# Unique Fixed Point Results and Its Applications in Complex-Valued Fuzzyb-Metric Spaces

Authors: Humaira; Muhammad Sarwar; Nabil Mlaiki
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The goal of this paper is to extend the concept of complex-valued fuzzy metric space to complex-valued fuzzyb-metric spaces and to discuss various existence results for fixed points to ensure their existence and uniqueness. To demonstrate the viability of the proposed strategies, a nontrivial example is used. Finally, applications to integral equations and initial value problems in mechanical engineering are discussed to demonstrate the superiority of the obtained results.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory combines topology, geometry, and analysis in an amazing way. Fixed point theory has emerged as a powerful tool in the study of nonlinear analysis in recent years. In fixed point theory and many other mathematical subjects, multiple separate objects are considered. As a result, mathematics is not only about numbers and shapes but also about prepositions, fluid flows, vector connections, and chemical interactions, among other things. Many researchers investigated the significance of various features of symmetry and demonstrated how they might be applied to many types of mathematical problems [1, 2]. There are several generalizations of the concept of metric spaces in the literature. Azam et al. developed the idea of complex-valued metric space and discovered that the Banach contraction principle may be applied to complex-valued metric spaces [3]. They studied its applications to complex integral equations. After that, fixed point theorems have been studied by many authors in complex-valued metric spaces [4–8].The concept ofb-metric spaces has been introduced by Bakhtin and Czerwik [9, 10]. Later on, many authors studied fixed point theorems for single and multivalued mappings in b-metric spaces for instance [11, 12]. In [13], the author generalized the concept of b-metric spaces by introducing the setting of complex-valued b-metric spaces. Many other researchers worked on complex-valued b-metric, and they extended generalized fixed point theorems in the sense of complex-valued b-metric spaces (see [14, 15] and the references therein).The concept of fuzzy sets was given by Zadeh [2] and opened the door of new direction in mathematical research. Pao-Ming and Ying-Ming established the notion of fuzzy metric spaces [16]. Afterwards, George and Veeramani improved the settings of fuzzy metric spaces [17]. Heilpern introduced the concept of fuzzy mapping and obtained fixed point results for fuzzy mappings [18]. Heilpern’s work was further extended by many authors, for instance, see [19–21]. Shukla et al. worked on the neighborhood structure of fuzzy fixed point [22]. Several other researchers worked on fuzzy metric spaces and obtained the generalizations of related results [23, 24].George and Veeramani generalized the concept of fuzzy metric to the context of complex-valued fuzzy metric and obtained the complex-valued fuzzy version of Banach contraction mapping result in different forms [17]. Also, they obtain some related fixed point results with valid examples.In this paper, we introduce the setting of complex-valued fuzzyb-metric spaces to generalize the setting of complex-valued b-metric space and establish the complex-valued fuzzy version of the Banach contraction principle. We also provide examples to back up our findings. The paper concludes with an application to integral and differential equation.All over the manuscript we have symbolized the set of complex numbers byC. We mark some shortcut representation used in this manuscript, as tc-norm for a complex-valued continuous triangular norm, CF b-metric for complex-valued fuzzy b-metric, and s.t. for such that.LetP=ξ,ρ:0≤ξ<∞,0≤ρ<∞⊂C. The elements 0,0,1,1∈P are denoted by ϑ and ℓ, respectively. The set Pϑ=ξ,ρ:0<ξ<∞,0<ρ<∞. Clearly for φ,ξ∈C,ξ⪯φ iff ξ−φ∈Pϑ. Let the unit closed complex interval be symbolized by I=ξ,ρ:0≤ξ≤1,0≤ρ≤1 and the open unit complex interval by I0=ξ,ρ:0≤ξ<1,0≤ρ<1.Definition 1 (see [17]).
