# A New Variant of Symmetric Distance Spaces and an Extension of the Banach Fixed-Point Theorem

Authors: Kushal Roy; Mantu Saha; Vahid Parvaneh; Maryam Khorshidi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The notion ofΔ-metric spaces has been proposed in this study as a generalization of b-metric spaces, extended b-metric spaces, and p-metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.


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## Body

## 1. Introduction and Preliminaries

Because of the wide application in numerous fields of mathematics, particularly, in nonlinear functional analysis, the metric fixed-point theory has emerged as an interesting subject in mathematics in recent decades. As it is today, the field of fixed-point theory is extensive and amenable to a wide range of applications using a number of approaches and ideas. Because of its many intriguing features, the metric structure plays a significant role in fixed-point theory. The most significant is the metric function’s continuity. Furthermore, in a metric space, the normal topology is alwaysT5, with the collection of open balls serving as the topology’s base. As a result, dealing with various sorts of contractive or expansive mappings over metric spaces is fairly simple. Researchers working on fixed-point theory, on the contrary, are interested in considering a variety of alternative modified metric structured spaces without taking into account the modified metrics’ continuity on the underlying spaces.By appropriately changing the notion of a metric in the normal sense, Bakhtin and Czerwik [1, 2] created a newly structured metric space termed b-metric space.Definition 1.
(b-metric space [1, 2]). Let X be a nonempty set and s be a real number satisfying s≥1. A function ρb:X×X⟶ℝ+ is a b−metric on X if the following conditions hold:(1)
ρbx,y=0 if and only if x=y(2)
ρbx,y=ρby,x, for all x,y∈X(3)
ρbx,z≤sρbx,y+ρby,z, for all x,y,z∈X (b-triangle inequality)
The spaceX,ρb is called a b−metric space.In 2017, Kamran et al. [3] used a generalized triangular inequality in their metric-type structure, replacing the constant coefficient of the b-triangle inequality with a two-variable function.Definition 2.
(extendedb-metric space) [3]. Let X be a nonempty set and θ:X×X⟶1,∞. A function ρθ:X2⟶0,∞ is called an extended b−metric if, for all x,y,z∈X, it satisfies(1)
ρθx,y=0 if and only if x=y(2)
ρθx,y=ρθy,x(3)
ρθx,z≤θx,zρθx,y+ρθy,z
The pairX,ρθ is called an extended b−metric space.Recently, Parvaneh and Ghoncheh [4] have defined a new generalized metric space by using an invertible, continuous, and strictly increasing function in the triangle inequality.Definition 3.
(p-metric space [4]). Let X be a nonempty set. A function ρp:X×X⟶0,∞ is said to be an extended b-metric or a 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 x,y,z∈X, the following conditions hold:(1)
ρpx,y=0 if and only if x=y(2)
ρpx,y=ρpy,x(3)
ρpx,z≤Ωρpx,y+ρpy,z
The pairX,ρp is called a p-metric space.The definitions of convergent sequence, Cauchy sequence, and completeness property in these spaces are analog to these definitions in a metric in the classical sense. In Section2, we define a generalized metric-type space.


