# An Extendedb-Metric-Type Space and Related Fixed Point Theorems with an Application to Nonlinear Integral Equations

Authors: Kushal Roy; Sayantan Panja; Mantu Saha; Vahid Parvaneh
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, the concept of sequentialp-metric spaces has been introduced as a generalization of usual metric spaces, b-metric spaces and specially of p-metric spaces. Several topological properties of such spaces have been discussed here. In view of this notion, we prove fixed point theorems for some classes of contractive mappings over such spaces. Supporting examples have been given in order to examine the validity of the underlying space and in respect to our proven fixed point theorems.


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## Body

## 1. Introduction and Preliminaries

In the last few decades, several generalizations of usual metric structure have been made by the researchers working in the area of fixed point theory. Different topological structured spaces play vital roles for obtaining fixed point theorems using several contractive or expansive or non-expansive type mappings. There are many interesting two-variable metric-type spaces such asb-metric space [1, 2], rectangular metric space [3], extended b-metric space [4], p-metric space [5], JS-metric space [6], modular metric space [7], multiplicative metric space [8], bipolar metric space [9, 10], cone metric space [11], and C∗-algebra-valued metric space [12]. Also, for considering and analyzing more generalizations of the concept of metric spaces, one can consider the following works dealing with (double) controlled metric spaces and generalized b-metric spaces [13–16]. Due to the presence of such interesting spaces and various types of applications of fixed point theorems therein, fixed point theory gains attention in the mathematical community specially to the new researchers working on functional analysis. In the context of various metric-type spaces which are the combination of the above-mentioned spaces, several authors have proved different types of fixed point theorems therein (see [17, 18]). Now, we give definitions of some generalized spaces which are relevant to our research work.Definition 1.b-metric space (see [1, 2]).
LetΛ be a nonempty set and s be a real number satisfying s≥1. A function ρb:Λ×Λ⟶ℝ+ is a b-metric on Λ if
(1)
ρbι,κ=0 if and only if ι=κ,(2)
ρbι,κ=ρbκ,ι for all ι,κ∈Λ,(3)
ρbι,z≤sρbι,κ+ρbκ,z for all ι,κ,z∈Λ.
The spaceΛ,ρb is called a b-metric space.LetΛ be a nonempty set and ρg:Λ×Λ⟶0,∞ be a mapping. For any ι∈Λ, let us define the set
(1)Cρg,Λ,ι=ιn⊂Λ:limn→∞ρgιn,ι=0.Definition 2. JS-metric space (see [6]).
Letρg:Λ×Λ⟶0,∞ be a mapping which satisfies
(1)
ρgι,κ=0 implies ι=κ,(2)
for everyι,κ∈Λ, we have ρgι,κ=ρgκ,ι,(3)
ifι,κ∈Λ×Λ and ιn∈Cρg,Λ,ι, then ρgι,κ≤plimsupn→∞ρgιn,κ, for some p>0.
The pairΛ,ρg is called a generalized metric space, usually known as a JS-metric space.Definition 3.p-metric space (see [5]).
LetΛ be a nonempty set. A function ρp:Λ×Λ⟶0,∞ is said to be p-metric if there exists a strictly increasing continuous function Ω:0,∞⟶0,∞ with Ω−1t≤t≤Ωt for all t≥0 and Ω−10=0=Ω0 such that for all ι,κ,z∈Λ(1)
ρpι,κ=0 if and only if ι=κ,(2)
ρpι,κ=ρpκ,ι,(3)
ρpι,z≤Ωρpι,κ+ρpκ,z.
The pairΛ,ρp is called a p-metric space.For various examples of the above spaces, one can see the research papers in the reference section of this manuscript regarding such notions and also the references cited in these papers.Now, we are ready to prove our main results. We define a new metric-type structure, which is the main concept of this paper.


