# On Two Banach-Type Fixed Points in Bipolar Metric Spaces

Authors: Yaé Ulrich Gaba; Maggie Aphane; Vizender Sihag
Journal: Abstract and Applied Analysis
 (2021)

---

## Abstract

In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.


---

## Body

## 1. Introduction and Preliminaries

Metric fixed point theory deals with the existence of a pointx∗ such that Tx∗=x∗ where T is a self-mapping defined on a metric space X,m. The notion of metric space has many generalizations in literature. One of the most recent is that of bipolar metric space which was introduced by Mutlu and Gürdal [1]. They explored the link between metric spaces and bipolar metric spaces and proved some well-known fixed point theorems in that new setting. Mutlu et al. [2] also proved coupled fixed point theorems in the setting of bipolar metric spaces. In [3], they also studied fixed point theorems for multivalued mappings on bipolar metric spaces. Finding fixed points for different type of contractions has become the focus of fruitful research activity, and numerous fixed point results in that theory can be found in the literature. Recently, many investigators have published various papers on the fixed point in bipolar metric spaces, and these can be read in [4–13] and the references therein, just to name a few. Bartwal et al. [14] even initiated the concept of fuzzy bipolar metric spaces. An account of the theory bipolar metric spaces can be read in detail in the paper by Mutlu and Gürdal [1]. Recent work in fixed-point characterization of completeness in bipolar metric spaces can be read in [15].In the present paper, we extend certain fixed point theorems, which we think are genuine generalizations of the Banach fixed point theorem. A similar work has been done by Gaba [16]. We give examples which present the applicability of our obtained results. In concluding this introductory part, we recall the definitions of some fundamental notions related to bipolar metric spaces.Definition 1 (see [1], Definition 8).
A bipolar metric space is a tripleX,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
Moreover,d is called a bipolar metric on the pair X,Y.Definition 2 (see [1], Definition 2.3).
LetX1,Y1 and X2,Y2 be pairs of sets and given a function f:X1∪Y1⟶X2∪Y2:
(1)
IffX1⊆X2 and fY1⊆Y2 , we call f a covariant map from X1,Y1 to X2,Y2 and denote this with f:X1,Y1⇉X2,Y2(2)
IffX1⊆Y2 and fY1⊆X2 , we call f a contravariant map from X1,Y1 to X2,Y2 and denote this with f:X1,Y1⇌X2,Y2(3)
Moreover, ifd1 and d2 are bipolar metrics on X1,Y1 and X2,Y2, respectively, the following notations are self-explanatory: f:X1,Y1,d1⇉X2,Y2,d2 and f:X1,Y1,d1⇌X2,Y2,d2Definition 3 (see [1], Definition 4.1).
LetX,Y,d be a bipolar metric space.
(1)
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(2)
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(3)
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(4)
A bisequencexn,yn is a Cauchy bisequence, if limn⟶∞,m⟶∞dxn,ym=0Remark 4 (see [1], Proposition 4.3).
In a bipolar metric space, every convergent Cauchy bisequence is biconvergent.Definition 5 (see [1], Definition 4.4).
A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.Example 6 (see [1], Example 5.7).
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
(1)dx,A=∣x−infA∣+x−supA.
Then, the tripleX,Y,d is a complete bipolar metric space.Definition 7 (see [1], Definition 3.4).
A covariant or a contravariant mapf 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⟶fvonX2,Y2,d2.