## 2. Introduction toΔ-Metric Space

In this section, we introduce the notion of a new generalized metric space known asΔ-metric space. First, we consider a nonempty set X and the following collection of functions:(1)SΔ≔Δ:X×X⟶0,∞:Δξ,ξ=0for allξ∈X.Definition 4.
LetX be a nonempty set and ρΔ:X×X⟶ℝ+ be a mapping. Then, ρΔ is said to be a Δ-metric if there exists some Δ∈SΔ such that, for all ξ,η,ζ∈X, it satisfies the following conditions:ρΔ10≤Δξ,η≤ρΔξ,ηρΔ2ρΔξ,η=Δξ,η if and only if ξ=ηρΔ3ρΔξ,η=ρΔη,ξρΔ4ρΔξ,ζ≤ρΔξ,η+ρΔη,ζ+Δξ,ζ
The pairX,ρΔ is called a Δ-metric space.Remark 1.
From conditionsρΔ1 and ρΔ2 and by the property of Δ, it follows that ρΔξ,η=0 if and only if ξ=η or Δξ,η<ρΔξ,η whenever ξ≠η.Remark 2.
Definition4 also holds for the function minΔx,y,Δy,x for all x,y∈X, and therefore, without loss of generality, we can assume that the function Δ is symmetric.Remark 3.
(i)
If, in Definition4, we set Δξ,η=0, for all ξ,η∈X, then ρΔ will give usual metric(ii)
If, in Definition4, we set Δξ,η=k−1/kρΔξ,η, for all ξ,η∈X and for some k>1, then ρΔ will give b-metric with coefficient k>1.(iii)
If we takeΔξ,η=θξ,η−1/θξ,ηρΔξ,η for all ξ,η∈X and for some function θ:X2⟶1,∞ in Definition 4, then ρΔ will give extended b-metric.(iv)
LetΩ:0,∞⟶0,∞ be a strictly increasing continuous function with Ω−1t≤t≤Ωt and Ω−10=0=Ω0. Then, if we consider Δξ,η=ρΔξ,η−Ω−1ρΔξ,η for all ξ,η∈X, then ρΔ will give p−metric.The following are some examples ofΔ metric spaces.Example 1.
LetX=ℕ and ρΔ:X×X⟶0,∞ satisfies ρΔξ,η=ρΔη,ξ, for all ξ,η∈X, and is given by(2)ρΔξ,η=0,ifξ=η,n,ifξ=1andη=n>1,en+m,ifξ=n∈ℕ∖1,η=m∈ℕ∖1andξ≠η.
Then,ρΔ is a Δ-metric space with Δ∈SΔ given by(3)Δξ,η=en+m−n+m,ifξ=n,η=m∈ℕ∖1andξ≠η,0,otherwise.
However,X is not a b-metric space, for any s≥1, since(4)limn,mn≠m⟶∞ρΔn,mρΔn,1+ρΔ1,m=limn,mn≠m⟶∞en+mn+m=∞.Example 2.
LetX=ℕ and ρΔ:X×X⟶0,∞ satisfy ρΔξ,η=ρΔη,ξ, for all ξ,η∈X, and is given by(5)ρΔξ,η=0,ifξ=η,1n,ifξ=1andη=n>1,2,ifξ=n∈ℕ∖1,η=m∈ℕ∖1andξ≠η.
Then,ρΔ is a Δ-metric space with Δ∈SΔ given by(6)Δξ,η=2−m+nmn,ifξ=n,η=m∈ℕ∖1andξ≠η,0,otherwise.
IfX is a b-metric space, for some s≥1, then, for all n,m∈ℕ∖1 with n≠m, we have(7)ρΔn,m≤sρΔn,1+ρΔ1,m⇒1≤s1n+1m⟶0asn,m⟶∞,a contradiction.
Hence,X cannot be a b-metric space, for any s≥1.Definition 5.
LetX,ρΔ be a Δ-metric space and ξn be a sequence in X. Then,(i)
The sequenceξn is said to be ρΔ-convergent and converges to some ξ∈X if ρΔξn,ξ⟶0 as n⟶∞.(ii)
The sequenceξn is called ρΔ-Cauchy if ρΔξn,ξm⟶0 as n,m⟶∞(iii)
X,ρΔ is said to be complete if every ρΔ-Cauchy sequence in X is ρΔ-convergentDefinition 6.
LetX,ρΔ be a Δ-metric space and ξn be a sequence in X. Then,(i)
The sequenceξn is said to be Δ-convergent to an element ξ∈X if Δξn,ξ⟶0 as n⟶∞(ii)
The sequenceξn is called Δ-Cauchy if Δξn,ξm⟶0 as n,m⟶∞Remark 4.
The concepts ofρΔ-convergence and Δ-convergence coincide in the case of b-metric spaces and p-metric spaces. A ρΔ-Cauchy sequence is also Δ-Cauchy in b and p metric spaces, and vice versa.Remark 5.
Clearly, in aΔ-metric space, any ρΔ-convergent sequence is also Δ-convergent, but the converse is not true in general.
Any sequence in the usual metric spaceℝ,du is Δ-convergent to every element of ℝ, but many sequences in ℝ are not du-convergent. Similarly, any sequence in ℝ is Δ-Cauchy, but there are many sequences in ℝ that are not du-Cauchy.Remark 6.
In aΔ-metric space, a ρΔ-convergent sequence may not be always a ρΔ-Cauchy sequence. In Example 2, we see that 2,3,4,…⊂ℕρΔ-converges to 1, but it is not a ρΔ-Cauchy sequence.Definition 7.
LetX,ρΔ and Y,ρΔ′ be two Δ-metric spaces. A mapping f:X⟶Y is said to be continuous at ξ∈X if, for any ϵ>0, there exists δ>0 such that(8)ρΔ′fη,fξ<εwheneverρΔη,ξ<δ,η∈X.Remark 7.
AΔ-metric is not always continuous because any b-metric space is also a Δ-metric space, and there are several examples of noncontinuous b-metrics.