## 2. Introduction to Sequentialp-Metric Space

In this section, we introduce a new type of extendedb-metric spaces. To develop such a notion, first, we define Sσ,Λ,a≔an⊂Λ:limn→∞σan,a=0, where σ:Λ×Λ⟶0,∞ is a given mapping.Definition 4.
LetΛ be a nonempty set. A mapping σ:Λ×Λ⟶0,∞ is said to be a sequential p-metric if for all a,b∈Λ(a)
σa,b=0 implies a=b,(b)
σa,b=σb,a,(c)
σa,b≤Ωlimsupn→∞σan,b, where an∈Sσ,Λ,a and Ω:0,∞⟶0,∞ is a strictly increasing continuous function with Ω−1t≤t≤Ωt for all 0≤t<∞ with Ω−1s=s=Ωs for s∈0,∞.
The tripletΛ,σ,Ω is called a sequential p-metric space. We express a sequential p-metric space simply as Λ,σ.Example 1.
LetΛ=N and the metric σ:Λ2⟶0,∞ be defined by
(2)σ1,1=0,σn,n=e−1,forn≥2,σ1,n=σn,1=e1/n+1−1,forn≥2,σn,m=σm,n=emn−1,foralln,m≥2withn≠m.Then, clearly,σm,n=0 implies m=n and σm,n=σn,m for all m,n∈Λ. Now, we show that σ satisfies condition (iii) of Definition 4.Forn≥2, Sσ,Λ,n=∅. Let nk∈Sσ,Λ,1. If all but finitely many terms of nk are 1 then we are done. So, suppose that nk only have finitely many 1’s. Without loss of generality, we can exclude such 1’s and then we get limsupk→∞σm,nk=limk→∞emnk−1=∞. Therefore, σ1,m≤limsupk→∞σm,nk for all m≥2.Hence,σ is a sequential p-metric on Λ for Ω1t=t for all t≥0 and Ω2t=et−1 for all t≥0.Proposition 5.
IfΛ,σ is a JS-metric space (see Definition 2) then σ is also a sequential p-metric on Λ.Proof.
IfΛ,σ is a JS-metric space then σ clearly satisfies the first two conditions of Definition 4. We just show that σ also satisfies the third condition of Definition 4.Sinceσ is a JS-metric then for all a,b∈Λ and for any sequence an∈Sσ,Λ,a, we have σa,b≤κlimsupn→∞σan,b, where κ>0.Then, if we chooseΩt=st for all t∈0,∞ with s≥max1,κ, then we have σa,b≤Ωlimsupn→∞σan,b for all a,b∈Λ and an∈Sσ,Λ,a. Therefore, Λ,σ is also a sequential p-metric space.Remark 6.
. (i) We know that any metric space,b−metric space [1, 2], dislocated metric space, and modular metric space with the Fatou property [7] are JS-metric spaces; therefore, these spaces are also sequential p-metric spaces. (ii) Any extended b-metric space or p-metric space is clearly a sequential p-metric space.Proposition 7.
LetΛ,σbeaJS-metric space with coefficient κ≥1. Let σ′a,b≔Γσa,b, where Γ is a strictly increasing continuous function with t≤Γt for all t≥0 and Γ0=0. Then, σ′ is a sequential p-metric for Ωt=Γκt=Γκt for all t≥0.Proof.
Here, we show thatσ′ satisfies all the conditions of Definition 4.
(a)
σ′a,b=0 gives Γσa,b=0. Then σa,b=Γ−10=0 implies a=b(b)
σ′a,b=σ′b,a holds trivially(c)
For alla,b∈Λ we have, σ′a,b=Γσa,b≤Γκlimsupn→∞σan,b where an∈Sσ,Λ,a=Sσ′,Λ,a
Now,σan,b≤Γσan,b=σ′an,b for all n∈N. Then, limsupn→∞σan,b≤limsupn→∞σ′an,b.
Therefore,σ′a,b≤Γκlimsupn→∞σ′an,b=Γκlimsupn→∞σ′an,b. This proves our proposition.Definition 8.
