# α,BK-Contractions in Bipolar Metric Spaces

Authors: Yaé Ulrich Gaba; Maggie Aphane; Hassen Aydi
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we introduce the concept ofα,BK-contractions and prove the existence of fixed points for contravariant mappings on bipolar metric spaces.


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## Body

## 1. Introduction and Preliminaries

The notion of a metric space has many generalizations in literature. One of the most recent generalizations is that of abipolar metric space, introduced by Mutlu and Gürdal [1], the motivation being that in many real-life applications, distances arise between elements of two different sets, rather than between points of a unique set. Hence, bipolar metrics came to formalize these types of distances. Some basic examples are distance between lines and points in an Euclidean space, distance between points and sets in a metric spaces, affinity between a class of students and a set of activities, lifetime mean distances between people and places, and many more. Many investigators have published various papers on fixed point in bipolar metric spaces, and these can be read in [2–6] and the references therein, just to name a few. This new notion of generalization and improvement of metric spaces leads to give existence and development of fixed-point theorems. In bipolar metric spaces, a lot of significant work has been done (see [3, 7–9]). More recently, Kishore et al. [7] established the existence and uniqueness of common coupled fixed-point results for three covariant mappings in bipolar metric spaces, while Mutlu [8] extended certain coupled fixed-point theorems, which can be considered as generalization of Banach fixed-point theorem, to bipolar metric spaces. Hence, bipolar metric fixed-point theory is an active research area which is capturing a lot of attention and seems to be a prospective flourishing domain.In the present paper, we introduce the so-calledα,BK-contractions and prove the existence of fixed points for contravariant mappings on bipolar metric spaces. We illustrate, how, under the choice of convenient constants, we can easily recover some well-known classical results. A detailed account of the theory of bipolar metric spaces can be read in the paper by Mutlu and Gürdal [1], but for the sake of completeness, we recall the definitions of some fundamental notions related to bipolar metric spaces.Definition 1.
(see [1], Definition 2.1). A bipolar metric space is a triple X,Y,d such that X,Y are two nonempty sets and d:X×Y⟶ℝ+=0,+∞ is a function satisfying the following properties:(1)
dx,y=0⇔x=y, whenever x,y∈X×Y(2)
dx,y=dy,x, whenever x,y∈X∩Y(3)
dx1,y2≤dx1,y1+dx2,y1+dx2,y2, whenever x1,y1,x2,y2∈X×Y
The pairX,Y is called a bipolar metric.Definition 2.
(see [1], Definition 2.3). Let X1,Y1 and X2,Y2 be pairs of sets and given a function f:X1∪Y1⟶X2∪Y2.(1)
IffX1⊆Y2 and fY1⊆X2, we call f a contravariant map from X1,Y1 to X2,Y2 and denote this with f:X1,Y1X2,Y2.(2)
Moreover, ifd1 and d2 are bipolar metrics on X1,Y1 and X2,Y2, respectively, then the notation f:X1,Y1,d1⤨X2,Y2,d2 denotes a contravariant map from X1,Y1,d1 to X2,Y2,d2.Definition 3.
(see [1], Definition 4.1). Let X,Y,d be a bipolar metric space.(1)
We have(1)X=set of left points;Y=set of right points;X∩Y=set of central points.In particular; ifX∩Y=∅, the space is called disjoint, and otherwise it is called joint. Unless otherwise stated, we shall work with joint spaces.(2)
A sequencexn on the set X is called a left sequence and a sequence yn on Y is called a right sequence. In a bipolar metric space, a left or a right sequence is called simply a sequence.(3)
A sequenceun is said to be convergent to a point u, if and only if un is a left sequence, limn⟶∞dun,u=0 and u∈Y, or un is a right sequence, limn⟶∞du,un=0 and u∈X.(4)
Abisequencexn,yn on X,Y,d is a sequence on the set X×Y. Furthermore, if the sequences xn and yn are convergent, then the bisequence xn,yn is said to be convergent. In addition, if xn and yn converge to a common point t∈X∩Y, then xn,yn is called biconvergent.(5)
A bisequencexn,yn is a Cauchy bisequence, if limn⟶∞,m⟶∞dxn,ym=0.Remark 1.
(see [1], Proposition 4.3). In a bipolar metric space, every convergent Cauchy bisequence is biconvergent.Definition 4.
(see [1], Definition 4.4). A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.Proposition 1 (see [1]).
If a central point is a limit of a sequence, then this sequence has a unique limit.Example 1.
(see [1], Example 5.7). Let X be the class of all singleton subsets of ℝ and Y be the class of all nonempty compact subsets of ℝ. We define d:X×Y⟶ℝ+ as(2)dx,A=x−infA+x−supA.
Then, the tripleX,Y,d is a complete bipolar metric space.Definition 5.
(see [1], Definition 3.4). A covariant or a contravariant map f from the bipolar metric space X1,Y1,d1 to the bipolar metric space X2,Y2,d2 is continuous, if and only if un⟶v on X1,Y1,d1 implies fun⟶fv on X2,Y2,d2.Proposition 2 (see [1]).
If the point to which a covariant or contravariant mapfis left and right continuous is central point, then the map fis continuous at this point.