## 2. Some Interesting Facts around Banach Fixed Point

In this section, we show that ifT is a self-map on a bipolar metric space X,Y,b and has a power which is a contraction, i.e., there exists n∈ℕ,n>1 and 0≤λ<1 such that
(2)bTnx,Tny≤λbx,y,then there is a transformation another bipolar metric b′ such that T a contraction on X,Y,b′.We introduce it with the following definitions.Definition 8.
Two bipolar metricsb1 and b2 on a set X×Y are said to be equivalent if there exist α,β≥0 such that
(3)αb1x,y≤b2x,y≤βb1x,y,forallx,y∈X×Y.Theorem 9 (see [1], Theorem 5.1).
LetX,Y,b be a complete bipolar metric space and covariant contraction f:X,Y,b⇉X,Y,b a covariant contraction. Then, the function f:X∪Y⟶X∪Y has a unique fixed point.Theorem 10 (see [1], Theorem 5.2).
LetX,Y,b be a complete bipolar metric space given a contravariant contraction f:X,Y,b⇌X,Y,b. Then, the function f:X∪Y⟶X∪Y has a unique fixed point.We give the following natural corollary, a consequence of the above theorems:Corollary 11.
LetX,p be a complete bipolar metric type space.
Assume thatf1:X,Y,b⇉X,Y,b and f2:X,Y,b⇌X,Y,b be mappings such that there exists λ,β∈0,1 satisfying
(4)bf1nx,f1ny≤λbx,y,bf2mx,f2my≤βbx,y,for some n,m>1, whenever x,y∈X×Y. Then, the functions f1:X∪Y⟶X∪Y and f2:X∪Y⟶X∪Y have a unique fixed point.Proof.
By Theorem9, f1n has a unique fixed point, say x∈X∩Y with f1nx=x. Since
(5)f1n+1x=f1f1nx=f1x=f1nf1x,it follows that f1x is a fixed point of f1n, and thus, by the uniqueness of x, we have f1x=x, that is, f1 has a fixed point. Since, the fixed point of f1 is necessarily a fixed point of f1n, so it is unique.
By Theorem10, and using the same reasoning as above, one can easily see that f2 has a unique fixed point.☐Moving in the same line as the Banach contraction, we introduce the following.Definition 12.
LetX,Y,b be a complete bipolar metric space and f:X,Y,b⇉X,Y,b a covariant contraction. f is called Lipschitzian if there exists a constant λ≥0 such that
(6)bfx,fy≤λbx,y,whenever x,y∈X×Y. The smallest constant λ verifying (6) will be denoted Lipf.Definition 13.
LetX,Y,b be a complete bipolar metric space and A⊆X×Y. The subset A is said to be b-bounded, if there exists M>0 such that
(7)bx,y≤M,
wheneverx,y∈A⊆X×Y.Theorem 14.
LetX,Y,b be a complete bipolar metric space and f:X,Y,b⇉X,Y,b a covariant map such that fn is Lipschitzian for any n≥0 and limn⟶∞Lipfn=0. Then, f has a unique fixed point ω∈X∩Y if and only if there exists x0,y0∈X×Y such that bfix0,fjy0,i,j≥0 is b-bounded. Moreover, if f has a fixed point ω, then for any x0,y0∈X×Y, the bisequence fnx0,fny0,n≥1 is biconvergent to ω.Proof.
Clearly, iff has a fixed point, say ω∈X∩Y, then the subset fnω,fmω,n,m≥1=ω,ω is b-bounded.
Conversely, letx0,y0∈X×Y such that A=fix0,fjy0,i,j≥0 is b-bounded, that is, there exists M≥0 such that
(8)bfix0,fjy0≤M,for any i,j≥0.
Lettingn,h≥0, we have
(9)bfn+hx0,fny0≤Lipfnbfhx0,y0≤LipfnM.
Sincelimn⟶∞Lipfn=0, then fnx0,fny0 is a Cauchy bisequence. Hence, fnx0 converges to some point ω∈X∩Y since X,Y,b is complete. fny0=yn+1⟶ω∈X∩Y guarantees that fny0 has a unique limit. Since the map f is continuous, xn⟶ω implies that fxn⟶fω and combining this with xn⟶ω gives fω=ω. Hence, ω is a fixed point of f.☐We are now in position to state the first main result of this paper.Theorem 15.
LetX,Y,b be a complete bipolar metric space and given a covariant contraction f:X,Y,b⇉X,Y,b be a self mapping such that
(10)bfnx,fny≤Kbx,y,
for somen>1 and 0<K<1, whenever x,y∈X∪Y. If λ is a nonnegative real such that
(11)K1/n<1λ<1,then the application b′:X×Y⟶0,∞ defined by
(12)b′x,y=∑i=0n−1λibfix,fiy,wheneverx,y∈X×Y,satisfies the following:
(i)
b′ is a bipolar metric on the space X×Y(ii)
T:X,Y,b′⟶X,Y,b′ a self-mapping such that(13)b′fx,fy≤1λb′x,y,that is, f is a contraction with constant 1/λ with respect to b′.Proof.
We first prove thatb′ is a bipolar metric:
(b0) Forx,y∈X×Y, if b′x,y=0, then bfix,fiy=0 for i=0,⋯,n. In particular, for i=0, we have bx,y=0, which implies that x=y since b is a bipolar metric.
(b1) Forx,y∈X×Y, if x=y, then fix=fiy and then bfix,fiy=0, i.e.,
(14)b′x,y=0.
(b2) Forx,y∈X×Y,(15)b′x,y=∑i=0n−1λibfix,fiy=∑i=0n−1λibfiy,fix=b′y,x,that is,
(16)b′x,y=b′y,x,for all x,y∈X×Y.
(b3) For allx1,x2∈X and y1,y2∈Y, since
(17)λibfix1,fiy2≤λibfix1,fiy1+bfix2,fiy1+bfix2,fiy2,we get
(18)b′x1,y2=∑i=0n−1λibfix1,fiy2≤∑i=0n−1λibfix1,fiy1+bfiy1,fix2+bfix2,fiy2=∑i=0n−1λibfix1,fiy1+bfix2,fiy1+bfix2,fiy2=b′x1,y1+b′fix2,y1+b′x2,y2.
So (b3) holds.
Hence,b′ is a bipolar metric on X×Y.
We now prove thatf:X,Y,b′ÃX,Y,b′ is a covariant contraction with constant 1/λ.