LetΛ,σ be a sequential p-metric space. Also let an be a sequence in Λ and a∈Λ.(i)
an is said to be convergent and converges to a if an∈Sσ,Λ,a,(ii)
an is said to be Cauchy if limn,m→∞σan,am=0,(iii)
Λ is called complete if any Cauchy sequence in Λ is convergent.Definition 9.
LetΛ,σ and Δ,σ∗ be two sequential p-metric spaces. A mapping T:Λ⟶Δ is called continuous at a point a∈Λ if for any ε>0 there exists δε>0 such that for any ι∈Λ,σ∗Tι,Ta<ε whenever σι,a<δε.T is said to be continuous on Λ if T is continuous at each point of Λ.Proposition 10.
In a sequentialp-metric space Λ,σ if a sequence an is convergent then it converges to a unique element in Λ.Proof.
Supposea,b∈Λ be such that an⟶a and an⟶b as n⟶∞. Then, we have, σa,b≤Ωlimsupn→∞σan,b implying that σa,b≤Ω0=0, i.e., a=b.Proposition 11.
LetΛ,σ be a sequential p-metric space and an⊂Λ converges to some a∈Λ then σa,a=0.Proof.
Sincean converges to a∈Λ, so limn→∞σan,a=0. Therefore, we have σa,a≤Ωlimsupn→∞σan,a=Ω0=0 which implies σa,a=0.Proposition 12.
Letan be a Cauchy sequence in a sequential p-metric space Λ,σ,Ω such that Ω−1 is continuous. Ifan has a convergent subsequence ank which converges to a∈Λ, then an also converges to a∈Λ.Proof.
From condition (c) of Definition4, we have σan,a≤Ωlimsupk→∞σan,ank which implies that Ω−1σan,a≤limsupk→∞σan,ank for all n∈N.
Due to the Cauchyness ofan, it follows that limn,k→∞σan,ank=0and thus Ω−1σan,a⟶0 as n⟶∞ which implies that σan,a⟶0 as n⟶∞, since Ω−1 is continuous. Hence, an converges to a∈Λ.Proposition 13.
In a sequentialp-metric space Λ,σ, if a self-mapping T is continuous at a∈Λ then Tan∈Sσ,Λ,Ta for any sequence an∈Sσ,Λ,a.Proof.
Letε>0 be given. Since T is continuous at a, then for any ε>0 there exists δε>0 such that σc,a<δε implies σTc,Ta<ε.
Asan converges to a, so for δε>0, there exists N∈ℕ such that σan,a<δε for all n≥N. Therefore, for any n≥N, σTan,Ta<ε and thus Tan⟶Ta as n⟶∞.Some observations regarding sequentialp-metric spaces are as follows:
(1)
In a metric space, a convergent sequence is always Cauchy, but it is not true in a sequentialp-metric space. In Example 1, the sequence nn≥2 converges to 1, but σn,m=enm−1⟶0 whenever n,m⟶∞(2)
In a metric space, ifan and bn are two sequences converging to a and b, respectively, then σan,bn⟶σa,b as n⟶∞. In particular, if a=b then σan,bn⟶0 as n⟶∞. But this does not always hold in a sequential p-metric space. In Example 1, let us consider two sequences 2nn≥1 and 2n+1n≥1 in Λ. Then, both of these two sequences converge to 1∈Λ, but σ2n,2n+1⟶0 as n⟶∞(3)
Ap-metric is always a sequential p-metric space, but the converse is not true in general. The metric σ defined in Example 1 is not a p-metric for any Ω. Otherwise, σn,m≤Ωσn,1+σ1,m for all n,m≥2 with n≠m, which at once implies that enm−1≤Ωe1/n+1+e1/m+1−2 for all n,m≥2 with n≠m, arrives at a contradiction