Define an ordered relation⪯ on C by ς1⪯ς2 if and only if ς2−ς1∈P. The relations ς1⪯ς2 and ς1≺ς2 indicate that Reς1≤Reς2,Imς1≤Imς2 and Reς1<Reς2,Imς1<Imς2, respectively.
LetB⊂C. If there exists infB such that it i the lower bound of B, that is, infB⪯a∀a∈B and v⪯infB for every lower bound v of B, then infB is called the greatest lower bound of B.Definition 2 (see [25]).
LetX be a nonempty set. A complex fuzzy set M is characterized by a mapping such that domain is X and the range in the closed unit complex interval I.Definition 3 (see [17]).
A binary equation⋆:I×I⟶I is said to be complex-valued t-norm if the following conditions hold:
(1)
ξ1⋆ξ2=ξ2⋆ξ1(2)
ξ1⋆ξ2⪯ξ3⋆ξ4 whenever ξ1⪯ξ3,ξ2⪯ξ4(3)
ξ1⋆ξ2⋆ξ3=ξ1⋆ξ2⋆ξ3(4)
ξ⋆ϑ=ϑ,ξ⋆ℓ=ξfor all ξ,ξ1,ξ2,ξ3,ξ4∈I.Some fundamental examples of atc-norm are as follows:
(1)
ξ1⋆aξ2=e1e2,e3,e4, for all ξ1=e1,e3,ξ2=e2,e4∈I(2)
ξ1⋆bξ2=mine1,e2,mine3,e4, for all ξ1=e1,e3,ξ2=e2,e4∈I(3)
ξ1⋆cξ2=maxe1+e2−1,0,maxe3+e4−1,0, for all ξ1=e1,e3,ξ2=e2,e4∈IDefinition 4 (see [17]).
LetX,M,⋆ be a complex-valued fuzzy metric space. A sequence φq in X is known as a Cauchy sequence if
(1)limq⟶∞infd>qMφq,φd,t=ℓ∀t∈Pϑ.
The complex-valued fuzzy metric spaceX,M,⋆ is complete if every Cauchy sequence is convergent in X.Definition 5 (see [17]).
A sequence is monotonic with respect to⪯ if either ςb⪯ςb+1 or ςb+1⪯ςb∀b∈N.Lemma 6 (see [17]).
LetX,M,⋆ be a complex-valued fuzzy metric space. If t,t′∈Pϑ and t⪯t′, then Mφ,u,t⪯Mφ,u,t′∀φ,u∈X.Lemma 7 (see [17]).
LetX,M,⋆ be complex-valued fuzzy metric space. A sequence φq in X converges to v∈X iff limq⟶∞Mφq,v,t=ℓ holds ∀t∈Pϑ.Remark 8 (see [17]).
Letφq∈P∀n∈N then:
(a)
If the sequenceφq is monotonic with respect to ⪯ and there exist γ,η∈P with γ°φq⪯η,∀q∈N, then there exists φ∈P such that limq⟶∞φq=φ(b)
Although the partial ordering⪯ is not a linear order on C, the pair C,⪯ is a lattice(c)
IfX⊂C and there exists γ,η∈C with γ⪯s⪯η∀s∈X, then infX and supX both existRemark 9 (see [17]).
Letφq,φ′q,ξ∈P,∀q∈N, then
(a)
Ifφq⪯φ′q⪯ℓ∀q∈N and limq⟶∞φq=ℓ, then limb⟶∞φ′q=ℓ(b)
Ifφq⪯ξ∀q∈N and limb⟶∞φq=φ, then φ⪯ξ(c)
Ifξ⪯φq∀q∈N and limq⟶∞φq=φ, then ξ⪯φDefinition 10 (see [15]).
LetX be a nonempty set and let b≥1 be a given real number. A function D:X×X⟶C is called a complex-valued b-metric on X if, for all ξ,φ,v∈C, the following conditions are satisfied:
(i)
Dξ,φ⪰0(ii)
Dξ,φ=0 if and only if ξ=φ(iii)
Dξ,φ=Dφ,ξ(iv)
bDξ,v+Dv,φ⪰Dξ,φ
The pairX,D is called a complex-valued b-metric space.Example 1 (see [15]).