## 2. First Results

In this section, we express a theorem based on Reich-type contractions on bipolar metric spaces. The results presented here will be generalized in the next section.Theorem 1(Reich-type contraction).
LetX,Y,d be a complete bipolar metric space and f:X,Y,d⤨X,Y,d be a contravariant map such that there exist constants a1,a2,a3≥0 with a1+a2+a3<1 so that(3)dfy,fx≤a1dx,y+a2dx,fx+a3dfy,y,whenever x,y∈X×Y. Then the function f:X∪Y⟶X∪Y has a unique fixed point.Proof.
Letx0∈X; for each nonnegative integer n, we define yn=fxn and xn+1=fyn. Then, we have(4)dxn,yn=dfyn−1,fxn≤a1dxn,yn−1+a2dxn,fxn+a3dfyn−1,yn−1=a1dxn,yn−1+a2dxn,yn+a3dxn,yn−1=a1+a3dxn,yn−1+a2dxn,yn,for all integers n≥1. Then,(5)dxn,yn≤a1+a31−a2dxn,yn−1,dxn,yn−1=dfyn−1,fxn−1≤a1dxn−1,yn−1+a2dxn−1,fxn−1+a3dfyn−1,yn−1=a1dxn−1,yn−1+a2dxn−1,yn−1+a3dxn,yn−1,so that(6)dxn,yn−1≤a1+a21−a3dxn−1,yn−1.
By setting(7)λ=a1+a31−a2,β=a1+a21−a3,we see that λ<1 and β<1 since a1+a2+a3=1.
Moreover, it is easy to see that(8)dxn,yn≤λ2ndx0,y0,dxn,yn−1≤β2n−1dx0,y0.
Hence, for all positive integersm and n, we have(1)
Ifm>n,(9)dxn,ym≤dxn,yn+dxn+1,yn+dxn+1,ym≤λ2n+β2n+1dx0,y0+dxn+1,ym⋮≤λ2n+β2n+1+⋯+β2mdx0,y0.(2)
Ifm<n,(10)dxn,ym≤dxm+1,ym+dxm+1,ym+1+dxn,ym+1≤β2m+1+λ2m+2dx0,y0+dxn,ym+1⋮≤β2m+1+λ2m+2+β2m+1+⋯β2ndx0,y0+dxn,yn≤β2m+1+λ2m+2+β2m+1+⋯β2n+λ2ndx0,y0.
Sinceλ<1 and β<1, this means that dxn,ym can be made arbitrarily small by larger m and n, and hence xn,ym is a Cauchy bisequence. Since X,Y,d is complete, xn,ym is convergent, and in fact biconvergent, since it is a convergent Cauchy bisequence. Let u be the point to which xn,ym biconverges. Then, xn⟶u, yn⟶u, and u∈X∩Y. Also, yn=fxn⟶Tu. Since yn has a limit in XcapY, this limit is unique. Hence, fu=u, and so f has a fixed point.
Ifv is any fixed point of f, then fv=v implies that v belongs to X∩Y. Then,(11)du,v=dTu,Tv≤a1dv,u+a2du,Tu+a3dTv,v=a1dv,u.
It is a contradiction, exceptu=v.Example 2.
LetX be the class of all singleton subsets of ℝ and Y be the class of all nonempty compact subsets of ℝ. We define d:X×Y⟶ℝ as(12)dx,A=x−infA+x+supA.
Then,X,Y,d is a complete bipolar metric space. The contravariant map f:X,Y,d⤨X,Y,d, defined as(13)fA=infA+supA+68⊂ℝ,for all A∈X∪Y satisfies inequality (3) with a1=11/15,a2=1/3,a3=2/5. Hence, f must have a unique fixed point, which in fact is the set 1∈X∩Y.
Now, we present some natural corollaries derived from Theorem1.Corollary 1 (Reich–Rus–Ćirić-type contraction).
LetX,Y,d be a complete bipolar metric space and f:X,Y,d⤨X,Y,d be a contravariant map such that for some α<1/3,(14)dfy,fx≤αdx,y+dx,fx+dfy,y,whenever x,y∈X×Y. Then, the function f:X∪Y⟶X∪Y has a unique fixed point.Proof.
Takea1=a2=a3=α in Theorem 1.Corollary 2 (Kannan-type contraction).
LetX,Y,d be a complete bipolar metric space and f:X,Y,d⤨X,Y,d be a contravariant map such that for some α<1/2,(15)dfy,fx≤αdx,fx+dfy,y,whenever x,y∈X×Y. Then, the function f:X∪Y⟶X∪Y has a unique fixed point.Proof.
Takea1=0,a2=a3=α in Theorem 1.
We conclude this section with the following natural extension of Theorem1. The result came from the observation that(16)dfy,fx≤a1dx,y+a2dx,fx+a3dfy,y≤a1+a2+a3maxdx,y,dx,fx,dfy,y,i.e.,(17)dfy,fx≤a1+a2+a3maxdx,y,dx,fx,dfy,y.Theorem 2.
LetX,Y,d be a complete bipolar metric space and f:X,Y,d⤨X,Y,d be a contravariant map such that there exists 0<λ<1 so that(18)dfy,fx≤maxλdx,y,dx,fx,dfy,y,whenever x,y∈X×Y. Then, the function f:X∪Y⟶X∪Y has a unique fixed point.Proof.
The proof follows closely the steps of the proof of Theorem1.
Letx0∈X; for each nonnegative integer n, we define yn=fxn and xn+1=fyn. Then, we have(19)dxn,yn=dfyn−1,fxn≤λmaxdxn,yn−1,dxn,fxn,dfyn−1,yn−1=λmaxdxn,yn−1,dxn,yn,dxn,yn−1=λdxn,yn−1.
Similarly,(20)dxn,yn−1=dfyn−1,fxn−1≤λmaxdxn−1,yn−1,dxn−1,fxn−1,dfyn−1,yn−1=λmaxdxn−1,yn−1,dxn−1,yn−1,dxn,yn−1=λdxn−1,yn−1.
Therefore, it is easy to see that(21)dxn,yn≤λ2ndx0,y0,dxn,yn−1≤λ2n−1dx0,y0.
The remaining part of the proof follows from that of Theorem1.