## 3. Some Topological Properties

LetX,ρΔ be a Δ-metric space. Define(9)ℬΔξ,t≔η∈X:ρΔξ,η<t,ℬΔξ,t≔η∈X:ρΔξ,η≤t,for all ξ∈X and t>0.Remark 8.
One can easily check that the collectionτΔ≔∅∪U≠∅⊂X:for anyξ∈U,there existst>0such thatℬΔξ,t⊂U forms a topology on X.We will now go through some key definitions.Definition 8.
(i)
A subsetA⊂X of a Δ-metric space X,ρΔ is called Δ-open if A∈τΔ(ii)
A subsetℱ⊂X of a Δ-metric space X,ρΔ is said to be Δ-closed if ℱc∈τΔ(iii)
TheΔ-closure of a subset ℋ⊂X in a Δ-metric space X,ρΔ is denoted by ℋ¯ and is defined by the smallest Δ-closed set containing ℋ(iv)
A subsetE⊂X of a Δ-metric space X,ρΔ is called Δ-bounded if supρΔξ,η:ξ,η∈Ε<∞Remark 9.
An open ballℬΔξ,t is not always Δ-open in a Δ-metric space X,ρΔ. In Example 2, we see that ℬΔ2,1=1,2, but ℬΔ1,δ=1∪η>1:η∈1/δ,+∞⊈ℬΔ2,1, for any δ>0.Definition 9.
The topologyτΔ defined on a Δ-metric space X,ρΔ is called regular if, for any ξ∈X, ℬΔξ,r¯⊂ℬΔξ,s whenever 0<r<s.Definition 10.
A subsetξλ:λ∈∧ is called an ϵ-net of a subset G⊂X of a Δ-metric space X,ρΔ, for some ϵ>0, if(10)G⊂∪λ∈∧ℬΔξλ,ϵ.Definition 11.
AΔ-bounded subset K⊂X of a Δ-metric space X,ρΔ is said to be Δ-precompact if, for any ϵ>0, there exists a finite ϵ-net for K.Definition 12.
AΔ-bounded subset ℳ⊂X of a Δ-metric space X,ρΔ is said to be Δ-sequentially compact if any sequence in ℳ has a ρΔ-convergent subsequence.Definition 13.
AΔ-metric space X,ρΔ is said to preserve ρΔ-Cauchyness if any sequence ξn satisfying(11)limn,m⟶∞ρΔξn,ξm−Δξn,ξm=0is ρΔ-Cauchy in X.Remark 10.
Any metric space,b-metric space, and p-metric space always preserve ρΔ-Cauchyness.Some topological properties of aΔ-metric space are stated below.Corollary 1.
LetX,ρΔ be a Δ-metric space, f:X⟶X be a mapping, and ξ∈X. Then, f is continuous at ξ if and only if, for all sequence ξnρΔ-converging to ξ, we have fξnρΔ-converges to fξ.Corollary 2.
In aΔ-metric space X,ρΔ, the following conditions are equivalent:(i)
ℱ⊂X is Δ-closed(ii)
For all sequenceξn⊂ℱρΔ-converging to ξ∈X, we have ξ∈ℱCorollary 3.
In aΔ-metric space X,ρΔ, if a sequence ξn⊂A⊂XρΔ-converges to ξ∈X, then ξ∈A¯.Corollary 4.
AΔ-closed subset of a complete Δ-metric space X,ρΔ is also complete.Now, we will prove a few propositions that will be utilized to support our next set of conclusions.Proposition 1.
LetX,ρΔ be a Δ-metric space. Then, for any ξ,η, and ζ in X,(12)ρΔξ,ζ−ρΔη,ζ≤ρΔξ,η+Δξ,ζ+Δη,ζ.Proof.
Using conditionρΔ4 in Definition 4, we obtain(13)ρΔξ,ζ≤ρΔξ,η+ρΔη,ζ+Δξ,ζ⇒ρΔξ,ζ−ρΔη,ζ≤ρΔξ,η+Δξ,ζ,and also,(14)ρΔη,ζ≤ρΔη,ξ+ρΔξ,ζ+Δη,ζ⇒ρΔη,ζ−ρΔξ,ζ≤ρΔξ,η+Δη,ζ.
Then, by combining (13) and (14), the result follows immediately.Proposition 2.
AΔ-metric space is always Hausdorff, that is, if in a Δ-metric space X a sequence ξn is ρΔ-convergent to two elements ξ and η, then ξ=η.Proof.
From conditionρΔ4 of Definition 4, we see that(15)ρΔξ,η≤ρΔξ,ξn+ρΔξn,η+Δξ,ηfor alln∈ℕ.
Takingn⟶∞, we get ρΔξ,η≤Δξ,η. Therefore, by using condition ρΔ1, it follows that ρΔξ,η=Δξ,η, and hence, from ρΔ2, it follows that ξ=η.Proposition 3.
If, in aΔ-metric space X, a sequence ξn is ρΔ-convergent to some ξ∈X, then limn,m⟶∞ρΔξn,ξm−Δξn,ξm=0. Moreover, if ξn is a Δ-Cauchy sequence, then ξn will be a ρΔ-Cauchy sequence in X.Proof.
For anyn,m≥1, we have(16)ρΔξn,ξm≤ρΔξn,ξ+ρΔξ,ξm+Δξn,ξm.