LetX=C. Define the mapping D:C×C⟶C by Dξ,φ=ξ−φ2+iξ−φ2 for all ξ,φ,v∈C. Then, C,X is complex-valued b-metric space with b=2.Definition 11 (see [17]).
LetX be a nonempty set, ⋆ a continuous complex-valued tC-norm, and M a complex fuzzy set on X×X×Pθ⟶I satisfying conditions:
(1)
0⪯Mξ,φ,t(2)
Mξ,φ,t=ℓ for every t∈Pϑ if and only if ξ=φ(3)
Mξ,φ,t=Mφ,ξ,t(4)
Mξ,φ,t⋆Mφ,ρ,t′⪯Mξ,ρ,t+t′(5)
Mξ,φ,⋆:Pϑ⟶I is continuous for all ξ,φ,ρ∈X and t,t′∈Pϑ
Then, the tripletX,M,⋆ is said to be a complex-valued fuzzy metric space, and M is called a complex-valued fuzzy metric on X. The functions Mξ,φ,t denote the degree of nearness and the degree of nonnearness between ξ and φ with respect to the complex parameter t, respectively.Example 2 (see [17]).
LetX=ℵ. Define ⋆ by ς′⋆ς′′=s′s′′,u′u′′ for all ς′=s′,u′,ς′′=s′′,u′′∈I. Define complex fuzzy set M as
(2)Mξ,φ,t=ξφℓifξ≤φ,φξℓifφ≤ξ,for each ξ,φ∈X,ς∈Pθ. Then, X,M,⋆ is complex-valued fuzzy metric spaces.


## 2. Fixed Point Results in Complex-Valued Fuzzyb-Metric Spaces

We start this section with the following definition.Definition 12.
X,M,⋆,b is said a complex-valued fuzzy b-metric space if X is an arbitrary set, ⋆ is a tC-norm, and M is a fuzzy set on X×X⟶P meeting the points below for all ξ,φ∈X,t,s>ϑ and provided a number b±1:
(1)
0⪯Mξ,φ,t(2)
Mξ,φ,t=ℓ for every t∈Pϑ if and only if ξ=φ(3)
Mξ,φ,t=Mφ,ξ,t(4)
Mξ,φ,t/b⋆Mφ,ρ,t′/b⪯Mξ,ρ,t+t′(5)
Mξ,φ,⋆:Pϑ⟶I is continuous for all ξ,φ,ρ∈X and t,t′∈Pϑ
Then, the tripletX,M,⋆ is said to be a complex-valued fuzzy metric space, and M is called a complex-valued fuzzy metric on X.Example 3.
LetMξ,φ,t be a complex-valued fuzzy metric defined by e−φ−ξr/tℓ such that t>1 be a real number. Then, M is CF b-matric space with b=2r−1.Proof.
(1), (2), (3), and (5) are obvious. Here, we prove (4). For an arbitrary integerb, we have
(3)ξ−ρ⪯bt+t′tξ−φ+bt+t′t′φ−ρξ−ρt+t′⪯btξ−φ+bt′φ−ρ⪯ξ−φt/b+φ−ρt′/b.
Sinceeξ is an increasing function for ξ, one can write
(4)eξ−ρ/t+t′⪯eξ−φ/t/b+eφ−ρ/t′/b.
Thus, we have(5)e−ξ−ρ/t+t′ℓ⪰e−ξ−φ/t/b+e−φ−ρ/t′/bℓ,Mξ,ρ,t+t′⪰Mξ,φ,tb⋆Mφ,ρ,t′b.Remark 13.