## 3. Cantor’s Intersection-Like Theorem on Sequentialp-Metric Spaces

LetΛ,σ be a sequential p-metric space with supporting function Ω. Define
(3)Ba,η≔b∈Λ:σa,b<σa,a+η,Ba,η≔b∈Λ:σa,b≤σa,a+η,for all a∈Λ and η>0.Remark 14.
One can easily check that the collectionτσ≔∅∪U≠∅⊂Λ: for anya∈U, there exists η>0 such that Ba,η⊂U forms a topology on Λ.Definition 15.
A setF is said to be closed if there exists an open set U⊂Λ such that F=Uc, where Uc denotes the complement of U in Λ.Proposition 16.
LetΛ,σ be a sequential p-metric space and F⊂Λ be closed. Let an⊂F such that an⟶a as n⟶∞. Then, a∈F.Proof.
Suppose thata∈F. Then, a∈Fc. Since Fc∈τσ, so there exists η>0 such that Ba,η⊂Fc. Again, since σan,a⟶0 as n⟶∞, so for η>0 there exists N∈N such that σan,a<σa,a+η for all n≥N. That is, an∈Ba,η⊂Fc for all n≥N, which is a contradiction. Hence, a∈F.Proposition 17.
LetΛ,σ be a complete sequential p-metric space and F⊂Λ be closed. Then, F,σ is also complete.Proof.
Letan be a Cauchy sequence in F⊂Λ. Then, an is convergent in Λ, since Λ is complete. Let limnan=a∈Λ. Then, by Proposition 16, it follows that a∈F. Consequently, F is complete.Definition 18.
In a sequentialp-metric space Λ,σ, for A⊂Λ, we define
(4)diamA≔supσa,b:a,b∈A.Theorem 19.
LetΛ,σ be a complete sequential p-metric space and Fn be a decreasing sequence of nonempty closed subsets of Λ such that diamFn⟶0 as n⟶∞. Then, the intersection ∩n=1∞Fn contains exactly one point.Proof.
Letan∈Fn be chosen as arbitrary for all n∈ℕ. Since Fn is decreasing, we have an,an+1,⋯⊂Fn for all n∈ℕ.Now, for anyn,m∈ℕ with n,m≥k, we have σan,am≤diamFk, k≥1. Let ε>0 be given. Then, there exists some l∈ℕ such that diamFl<ε, since diamFn⟶0 as n⟶∞. From this, it follows that σan,am<ε whenever n,m≥l. Therefore, an is Cauchy in Λ. By the completeness of Λ, there exists some a∈Λ such that an converges to a. Since an,an+1,⋯⊂Fn and Fn is closed for each n∈ℕ, using Proposition 16, we have a∈∩n=1∞Fn.Next, we prove the uniqueness of the pointa. Let c∈∩n=1∞Fn be another point, then σa,c>0. As diamFn⟶0, there exists N0∈ℕ such that
(5)diamFn<σa,c≤diamFn,for all n≥N0, a contradiction. Hence, ∩n=1∞Fn=a and this completes the proof of our theorem.