Therefore, from (16), we get limn,m⟶∞ρΔξn,ξm−Δξn,ξm=0. Clearly, if ξn is Δ-Cauchy, then limn,m⟶∞Δξn,ξm=0, which implies that limn,m⟶∞ρΔξn,ξm=0 and ξn will be a ρΔ-Cauchy sequence in X.Proposition 4.
If, in aΔ-metric space X, two sequences ξn and ηn both are ρΔ-convergent to an element ξ∈X, then limn⟶∞ρΔξn,ηn−Δξn,ηn=0. Moreover, if limn⟶∞Δξn,ηn=0, then limn⟶∞ρΔξn,ηn=0.Proof.
For anyn∈ℕ, from condition ρΔ4 in Definition 4, we obtain(17)ρΔξn,ηn≤ρΔξn,ξ+ρΔξ,ηn+Δξn,ηn.
Hence,limn⟶∞ρΔξn,ηn−Δξn,ηn=0. Therefore, it is clear that if limn⟶∞Δξn,ηn\\=0, then limn⟶∞ρΔξn,ηn=0.Remark 11.
Because of Remark3, the additional conditions imposed in Propositions 3 and 4 are not required in the case of a metric space, a b-metric space, or a p-metric space.Proposition 5.
LetA be a Δ-precompact subset of a Δ-metric space X,ρΔ and ξn⊂A be a sequence. Then, ξn has a subsequence ξnk such that(18)limk,l⟶∞ρΔξnk,ξnl−Δξnk,ξnl=0.Proof.
SinceA is Δ-precompact, then A has a finite 1/2-net. So, there exists A1⊂A which contains infinitely many terms of ξn, where A1=A∩ℬΔν2,1/2, ν2∈X. Clearly A1 is Δ-precompact, and therefore, A1 has a finite 1/3-net. So, there exists A2⊂A1 which contains infinitely many terms of ξn, where A2=A1∩ℬΔν3,1/3, ν3∈X. Proceeding in a similar way, we can construct a sequence of subsets An of A such that An+1=An∩ℬΔνn+2,1/n+2, νn+2∈X, for all n≥0, and each An contains infinitely many terms of ξn, n≥1. Let us choose ξni∈Ai such that ni+1>ni, for all i∈ℕ. Then, for any 1≤k<l, we have ξnk,ξnl∈Ak, and we obtain(19)ρΔξnk,ξnl≤ρΔξnk,νk+1+ρΔξnl,νk+1+Δξnk,ξnl<21k+1+Δξnk,ξnl.
Thus,ξn has a subsequence ξnk such that limk,l⟶∞ρΔξnk,ξnl−Δξnk,ξnl=0.Remark 12.
If aΔ-metric space X preserves ρΔ-Cauchyness, then any sequence in a Δ-precompact subset of X has a ρΔ-Cauchy subsequence.Definition 14.
In aΔ-metric space X,ρΔ, for A⊂X, we define(20)diamA≔supρΔθ,ϑ:θ,ϑ∈A.Theorem 1.
LetX,ρΔ be a complete Δ-metric space and ℱn be a decreasing sequence of nonempty Δ-closed subsets of X such that diamℱn⟶0 as n⟶∞. Then, the intersection ∩n=1∞ℱn contains exactly one point.Proof.
Letηn∈ℱn be chosen as arbitrary, for all n∈ℕ. Since ℱn is decreasing, we have ηn,ηn+1,…⊂ℱn, for all n∈ℕ.
Now, for anyn,m∈ℕ with n,m≥k, we have ρΔηn,ηm≤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 ρΔηn,ηm<ϵ whenever n,m≥l. Therefore, ηn is ρΔ-Cauchy sequence in X. By the completeness of X, there exists some η∈X such that ηn is ρΔ-convergent to η. Since ηn,ηn+1,…⊂ℱn and ℱn is Δ-closed for each n∈ℕ, using Corollary 2, we have η∈∩n=1∞ℱn.
Next, we prove the uniqueness of the pointη. Let ζ∈∩n=1∞ℱn be another point; then, ρΔη,ζ>0. As diamℱn⟶0, there exists N0∈ℕ such that(21)diamℱn<ρΔη,ζ≤diamℱn,for all n≥N0, a contradiction. Hence, ∩n=1∞ℱn=η, and this completes the proof of our theorem.Proposition 6.
AΔ-closed, complete, and Δ-precompact subset of a Δ-metric space X preserving ρΔ-Cauchyness is Δ-sequentially compact.Proof.
LetA be a Δ-closed, complete, and Δ-precompact subset of a Δ-metric space X,ρΔ which preserves ρΔ-Cauchyness and ξn⊂A be any sequence. Then, by Remark 12, ξn has a ρΔ-Cauchy subsequence. Since A is complete, it follows that ξn is ρΔ-convergent to some ξ∈X. Here, A is Δ-closed, and therefore, ξ∈A. So, ξn has a ρΔ-convergent subsequence. As ξn is chosen as arbitrary, it follows that A is Δ-sequentially compact.