CFb-metric is the generalization of complex-valued fuzzy metric space. It is obvious from example that is every CF b-metric is complex-valued fuzzy metric for b=1. Similarly, some important results like Lemmas 6 and 7 and definitions of convergence and Cauchy presented in Section 1 can also be defined in the same manner in CF b-metric space as mentioned in complex-valued fuzzy metric space.Theorem 14.
LetX,M,⋆,b be a complete CF b-metric space and let ς:X⟶X be mapping enjoying the following condition:
(6)ℓMςξ,ςρ,t−ℓ⪯qℓMξ,ρ,t−ℓ,for all ξ,ρ∈X and q∈0,1. Then, ς has a unique fixed point τX, for all τ∈Pϑ.Proof.
Letφ0∈X. Define a sequence φr in X by
(7)φr=ςφr−1forallr∈N.
Ifφ0=φr−1 for some r∈N. Then clearly, ς has a fixed point. Suppose φ0=φr−1 for all r∈N. To show that φr is a Cauchy sequence, let define
(8)Br=Mφi,φj,t:j>i⊂I.
Sinceϑ≺Mφi,φj,t, by Remark 8, the infBr=βr exists. For j,i∈N, using (6), we get
(9)ℓMφi+1,φj+1,t−ℓ=ℓMςφi,ςφj,t−ℓ⪯qℓMφi,φj,t−ℓ⪯ℓMφi,φj,t−ℓ,which implies
(10)ℓMφi+1,φj+1,t⪯ℓMφi,φj,t.
Therefore, by definition, we get(11)ℓ⪯βr⪯βr+1⪯ϑ,forallr∈N.
Thus,φr is monotonic in P. Using Remark 8 and from (11), there exists ℓ⋆∈P, with
(12)limr∞βr=ℓ⋆.
From inequality (9), we have
(13)ℓMφi+1,φj+1,t⪯qℓMφi,φj,t+1−qℓ,for all i,j and so ℓ/βi+1⪯qℓ/βi+1−qℓ for every i∈N, which yields from (12)
(14)1−qℓ⪯1−qℓ⋆ℓ⋆.
Sinceq∈0,1 and applying Remark 9, we must obtained ℓ=ℓ⋆. Thus,
(15)limr⟶∞βr=ℓ.
Hence,(16)limr⟶∞infj>iMφi,φj,t=ℓ,forallt∈Pϑ.
Therefore, from (16), we have that φr is a Cauchy sequence. From the completeness of X and Lemma 7, we get that there exists τ∈X such that
(17)limr⟶∞Mφr,τ,t=ℓ,forallt∈Pϑ.
Now fort∈Pϑ and r∈R, it yields from (6) that
(18)ℓMςφr,ςτ,t−ℓ⪯qℓMφr,τ,t−ℓ,that is
(19)Mςφr,ςτ,t⪰1q/Mφr,τ,t+1−q.
Now, for anyt∈Pϑ,
(20)Mτ,ςτ,t⪰Mτ,φr+1,t2b⋆Mφr+1,ςφτ,t2b=Mτ,φr+1,t2b⋆Mςφr,ςφτ,t2b.
Takingr⟶∞ and using (17), (19), and Remark 9, we get that Mτ,ςτ,t=ℓ for all t∈Pϑ; that is, ςτ=τ.
Now, we have to show the uniqueness of fixed pointτ of ς. On contrary, suppose v be another fixed point of ς. Then, there exists t∈Pϑ such that Mτ,v,t<ℓ, than from (6) we have
(21)ℓMτ,v,t−ℓ=ℓMςτ,ςv,t−ℓ⪯qℓMτ,v,tℓ,which is a contradiction. Therefore, we must obtain Mτ,v,t=ℓ for all t∈Pϑ. Hence, τ=v.Corollary 15.
LetX,M,⋆,b be a complete CF b-metric space and letς:X⟶X be mapping enjoying the following condition:
(22)ℓMςrξ,ςrρ,t−ℓ⪯qℓMξ,ρ,t−ℓ,for all ξ,ρ∈X and q∈0,1. Then, ς has a unique fixed point τX, for all t∈Pϑ.Proof.