## 3. Main Results

In this section, we present the generalization of the Reich-type contraction proved in the previous section. As we have pointed out in the introduction, the contractivity condition we will employ is quite general. Our framework is that of bipolar metric spaces. In the what comes next, we introduce the following fundamental definition.Definition 6.
LetX,Y,d be a bipolar metric space, T:X,Y,d⤨X,Y,d be a contravariant map, and α=a1,a2,a3,a1,a2,a3≥0 be constants. Then, T is called an α,BK-contraction if there exist two functions ψ,ϕ:0,∞⟶ℝ:(22)ψdTy,Tx≤ϕa1dx,y+a2dx,Tx+a3dTy,y,whenever x,y∈X×Y and dTy,Tx>0.For simplicity, we shall put(23)BKTα,x,y=α1dx,y+α2dx,Tx+α3dTy,y,where α=α1,α2,α3 so that inequality (22) amounts to(24)ψdTy,Tx≤ϕBKTx,y,whenever x,y∈X×Y and dTy,Tx>0. In the first part of this section, we shall study, in relation with the functions ψ and ψ, the existence of a fixed point for an α,BK-contraction with respect to the sign of BKTx,y.We begin with the following proposition.Proposition 3.
LetX,Y,d be a bipolar metric space and T:X,Y,d⤨X,Y,d be a contravariant map. Suppose that there are two distinct points x0∈X,y0∈Y such that BKTα,x0,y0=0, where BKT is defined in (23). Then, α1=0, and at least one of the following three statements holds:(1)
αi=0 for all i∈1,2,3. Then, BKTα,x,y=0 whenever x,y∈X×Y.(2)
α2≠0 and x0 is a fixed point of T.(3)
α3≠0 and y0 is a fixed point of T.Proof.
(1)
Ifαi=0 for all i∈1,2,3, then the first case holds.(2)
By the way of contradiction, assume that someαi is distinct from zero. Since BKTα,x0,y0=0 and αi≥0 for all i∈1,2,3, then(25)α1dx0,y0=α2dx0,Tx0=α3dTy0,y0=0.Sincedx0,y0>0, then necessarily α1=0 (observe that this is already an hypothesis). If α2≠0, then dx0,Tx0=0, so x0 is a fixed point of T and the second case holds.(3)
Ifα2=0 and α3≠0, then dTy0,y0=0, so y0 is a fixed point of T and the third case holds.
From the above discussion, the following corollaries arise naturally.Corollary 3.
LetX,Y,d be a bipolar metric space, T:X,Y,d⤨X,Y,d be a contravariant map, and BKTα,x,y be as defined in (23). Then, the following hold:(1)
Ifα1>0, then BKTα,x,y>0 for all distinct points x,y∈X×Y(2)
Ifα2>0 and there are x0∈X,y0∈Y such that BKTα,x0,y0=0, then x0 is a fixed point of T(3)
Ifα3>0 and there are x0∈X,y0∈Y such that BKTα,x0,y0=0, then y0 is a fixed point of T(4)
Ifα1+α2+α3>0, then either T admits a fixed point or BKTα,x,y>0 for all distinct points x,y∈X×Y
The two preceding results give us a measure of the cardinality ofFT=x∈X∪Y:Tx=x, although inequality (22) could be satisfied. Indeed, if we let X=x0,Y=y0, where x0≠y0, dx,y=2, and let us define T:X,Y,dX,Y,d by Tx0=y0 and Ty0=x0. Then, T is fixed-point-free. However, if α1=α2=α3=0, then BKTα,x0,y0=0.Corollary 4.