## 4. A Hausdorff Measure of Noncompactness

The degree of noncompactness of a set is measured with the help of mappings known as measures of noncompactness. Noncompactness measures are widely used in fixed-point theory, differential equations, integral equations, optimization theory, etc. (for more details, see [5]). Here, we define a Hausdorff noncompactness measure in the setting of a Δ-metric space by means of its topology. We denote the collection of all Δ-bounded subsets in a Δ-metric space (X,ρΔ by ℬDΔX. We begin this section with the definition of the Hausdorff measure of noncompactness.Definition 15.
LetX,ρΔ be a Δ-metric space and Q∈ℬDΔX. Then, the Hausdorff measure of noncompactness of the set Q is denoted by χΔQ and is given by(22)χΔQ=infε>0:Q⊂∪i=1mℬΔξi,ri,ξi∈X,ri≤ε2,i=1,…,m,m∈ℕ.Lemma 1.
Letℳ, P, and Q be in ℬDΔX of the Δ-metric space X,ρΔ. Then,(i)
χΔℳ=0 if and only if ℳ is Δ-precompact(ii)
χΔℳ=χΔℳ¯ provided the topology on X is regular(iii)
P⊂Q implies χΔP≤χΔQ(iv)
χΔP∪Q=maxχΔP,χΔQ(v)
χΔP∩Q≤minχΔP,χΔQProof.
(i)
The proof is straightforward.(ii)
Letδ>0 be such that ℳ¯⊂∪i=1nℬΔξi,ri,ξi∈X,ri≤δ/2,i=1,…,n. Then, ℳ⊂ℳ¯⊂∪i=1nℬΔξi,ri. Therefore, δ∈ε>0:ℳ¯⊂∪i=1mℬΔξi,ri,ξi∈X,ri≤ε/2,i=11m,m∈ℕ implies δ∈ε>0:ℳ⊂∪i=1mℬΔξi,ri,ξi∈X,ri≤ε/2,i=11m,m∈ℕ. Hence, χΔℳ≤χΔℳ¯.Now, for the converse part, letχΔℳ=t≥0, and we choose δ>0 as arbitrary. Then, there exist an ϵ>0 and ξ1,ξ2,…,ξn⊂X such that ℳ⊂∪i=1nℬΔξi,ri,ξi∈X,ri≤ϵ<2ϵ<t+δ, for all i=1,2,…,n. So, ℳ¯⊂∪i=1nℬΔξi,ri¯⊂∪i=1nℬΔξi,t+δ/2 since the topology defined on X is regular. Therefore, we have χΔℳ¯≤t+δ. Due to the arbitrariness of δ, we have χΔℳ¯≤t=χΔℳ. Thus, χΔℳ=χΔℳ¯.(iii)
The proof can be done easily.(iv)
SinceP⊂P∪Q and Q⊂P∪Q, then, from (iii), it follows that maxχΔP,χΔQ≤χΔP∪Q.Now, letmaxχΔP,χΔQ=h≥0, and we choose a δ>0 as arbitrary. Then, there exists ϵ1>0 such that ϵ1<h+δ and ξ1,ξ2,…,ξn⊂X such that P⊂∪i=1nℬΔξi,ri,ξi∈X,ri≤ϵ1/2, for all i=1,2,…,n. Similarly, we can get an ϵ2>0 such that ϵ2<h+δ and η1,η2,…,ηm⊂X such that Q⊂∪i=1mℬΔηi,si,ηi∈X,si≤ϵ2/2, for all i=1,2,…,m. Let us take ϵ=maxϵ1,ϵ2; then, we have P∪Q⊂∪i=1nℬΔξi,ϵ/2∪∪i=1mℬΔηi,ϵ/2. Therefore, χΔP∪Q≤ϵ<h+δ. Since δ>0 is arbitrarily chosen, then we have χΔP∪Q≤h=maxχΔP,χΔQ. Hence, the result is as follows.(v)