By the use of Theorem14, ςr has a fixed point τ as ςr observes all conditions. But ςrςτ=ςςrτςτ, implies that ςτ is another fixed point of ςr. By uniqueness of fixed point, we have ςτ=τ.As fixed point of ς is also a fixed point ofς. Thus, ς has a unique fixed point.Corollary 16.
LetX,M,⋆,b be a complete CF b-metric space and let ς:X⟶X be mapping enjoying the following condition:
(23)ℓMςrξ,ςrρ,t−ℓ⪯qtℓMξ,ρ,t−ℓ,for all ξ,ρ∈X and q:Pϑ⟶0,1. Then, ς has a unique fixed point τX, for all t∈Pϑ.Example 4.
LetX=0,∞ and t-norm be defined by c1⋆c2=c1c2 for all c1=a1,a2,c2=a1,a2∈I. Define M as
(24)Mξ,ρ,t=expξ−ρ2/t−1ℓforallξ,ρ∈X,t∈Pϑ.
Then,X,M,⋆ is a CF b-metric space. Define ς:X⟶X as
(25)ςξ=0,ifξ=m,ξ4,ifξ∈0,m,ξ8,ifξ∈m,∞.Then, we have the following cases.Case 1.
Ifξ,ρ=m, then ςξ,ςρ=0.Case 2.
Ifξ=m and ρ∈0,m, then ςξ=0 and ςρ=ρ/4.Case 3.
Ifξ=m and ρ∈m,∞, then ςξ=0 and ςρ=ρ/8.Case 4.
Ifξ∈0,m and ρ∈m,∞, then ςξ=ξ/4 and ςρ=ρ/8.Case 5.
Ifξ∈0,m and ρ∈0,m, then ςξ=ξ/4 and ςρ=ρ/4.Case 6.
Ifξ∈0,m and ρ=m, then ςξ=ξ/4 and ςρ=0.Case 7.
Ifξ∈m,∞ and ρ=m, then ςξ=ξ/8 and ςρ=0.Case 8.
Ifξ∈m,∞ and ρ∈m,∞, then ςξ=ξ/8 and ςρ=ρ/8.The above-mentioned cases observe all conditions of Theorem14 with q∈1/2,1. Thus, the fuzzy contractive mapping ς has a unique fixed point, which is 0,0.Theorem 17.
LetX,M,⋆,b be a complete CF b-metric space with t⪯t⋆t for t∈Iϑ. Let ς:X⟶X be mapping enjoying the following conditions:
(i)
There existsφ0∈X and ε∈Iϑ such that ℓ−ε⪯Mφ0,ςφ0,t for all t∈Pϑ(ii)
There existsq∈0,1 such that for all ξ,ρ∈Bφ0,ε,t,(26)ℓMςξ,ςρ,t−ℓ⪯qℓMξ,ρ,t−ℓ.
Then,ς has a unique fixed point in Bφ0,ε,t.Proof.
It is enough to proof thatBφ0,ε,t is complete and ςφ∈Bφ0,ε,t for all φ∈Bφ0,ε,t. Let φr be a Cauchy sequence in Bφ0,ε,t. Since X is complete thus by the use of Lemma 7, there exists u∈X such that
(27)limr⟶∞Mφr,u,t=ℓ,for all t∈Pϑ. Now for all i,r∈N,(28)Mφ0,u,t+ti⪰Mφ0,φr,tb⋆Mφ0,φr,tib.
Sinceφr∈Bφ0,ε,t for every r∈N, also limr⟶∞Mφr,u,t=ℓ. By using the properties of t-norm and Remark 9, we obtain
(29)Mφ0,u,t+ti⪰ℓ−r⋆ℓ=ℓ−r,foreveri∈N.
Takinglimi⟶∞ and using Remark 9, we get Mφ0,u,t±ℓ−r. Therefore, u∈Bφ0,ε,t.