## 4. Some Fixed Point Theorems

Theorem 20.
LetΛ,σ be a complete sequential p-metric space and Y:Λ⟶Λ be a mapping so that
(i)
σYa,Yb≤ασa,b for all a,b∈Λ and for some α∈0,1(ii)
there existsa0∈Λ such that δσ,Y,a0≔supσYia0,Yja0:i,j=1,2,⋯<∞
Then,Y has at least one fixed point in Λ. Moreover, if a and b are two fixed points of Y in Λ with σa,b<∞ then a=b.Proof.
Let us defineδσ,Yp+1,a0≔supσYp+ia0,Yp+ja0:i,j=1,2,⋯, for all p≥1. Clearly, δσ,Yp+1,a0≤δσ,Y,a0<∞ for all p≥1.Then, for allp≥1 and for all i,j=1,2,⋯,
(6)σYp+ia0,Yp+ja0≤ασYp−1+ia0,Yp−1+ja0≤αδσ,Yp,a0,implies, for all p≥1,
(7)δσ,Yp+1,a0=supi,j≥1σYp+ia0,Yp+ja0≤αδσ,Yp,a0≤α2δσ,Yp−1,a0⋮≤αpδσ,Y,a0.Denoteai=Yai−1=Yia0 for all i∈ℕ. For 1≤n<m, we have
(8)σan,am=σYna0,Yma0=σYn−1+1a0,Yn−1+m−n+1a0≤δσ,Yn,a0≤αn−1δσ,Y,a0⟶0asn⟶∞.Therefore,an is a Cauchy sequence in Λ. Due to the completeness of Λ, an is convergent and let limnan=a∈Λ.Now,σYa,Yan≤ασa,an⟶0 as n⟶∞. Therefore, an+1⟶Ya as n⟶∞. Hence, by Proposition 10, it follows that Ya=a, i.e., a∈Λ is a fixed point of Y.Now, ifa and b are two fixed points of Y in Λ with σa,b<∞, then we have σa,b=σYa,Yb≤ασa,b which gives σa,b=0 implies a=b.Theorem 21.
LetΛ,σ be a complete sequential p-metric space and Y:Λ⟶Λ such that
(i)
σYa,Yb≤γσa,Ya+σb,Yb for all a,b∈Λ and for some γ∈0,1/2,(ii)
there existsa0∈Λ such that δσ,Y,a0≔supσYia0,Yja0:i,j=1,2,⋯<∞Then, the Picard iterating sequencean, an=Yna0 for all n∈ℕ, converges to some a∈Λ. If σa,Ya<∞ and γt<Ω−1t for all t>0, then a∈Λ is a fixed point of Y. Moreover, if b is a fixed point of Y in Λ such that σa,b<∞ and σb,b<∞ then a=b.Proof.
For allp≥1 and for all i,j=1,2,⋯,
(9)σYp+ia0,Yp+ja0≤γσYp−1+ia0,Yp+ia0+σYp−1+ja0,Yp+ja0≤2γδσ,Yp,a0.This implies thatδσ,Yp+1,a0=supi,j≥1σYp+ia0,Yp+ja0≤2γδσ,Yp,a0 for all p≥1. Then, proceeding in a similar way as in Theorem 20, it can be easily shown that an is a Cauchy sequence in Λ, and by the completeness of Λ, there exists some a∈Λ such that limnan=a.Now,σan+1,Ya=σYan,Ya≤γσan,an+1+σa,Ya for all n≥0, which implies that limsupn→∞σan+1,Ya≤γσa,Ya<∞. Then,
(10)σa,Ya≤Ωlimsupn→∞σan+1,Ya≤Ωγσa,Ya.Ifσa,Ya>0 then Ω−1σa,Ya≤γσa,Ya<Ω−1σa,Ya, a contradiction. Hence Ya=a, i.e., a∈Λ is a fixed point of Y.Now, ifb is a fixed point of Y in Λ with σa,b<∞ and σb,b<∞, then we have σa,b=σYa,Yb≤γσa,Ya+σb,Yb=0, as σb,b=0, implying that a=b.Theorem 22.
LetΛ,σ be a complete sequential p-metric space and Y:Λ⟶Λ be a mapping satisfying the following conditions:
(i)
σYa,Yb≤βσa,Yb+σb,Ya for all a,b∈Λ and for some β∈0,1/2(ii)
there existsa0∈Λ such that δσ,Y,a0≔supσYia0,Yja0:i,j=1,2,⋯<∞Then, the Picard iterating sequencean, an=Yna0 for all n≥1, converges to some a∈Λ. If limsupn→∞σan,Ya<∞ then a∈Λ is a fixed point of Y. Also, if b is a fixed point of Y in Λ such that σa,b<∞ then a=b.Proof.
by similar argument as in Theorem20, an is a Cauchy sequence in Λ, and by completeness of Λ, it converges to an element say a∈Λ.Now, for alln∈ℕ∪0, σan+1,Ya=σYan,Ya≤βσan,Ya+σan+1,a, which implies that limsupn→∞σan+1,Ya≤βlimsupn→∞σan,Ya and hence limsupn→∞σan,Ya=0. Therefore, σa,Ya≤Ωlimsupn→∞σan,Ya=Ω0=0, and consequently, Ya=a.Ifb is a fixed point of Y in Λ with σa,b<∞, then we have σa,b=σYa,Yb≤βσa,Yb+σb,Ya=2βσa,b which implies σa,b=0 that is a=b.Example 2.
ConsiderΛ=0,1 and σa,b=∣a∣+∣b∣+ln1+∣a∣+∣b∣ for all a,b∈Λ. Then, σ forms a sequential p-metric on Λ with the function Ωt=t+ln1+t for all t≥0.
(i)
DefineY:Λ⟶Λ by Ya=a/3 for all a∈Λ. Then, Y satisfies all the conditions of Theorem 20 for α=1/2 and clearly Y has a unique fixed point 0∈Λ(ii)
Define,Ya=a/11if0≤a<1/2a/12if1/2≤a≤1. Then, Y satisfies all the conditions of Theorem 21 for γ=2/5 and clearly Y has a unique fixed point 0∈Λ