The result follows from (iii).Example 3.
In Example2, we see that ℕ is Δ-bounded. Here, ℬΔ1,δ=1∪η>1:η∈1/δ,+∞, for all δ>0. Therefore, for any ϵ>0, the set ℕ∖ℬΔ1,ϵ/2 contains only finitely many points of ℕ. Therefore, any ϵ>0ℕ can be covered by finitely many open balls of radius ϵ/2>0. Hence, χΔℕ=0, and from Lemma 1, it follows that ℕ is Δ-precompact.Remark 13.
Since, in Example2, ℕ is Δ-closed, complete, and Δ-precompact, then, from Proposition 6, it follows that ℕ is Δ-sequentially compact.Lemma 2.
LetX,ρΔ be a complete Δ-metric space which preserves ρΔ-Cauchyness such that the topology τΔ is regular. If ℱn is a decreasing sequence of nonempty, Δ-closed, and Δ-bounded subsets of X such that limn⟶∞χΔℱn=0, then the intersection ℱ′=∩n=1∞ℱn is nonempty, Δ-closed, and Δ-sequentially compact subset of X.Proof.
Clearly,ℱ′ is Δ-closed. Since ℱ′⊂ℱn, for all n≥1, then χΔℱ′≤χΔℱn⟶0 as n⟶∞. So, χΔℱ′=0, and it follows from (i) of Lemma 1 that ℱ′ is Δ-precompact. Therefore, from Corollary 4 and Proposition 6, we get ℱ′ is Δ-sequentially compact. We have to prove that ℱ′≠∅. Let ξn∈ℱn be chosen as arbitrary. Then, En=ξn,ξn+1,…⊂ℱn, for all n∈ℕ. Now, E1=ξ1,…,ξn−1∪En, for all n>1. Therefore,(23)χΔE1=maxχΔEn,χΔξ1,…,ξn−1=χΔEn≤χΔℱn⟶0asn⟶∞.As any finite set isΔ−precompact.
Therefore,χΔE1=0. So, due to Lemma 1, χΔE1¯=0, implying that E1¯ is Δ-closed and Δ-precompact. Here, X is complete, and therefore, E1¯ is also complete. Then, from Proposition 6, we see that E1¯ is Δ-sequentially compact. So, ξn has a ρΔ-convergent subsequence in E1¯. Let it be convergent to ξ∈X. Since En=ξn,ξn+1,…⊂ℱn and each ℱn is Δ-closed, it follows that ξ∈ℱn, for all n≥1, that is, ξ∈ℱ′. Hence, ℱ′ is nonempty.Remark 14.
In the case of a metric space,b-metric space, or p-metric space, the assumption that X preserves ρΔ-Cauchyness in Lemma 2 is not required.


