe that a spirit has been raised which one would hardly imagine on looking at the surface of things, though that is troubled enough." This letter was dated April 22, 1843. I had just arrived in Baltimore, at the invitation of Dr. Whittingham, the Protestant Episcopal Bishop of Maryland, and been received as a candidate for orders in his diocese. Mr. Baker, who was also a candidate for orders, lived just opposite the Bishops's residence, in Courtlandt street, and was pursuing his theological studies in private. I lived in the Bishop's house, and I think I met Mr. Baker
- 76. there on the first evening of my arrival. We were nearly of the same age, and soon found that our tastes and opinions were very congenial to each other. Of course, I returned his visit very soon, and I became at once very intimate with his family. It was a charming place and a delightful circle. Francis, as the eldest brother, was the head of the house. His aunt, Miss Dickens, fulfilled the office of a mother to her orphaned nephews and nieces with winning grace and gentleness. A younger brother, Alfred, then about eighteen years of age, was at home, pursuing his medical studies. Two sisters completed the number of the family, all bound together in the most devoted and tender love, all alike in that charm of character which is combined from it fervent and genial spirit of religion, amiability of temper, and a high-toned culture of mind and manners, chastened and subdued by trial and sorrow. I must not pass by entirely without mention another inmate of the family, whose good-humored, joyous countenance was always the first to greet me at the door—little Caroline, the last of the family servants, who was manumitted as soon as she arrived at a proper age, always devotedly attached to her young master, and afterward one of the most eager and delighted spectators at his ordination as a Catholic priest. The house was one of those places where every article of furniture and the entire spirit that pervades its arrangement speaks eloquently of the past family history, and recalls the memory of its departed members and departed scenes of domestic happiness. Dr. Baker had left his children a competent but moderate fortune, which was managed with the utmost prudence by Francis, who possessed at twenty-one all the wisdom of a man of fifty. There was nothing of the splendor and luxury of wealth to be seen in the household, but a modest simplicity and propriety, a home-like comfort, and that perfection of order and arrangement, regulated by a pure and exquisite taste, which is far more attractive. Mr. Baker's home was always the mirror of his mind. In later years, when he lived in his own rectory, although his family circle had lost two of its precious links, the same charm pervaded every nook and
- 77. corner of the home of the survivors, the young and idolized pastor and his two sisters. His study at St. Luke's rectory was the beau ideal of a clergyman's sanctuary of study and prayer, after the Church of England model; with something added, which betokened a more recluse and sacerdotal spirit, and a more Catholic type of devotion. One might have read in it Mr. Baker's character at a glance, and might have divined that the inhabitant of that room was a perfect gentleman, a man of the most pure intellectual tastes, a pastor completely absorbed in the duties of his state, a recluse in his life, and very Catholic in the tendencies and aspirations of his soul. Of Mr. Baker's family, only one sister has survived him. Alfred Baker died first. Like his brother, he was a model of manly beauty, although he did not in the least resemble him in form or feature. Francis Baker, as all who ever saw him know, was remarkably handsome. Those who only knew him after he reached mature age, and remember him only as a priest, will associate with his appearance chiefly that impress of sacerdotal dignity and mildness, of placid, intellectual composure, of purity, nobility, and benignity of character, which was engraven or rather sculptured in his face and attitude. Dressed in the proper costume, he might have been taken as a living study for a Father of the Church, a holy hermit of the desert, or a mediæval bishop. He was cast in an antique and classic mould. There was not a trace of the man of modern times or of the man of the world about him. His countenance and manner in late years also bore traces of the fatiguing, laborious life which he led, and the hard, rough work to which he was devoted. On account of these things, and because he was so completely a priest and a religious, one could scarcely think of admiring him as a man. His portrait was never painted, and the photographs of him which were taken were none of them very successful, and most of them mere caricatures. An ambrotype in profile was taken at Chicago for Mr. Healy the artist, which is admirable, and from this the only good photographs have been taken; but the adequate image of Father Baker, as he appeared at the altar, or when his face was lit up in
- 78. preaching the Divine word, will live only in the memory of those who knew him. At the period of which I speak, he had just attained the maturity of youthful and manly beauty, which was heightened in its effect by his perfect dignity and grace of manner. His brother Alfred was cast in a slighter mould, and had an almost feminine loveliness of aspect, figure, and character. He was as modest and pure as a young maiden, with far more vivacity of feature and manner than his brother, and a more vivid and playful temperament. There was nothing, however, effeminate in his character or countenance. He was full of talent, high-spirited, generous and chivalrous in his temper, conscientious and blameless in his religious and moral conduct. He graduated at the Catholic College of St. Mary's in Baltimore, and was a great favorite of the late Archbishop Eccleston and several others of the Catholic clergy. His High Church principles had a strong dash of Catholicity in them, and he used often to speak of the "ignominious name, Protestant," which is prefixed to the designation of the Episcopal Church in this country. He was a devoted admirer of Mr. Newman, and followed him, like so many others, to the verge of the Catholic Church, but drew back, startled and perplexed, when he passed over. Two or three years after the time I am describing, he began the practice of his profession, with brilliant prospects. The family removed to a larger and more central residence, for his sake, near St. Paul's Church, where Francis was Assistant Minister. All things seemed to smile and promise fair, but this beautiful bud had a worm in it. A slow and lingering but fatal attack of phthisis seized him, just as he was beginning to succeed in his professional career. His brother accompanied him to Bermuda, but the voyage was rather an additional suffering than a benefit, and on the 9th of April, 1852, he died. It was Good Friday. He had prayed frequently that he might die on that day, and before his departure, he called his brother to him, made a general confession, desired him to pronounce over him the form of absolution prescribed in the English Prayer-Book, and received the communion of the Episcopal Church. These acts were sacramentally valueless, but I trust, without presuming to decide positively on a secret matter which
- 79. God alone can judge, that his intention was right before God, and his error a mistake of judgment without perversity of will. His brother afterward felt deeply solicitous lest he might have been himself blamable for keeping him in the Episcopal communion, and grieved that he had died out of the visible communion of the Catholic Church. Still, as he was conscious of his own integrity of purpose, he tranquillized his mind with the hope that his brother had died in spiritual communion with the true Church and in the charity of God, and endeavored to aid him, as far as he was still within the reach of human assistance, by having many masses offered for the repose of his soul. Miss Dickens died a little before Alfred, and Elizabeth Baker died some time after her brother became a Catholic, but before his ordination. I return now to the period when Mr. Baker and all these members of his family were living a retired and happy life together in the home on Courtlandt street. I remember this time with peculiar pleasure. Mr. Baker, whom I always called Frank, as he was usually called by his friends, partly from the peculiar affection they felt for him, and also because of its appropriateness as an epithet of his character, went every day with me once or twice to prayers; and every day we walked together. When the peculiar, tinkling bell of old St. Paul's, which will be remembered by many a reader of these pages, gave notice of divine service there, we resorted in company to that venerable and unique church. It was spacious and ecclesiastical, though not regularly beautiful in its architecture. A basso-relievo adorned its architrave, and a bright gilded cross graced its tall tower. It had a handsome altar of white marble, an object of our special pride and devotion, with the usual reading- desk and pulpit rising behind it. The pulpit was a light and graceful structure, surmounted by a canopy which terminated in a cross, and having another cross surrounded by a glory emblazoned on its ceiling, just over the preacher's head. The door was in the rear of the pulpit, which stood far out from the chancel wall, and in the
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