## 5. An Application to Nonlinear Integral Equations

In this section, we discuss about the existence of solutions for nonlinear integral equations as an application of Theorem20.LetΛ=Ca,b be the set of all real valued continuous functions on a,b and σ:Λ2⟶0,∞ be defined by
(11)σu,v=sinhmaxa≤t≤b∣ut−vtpforallu,v∈Λ,p≥1.Then,Λ,σ is a complete sequential p-metric space with Ωt=sinh2p−1t for all t≥0. Now, let us consider the integral equation
(12)ut=ht+∫abFt,sKt,s,usds,where h:a,b⟶ℝ, F:a,b×a,b⟶ℝ and K:a,b×a,b×ℝ⟶ℝ are continuous functions.Theorem 23.
Assume that the following hypotheses are satisfied:(i)
for allt,s∈a,b, we have
(13)Kt,s,us−Kt,s,vsp≤Au−vp,p≥1,A>0,where u−v=maxa≤t≤b∣ut−vt∣, u,v∈Λ,(ii)
maxa≤t≤b∫abFt,spds≤M, where M>0 is such that AMb−a<1
Then, integral equation (12) has a unique solution u∈Λ.Proof.
Let us defineY:Λ⟶Λ by
(14)Yut=ht+∫abFt,sKt,s,usds,u∈Λ,t,s∈a,b.Then, by conditionsi and ii, for all φ,ψ∈Λ, we get
(15)Yφt−Yψtp=∫abFt,sKt,s,φs−Kt,s,ψsdsp≤∫abFt,sqdsp/q∫abKt,s,φs−Kt,s,ψspds≤AMb−aφ−ψp≤μφ−ψpforallt∈a,b,where μ=AMb−a<1.Therefore,Yφ−Yψp≤μφ−ψp. Thus,
(16)σYφ,Yψ=sinhYφ−Yψp≤sinhμ∣φ−ψp≤μsinhφ−ψp=μσφ,ψ,for μ∈0,1 and for all φ,ψ∈Λ. Hence, the conditions of Theorem 20 are satisfied, and therefore, Y has a unique fixed point in Λ, provided δσ,Y,u0<∞ for some u0∈Λ, i.e., nonlinear integral equation (12) has a unique solution in Ca,b.Now, we give a numerical example in support of Theorem23.Example 3.
Let us consider the complete sequentialp-metric space Λ,σ defined in Theorem 23 for a=0, b=1, p=2, and the nonlinear integral equation given by
(17)ut=et+∫01tst+s+23usds,t∈0,1.Then, for allu,v∈Λ, we have
(18)Kt,s,us−Kt,s,vs2≤23u−v2,forallt,s∈0,1,max0≤t≤1∫01Ft,s2ds=12.Therefore,Y satisfies the contractive condition of Theorem 23 for μ=1/3, and also, we see that for u0∈Λ defined by u0t=0 for all t∈a,b, σYiu0,Yju0≤sinh36e+16/152<∞ for all i,j≥1. Hence, all the conditions of Theorem 20 are satisfied and therefore (17) has a unique solution in Λ.



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