## 5. Banach Contraction Principle inΔ-Metric Spaces

We prove the Banach fixed-point theorem in the context ofΔ-metric space in this section.Definition 16.
LetX be a Δ-metric space. A mapping F:X⟶X is said to be Banach Δ-contraction if, for all ξ,η∈X, F satisfies(24)ρΔFξ,Fη≤αρΔξ,η,whereα∈0,1.Theorem 2.
LetX be a complete Δ-metric space and F:X⟶X be a Banach Δ-contraction for α∈0,1. If, for some ξ0∈X, the Picard iterating sequence ξn, ξn=Fnξ0, for all n≥1, is Δ-Cauchy; then, F has a unique fixed point in X.Proof.
By repeated application of condition (24), we have(25)ρΔξn,ξn+1≤αρΔξn−1,ξn≤α2ρΔξn−2,ξn−1≤⋯≤αnρΔξ0,ξ1.
Now, for any1≤n<m, we see that(26)ρΔξn,ξm≤ρΔξn,ξn+1+ρΔξn+1,ξm+Δξn,ξm≤ρΔξn,ξn+1+ρΔξn+1,ξm+1+ρΔξm+1,ξm+Δξn+1,ξm+Δξn,ξm≤αnρΔξ0,ξ1+αρΔξn,ξm+αmρΔξ0,ξ1+Δξn+1,ξm+Δξn,ξm.
Therefore,ρΔξn,ξm≤αn+αm/1−αρΔξ0,ξ1+1/1−αΔξn+1,ξm+Δξn,ξm, for all n,m≥1. Since ξn is a Δ-Cauchy sequence, then, from (26), it follows that ξn is a ρΔ-Cauchy sequence in X. Due to the completeness of X, there exists ξ∈X such that ξn is ρΔ-convergent to ξ. Now, using condition (24), we obtain(27)ρΔξn+1,Fξ≤αρΔξn,ξ⟶0asn⟶∞.
Thus, from Proposition2, we have Fξ=ξ and ξ is a fixed point of F in X.
To prove the uniqueness of the fixed point ofF, let us assume that η is another fixed point of F. Then,(28)ρΔξ,η=ρΔFξ,Fη≤αρΔξ,η,implying that ρΔξ,η=0, and therefore, from Remark 1, it follows that ξ=η, a contradiction. Hence, the fixed point of F is unique in X.Example 4.
Let us consider the completeΔ-metric space defined in Example 1 and F:X⟶X be given by(29)Fξ=1,ifξ∈1,2,2,ifξ>2.
Then,F is a Banach Δ-contraction for α=2/3 and for ξ0=3; we have ξn=1, for all n≥2; therefore, ξn is clearly Δ-Cauchy sequence in X. Hence, all the conditions of Theorem 2 are satisfied and 1 is the unique fixed point of F in X.Remark 15.
It is not necessary to assume the additional condition in Theorem2 in the setting of a metric space, a b-metric space, or a p-metric space because, from the contractive condition (24), it follows immediately that any Picard iterating sequence is ρΔ-Cauchy or Δ-Cauchy.