LetX,Y,d be a bipolar metric space, T:X,Y,d⤨X,Y,d be a contravariant map, and BKTα,x,y be as defined in (23). Suppose that T is fixed-point-free. If α1+α2+α3>0, then BKTα,x,y>0 for all distinct x,y∈X×Y.Remark 2.
Whenα1=0, the equality BKTα,x0,y0=0 implies that x0 or y0 depending on whether α2≠0 or α3≠0. Hence, an idea to guarantee that BKTα,x,y>0 for all for all x,y∈X×Y such that dTy,Tx>0 follows from the assumption that α1≠0.
We have described the cautions to be observed when applying contractivity condition (22). In the last part of this section, we give and prove the two main theorems of this paper.Theorem 3.
LetX,Y,d be a bipolar metric space, T:X,Y,d⤨X,Y,d be a contravariant map, and α=α1,α2,α3,α1,α2,α3≥0 be constants. Let T be an α,BK-contraction, where the two functions ψ,ϕ:0,∞⟶ℝ are such that ϕt<ψt for any t>0 and ψ is nondecreasing and 0<α1≤1. Then, T admits at most one fixed point.Proof.
Observe that for the contravariant mapT, any fixed point x∗ necessarily belongs to X∩Y. Indeed if x∗∈X is a fixed point for T, then x∗∈X and x∗=Tx∗∈Y and similarly if we assume that x∗∈Y. For uniqueness of the fixed point, in case of existence, suppose that T admits two distinct fixed points, that is, there are a,b∈X∩Y such that Ta=a≠b=Tb. Then, da,b>0 and(26)BKTα,a,b=α1da,b+α2da,Ta+α3dTb,b=α1da,b.
Therefore,BKTα,a,b>0 because α1>0 and da,b>0. Hence, condition (22) can be applied because dTb,Ta=dTa,Tb>0, and it guarantees that(27)ψdTb,Ta≤ϕBKTα,a,b.
Since0<α1≤1 and using the hypotheses on ψ and ϕ, one has(28)ψdTb,Ta=ψdTa,Tb=ψda,b≤ϕBKTα,a,b=ϕα1da,b≤ψα1da,b≤ψda,b,which is a contradiction. Hence, we can conclude that T admits, at most, a unique fixed point.
In the following result, the uniqueness of the fixed point will be deduced from Theorem3.Theorem 4.
LetX,Y,d be a bipolar metric space, T:X,Y,d⤨X,Y,d be a contravariant map, and α=a1,a2,a3,a1,a2,a3≥0 be constants. Let T be an α,BK-contraction, where the two functions ψ,ϕ:0,∞⟶ℝ are such that ϕt<ψt for any t>0 and ψ is nondecreasing and 0<α1+α2+α3<1. Then, T admits exactly one fixed point.


## 4. Conclusion

In this paper, using the new framework of bipolar metric spaces, we introduced the concept ofα,BK-contractions and proved the existence of fixed points for contravariant mappings. It was pointed out that α,BK-contractions are a special extension of Reich-type contractions for which the existence of fixed point need not follow the traditional routine. An interesting future study could then be to look at the implications of these results in the induced metric space. We might get interesting feedback. Moreover, one could also be interested in the formulation of α,BK-contractions for covariant maps.



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