For everyφ∈Bφ0,ε,t, it yields from (26)
(30)ℓMςφ0,ςφ,t−ℓ⪯qℓMφ0,φ,t−ℓ,that is
(31)Mςφ0,ςφ,t⪰1q/Mφ0,φ,t+1−q.
Thus, for alli∈N, we get
(32)Mφ0,ςφ,t+ti±Mφ0,ςφ0,tib⪰Mςφ0,ςφ,tb⪰ℓ−ε⋆1q/Mφ0,φ,t/b+1−q⪰ℓ−ε⋆1q/ℓ−ε+1−q⪰ℓ−ε⋆ℓ−ε.
Takinglimi⟶∞ and using Remark 9, we have
(33)Mφ0,ςφ,t⪰ℓ−ε.
Therefore,ςφ∈Bφ0,ε,t.Theorem 18.
LetX,M,⋆,b be a complete CF b-metric space such that for any sequence tr∈Pϑ with limr⟶∞tr=∞, we get limr⟶∞infρ∈XMξ,ρ,tr=ℓ, for all ξ∈X. Let ς:X⟶X be a mapping observing that
(34)Mςξ,ςρ,δt⪰Mξ,ρ,t,for all t∈Pϑ, where 0<δ<1. Then, ς has a unique fixed point in X.Proof.
Letφ0∈X. Define a sequence φr in X by
(35)φr=ςφr−1forallr∈N.
Ifφ0=φr−1ξ for some r∈N. Then clearly, ς has a fixed point. Suppose φ0≠ for all r∈N. To show that φr is a Cauchy sequence, let define
(36)Br=Mφr,φs,t:s>r⊂I.
Sinceϑ≺Mφr,φs,t, by Remark 8, the infBr=βr exists. For s,r∈N, by the use of (??) and Lemma 6, we get
(37)Mφr+1,φs+1,t⪰Mφr+1,φs+1,δt⪰Mςφr,ςφs,δt⪰Mςφr,ςφs,t,which yields
(38)Mςφr,ςφs,t⪯Mφr+1,φs+1,tfors>r.
Therefore, by definition, we obtain(39)ϑ⪯βr⪯βr+1⪯ℓ,forallr∈N.
Hence,βr is monotonic in P, and by the use of Remark 8 and (39), there exists ℓ⋆ such that
(40)limr⟶∞βr=ℓ⋆.
Fort∈Pϑ, once again from (34), we have
(41)βr+1=infs>rMφr+1,φs+1,t⪰infs>rMφr,φs,tδ=infs>rMςφr,ςφs,tδ⪰infs>rMφr−1,sr−1,tδ2=infs>rMςφr−2,ςφs−2,tδ2⪰infs>rMςφr−2,ςφr−2,tδ3⪰⋯⪰infs>rMφ0,φs−r,tδr+1,for all r∈N and t∈Pϑ, we have
(42)βr+1=infs>rMφr+1,φs+1,t⪰infs>rMφ0,φs−r,tδr+1⪰infs>rMφ0,ρ,tδr+1.
Aslimr⟶∞t/δr+1=∞, using (40) and assumption, we get
(43)ℓ⋆±limr⟶∞infρ∈XMφr,φs,t⪰=ℓ.
From (40) and (43)
(44)limr⟶∞βr=ℓ.
Thus,φr is a Cauchy sequence in X. Since X is complete, by Lemma 7, there exists u∈X such that
(45)limr⟶∞Mφr,u,t=ℓ.
Fort∈Pϑ, (34) yields that
(46)Mu,ςu,t⪰Mu,φr+1,t2b⋆Mφr+1,ςu,t2b⪰Mu,φr+1,t2b⋆Mςφr,ςu,t2b⪰Mu,φr+1,t2b⋆Mφr,u,t2bδ.