## 6. Application to the Well-Posedness of Fixed-Point Problem

The well-posedness of the fixed-point problem is an intriguing study in fixed-point theory. The following is a definition of well-posedness for a fixed-point problem in metric spaces:Definition 17.
(see [6]). Let X,ρ be a metric space and S:X⟶X be a mapping. The fixed-point problem of S is said to be well-posed if (a) S has a unique fixed point ζ∈X and (b) for any sequence ξn in X with ρξn,Sξn⟶0 as n⟶∞, we have ρξn,ζ⟶0 as n⟶∞.The following is an analog definition of the well-posedness of a fixed-point problem for some mappings in aΔ-metric space.Definition 18.
(see [6]). Let X,ρΔ be a metric space and T:X⟶X be a mapping. The fixed-point problem of T is said to be well-posed if(i)
T has a unique fixed point ζ∈X(ii)
For anyΔ-convergent sequence ξnΔ-converges to ζ in X with ρΔξn,Tξn⟶0 as n⟶∞, we have ρΔξn,ζ⟶0 as n⟶∞Theorem 3.
IfX,ρΔ is a complete Δ-metric space and a mapping T:X⟶X satisfies all the conditions of Theorem 2, then the fixed-point problem of T is well-posed.Proof.
From Theorem2, it follows that T has a unique fixed point ζ in X. Now, let ξn be a Δ-convergent sequence which Δ-converges to ζ with ρΔξn,Tξn⟶0 as n⟶∞. Then,(30)ρΔξn,ζ≤ρΔξn,Tξn+ρΔTξn,ζ+Δξn,ζ=ρΔξn,Tξn+ρΔTξn,Tζ+Δξn,ζ≤ρΔξn,Tξn+αρΔξn,ζ+Δξn,ζfor alln≥1.
From (30), we have ρΔξn,ζ≤1/1−αρΔξn,Tξn+Δξn,ζ⟶0 as n⟶∞. Therefore, the fixed-point problem of T is well-posed.Remark 16.
In Definition18, it is not needed to assume the sequence ξn to be Δ-convergent in the setting of a metric space or b-metric space or p-metric space.


## 7. Conclusion

In this study, we have introduced a new metric-like structure, known asΔ-metric, as a generalization of the usual metric, b-metric, extended b-metric, and p-metric structures. We have established the topology of such spaces and introduced a noncompactness measure. A Banach-type fixed-point theorem has been proved here, and it has also been applied to the well-posedness of a fixed-point problem. There is a lot of scope for research to prove different fixed-point theorems in Δ-metric space. One can establish the Kannan fixed-point theorem, the Chatterjea fixed-point theorem, and many more in such spaces and apply those theorems to several branches of mathematics. Also, the idea of nontriangular metric is a very interesting idea. One can construct the notion of δ-nontriangular metric spaces which will be a real generalization of nontriangular metric spaces [7]. Nontriangular metric spaces are real generalizations of the JS-metric spaces [8]. Also, in [9], Deshmukh and Gopal have made an attempt to study the topology of nontriangular metric spaces.



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