Takinglimr⟶∞ and by (45) and Remark 9, we have Mu,ςu,t=ℓ; that is, ςu=u.
Now to investigate the uniqueness of fixed point, let on contrary thatv∈X be any other fixed point of ς. So there exist t∈Pϑ with Mu,v,t=ℓ; then, (34) yields
(47)Mu,v,t=Mςu,ςv,t⪰Mu,v,tδ.
Continuing this way, we obtain(48)Mu,v,t⪰Mu,v,tδr⪰infρ∈XMu,v,tδr.
Usinglimr⟶∞t/δr=∞, it follows that Mu,v,t⪰ℓ, which is contradiction. Thus, Mu,v,t=ℓ; that is, u=v.Example 5.
LetX=0,1 and t-norm be defined by c1⋆c2=c1c2 for all c1=a1,a2,c2=a1,a2∈I. Define M as
(49)Mξ,ρ,t=exp−ξ−ρ/tℓforallξ,ρ∈X,t∈Pϑ.
Then,X,M,⋆ is a CF b-metric space. Define ς:X⟶X as
(50)ςξ=0,ifξ∈0,12,ξ14,ifξ∈12,1.
Forlimt⟶∞Mξ,ρ,t=limt⟶∞exp−ξ−ρ/tℓ=ℓ, we obtain that for all values of X we have Mςξ,ςρ,δt±Mςξ,ςρ,t, and for only 0, we have limt⟶∞infρ∈XMξ,ρ,tr=exp0ℓ=ℓ. Thus, all conditions of Theorem 18 are satisfied so, 0,0 is a unique fixed point of ς.Example 6.
LetX=C1,3,R,A>0 and for every ξ,ρ∈X let
(51)Mξ,ρ,t=exp−ξ−ρ/tℓ.
Let defineς:X⟶X by
(52)ςξτ=4+∫1τξv+ρvev−1dv,t∈1,3.
For everyξ,ρ∈X(53)Mςξ,ςρ,t=exp−ςξτ−ςρτ/tℓ=exp−∫1τmaxτ∈1,3ςξτ−ςρτ/tℓ⪰exp−∫1τmaxτ∈1,3ςξv−ςρve2/tℓ⪰2e2Mξ,ρ,t.
Similarly(54)Mςrξ,ςrρ,t⪰2rr!e2rMξ,ρ,t.
Note that(55)e2r2rr!=537.9ifr=3,5,873.7ifr=5,1.31ifr=37,0.202ifr=39.Thus, all conditions of Corollary15 are satisfied for q=0.202 and r=39, so ς has a fixed point which is a solution of the integral equation
(56)ξτ=4+∫1τξv+ρvev−1dv,t∈1,3,or the differential equation
(57)ξ′τ=ξ+τ2eτ−1τ∈1,3,ξ1=4.


## 3. Application

Integral equations have plenty applications in many scientific fields. It is a ripely rising field in abstract theory. One of its significant approach in the study of integral equations is to apply fixed point results to the function defined by the right-hand side of the equation or to develop homotopy methods, which are highly considered in fixed point theory to find the approximate solution. In this section, firstly, we study application of our main Theorem14 the existence of unique solution to Fredholm integral equation.Theorem 19.
LetΞ=C0,m,R be the spaces of continuous real valued functions defined on interval 0,m, where m>0. The Fredholm integral equation is
(58)zt=∫0mKt,δ,zδdδ.
LetΞ=C0,m,R and M:Ξ×Ξ×I⟶I be a CF b-metric defined as follows:
(59)My,z,c=cc+y−z2ℓ,y,z∈X,c>0.
If there existsq∈0,1 with
(60)Θy,zt⪰1qΛy,zt,where
(61)Θy,zt=cc+∫0mKt,δ,yδdδ−∫0mKt,δ,zδdδ2ℓ,Λy,zt=cc+yt−zt2ℓ,holds. Then, (58) has a unique solution in X.Proof.
LetΓ:Ξ⟶Ξ define as
(62)Γzt=∫0mKt,δ,zδdδ.
Then(63)Γy−Γz2=∫0mKt,δ,yδdδ−∫0mKt,δ,zδdδ2.
For ally,z∈X, we have
(64)ℓΘy,zt⪯qℓΛy,zt,so,
(65)ℓΘy,zt−ℓ⪯qℓΛy,zt−ℓ⪯qℓΛy,zt−ℓ,which implies that
(66)ℓc/c+∫0mKt,δ,yδδ−∫0mKt,δ,zδdδ2−ℓ⪯qℓc/c+yt−zt2−ℓ.
Therefore,(67)ℓMΓy,Γz,c−ℓ⪯qℓMy,z,c−ℓ.
Since all conditions of Theorem14 are satisfied, thus (58) has a unique solution in X.Next, we study the application of Theorem18, in mechanical engineering, since the system of auto mobile suspension is an achievable application for the system of spring mass in the field of engineering. We are going to study the motion of an auto mobile spring when its motion is upon a craggy and cleft road, where the forcing term is the craggy road and bumps noticed provide the absorbing. Tension, gravity, and earth quick are the possible external forces acting on the system. We express spring mass by κ and the external force acting on it by Θ. Then the following initial value problem represents the damped motion of the spring mass system under the action of external force Θ.(68)κd2y¯dt2+πdy¯dt=Θt,y¯t=0,y¯0=0,y¯′0=0,where π>0 express the damping constant and Θ:0,ϕ×R¯+⟶R¯ is a continuous mapping. Clearly, the problem (68) is equivalent to the following integral equation
(69)y¯t=∫0ϕΛt,δΘδ,y¯δdδ,witht,δ∈0,ϕ,whereΛt,δ represents the corresponding Green’s function and defined as
(70)Λt,δ=1−eρt−δρ,for0≤δ≤t≤ϕ,0for0≤t≤δ,where ρ=π/κ is a constant ratio. Consider the set of real valued functions Y¯=C0,ϕ,R. For b>1, consider CF b-mertic space defined by
(71)My,z,c=e−supn∈0,1y¯t−z¯t2/c,for all y,z∈Y¯. WE have to show that problem (68) has a solution iff there exists y¯∗ in Y¯, a solution of the integral equation (69).Theorem 20.
Consider problem (68), suppose the following conditions are satisfied:
(i)
Θδ,y¯δ−Θδ,z¯δ2≤y¯δ,z¯δ2(ii)
∫0ϕΛt,δ≤1
Then, the integral equation (69) has a unique solution in Y¯.Proof.
Let define an operatorΓ:Y¯⟶Y¯(72)Γy¯t=∫0ϕΛt,δΘδ,y¯δdδ,witht,δ∈0,ϕ.
Now,(73)e−supn∈0,1Γy¯t−Γz¯t2/λc≥e−supn∈0,1∫0ϕΛt,δΘδ,y¯δ−Θδ,y¯δ2dδ/λc≥e−supn∈0,1Θδ,y¯δ−Θδ,y¯δ2dδ/λc≥e−supn∈0,1y¯δ,z¯δ2/λc,this yields that
(74)e−supn∈0,1Γy¯t−Γz¯t2/λcℓ⪰e−supn∈0,1y¯δ,z¯δ2/λcℓ.
Consequently, we get(75)MΓy¯,Γz¯,λc⪰My¯,z¯,c.
Thus, by Theorem18, we obtained the existence of unique solution to integral equation (69).


## 4. Conclusion

In this article, we presented the generalization of CFb-metric space and successfully obtained the generalization of Banach contraction principle to the new established setting herein. In support of our obtained results, we have constructed some examples, and with the help of derived result, we guaranteed the existence of unique solution to integral equation, which makes it possible for more integral equations to be verified in such conditions.



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*Source: 2132957-2022-09-01.xml*