It is readily seen, by a simple computation, that(19)b′fx,fy=1λb′x,y−bx,y+λn−1bfnx,fny.
Sincefn:X,Y,bÃX,Y,b is a covariant contraction with constant K, it follows that
(20)b′fx,fy≤1λb′x,y−bx,y+Kλn−1bx,y=1λb′x,y+K−1λnλn−1bx,y≤1λb′x,y,because of the choice K1/n<1/λ. This completes the proof.☐Remark 16.
The mapb′ can be thought of as an approximation of order n−1 of a certain bipolar metric h, equivalent to b′. Indeed, under the assumptions of Theorem 15, it is readily seen that
(21)b′x,y≤∑i=0∞λibfix,fiy≤b′x,y+λnKb′x,y+λ2nK2b′x,y+⋯=11−λnKb′x,y.The sumhx,y≔∑i=0∞λibfix,fiy therefore defines a bipolar metric h, equivalent to b′, as long as the series happen to converge for some λ>1. Moreover, whenever h is finite, the map f:X,Y,bÃX,Y,b is a contraction with contraction constant 1/λ.We introduce the following notion of completeness, namely, theB-completeness which seems quite more natural than the one introduced by Definition 4.4 of [1]. However, these two notions are equivalent, and to see this, it is enough to look at Proposition 4.3 of [1]. The main idea is that in a bipolar metric space, every convergent Cauchy bisequence is biconvergent and every biconvergent bisequence is a Cauchy bisequence (see Proposition 4.2 of [1]). So we rather express completeness in terms of biconvergent Cauchy bisequences.Definition 17 (compare [1], Definition 4.4).
A bipolar metric space is calledB-complete, if every Cauchy bisequence in this space is biconvergent.Remark 18.
As already observed by Mutlu and Gürdal [1], in bipolar metric spaces, the notion of completeness is defined via bisequences, rather than sequences. In the present manuscript, we observe that the newly introduced concept of completeness, namely, the B-completeness, is equivalent to the notion of completeness in metric spaces, when a bipolar metric space X,Y,b represents a metric space, that is, X=Y.Proposition 19.
A metric spaceX,b is complete if and only if the corresponding bipolar metric space X,X,d is B-complete.Proof.
LetX,b be a complete metric space and xn,yn be a Cauchy bisequence on X,X,b. By the fact that xn,yn is a Cauchy bisequence on X,X,b, we know that
(22)limn⟶+∞,m⟶+∞bxn,ym=limn⟶+∞,m⟶+∞bym,xn=0.
Following the proof of Proposition 4.5 of [1], it is easy to see that xn and yn are Cauchy sequences in X,b, and since X,b is complete, there exist x∗,y∗∈X such that
(23)limn⟶+∞bxn,x∗=0=limn⟶+∞by∗,yn.
Since(24)by∗,x∗≤by∗,yn+byn,xn+bxn,x∗,
taking the limit asn⟶∞, we get that
(25)by∗,x∗=0,i.e.x∗=y∗.
Therefore, the Cauchy bisequencexn,yn is biconvergent. Hence, the bipolar metric space X,X,b is B-complete.
Conversely, assume that the bipolar metric spaceX,X,b is B-complete. Let X,b be the metric space, such that the corresponding bipolar metric space is X,X,b. Let xn be a Cauchy sequence in X,b. Then, since dxn,xm can be made arbitrarily small, the bisequence xn,ym=xn,xm is a Cauchy bisequence on the complete bipolar metric space X,X,b, so it is biconvergent. So there exist x∗∈X such that
(26)limn⟶+∞bxn,x∗=0.
Therefore, the Cauchy sequencexn is convergent. Hence, the metric space X,b is complete.☐In order to move forward, we introduce the following notion ofuniform continuity.Definition 20.
Given two bipolar metric type spacesX1,Y1,b1 and X2,Y2,b2 , we say that f:X1,Y1,b1⇉X2,Y2,b2 is uniformly continuous if for every real number ε>0, there exists δ=δε>0 such that for every x∈X1,y∈Y1 with b1x,y<δ, we have that b2fx,fy<ε.Next, we establish that whenever the mappingf:X,Y,bÃX,Y,b is uniformly continuous and the bipolar metric b is complete, then so is the bipolar metric b′.Theorem 21.
We repeat the assumptions of Theorem15. If f is uniformly continuous and the bipolar metric b is B-complete, then so is the bipolar metric b′.Proof.
Since we know that(27)bx,y≤b′x,yforanyx,y∈X×Y,any Cauchy bisequence in X,Y,b′ is also a Cauchy bisequence in X,Y,b. It is therefore enough to prove that, under uniform continuity of f in X,Y,b, any biconvergent bisequence in X,Y,b is also biconvergent in X,Y,b′.
So letxn,yn be a biconvergent bisequence in the bipolar metric space X,Y,b such that xn right converges to some ξ∈X,Y,b,ξ∈X∩Y and yn left converges to the same ξ. Set M=maxλi,i=1,⋯,n−1 and observe that
(28)M≥λ>1.
Since all the powers off are also uniformly continuous in X,Y,b, we can write that, for any ε>0, there exists η>0 such that for all x∈X,y∈Y, and i=1,⋯,n−1,
(29)bx,y<η⇒dTix,Tiy<εMn.
Sinceyn left converges to some ξ∈X,Y,d, there exists n0∈ℕ such that
(30)p≥n0⇒bξ,xp<η.
Then,(31)p>n0⇒bfiξ,fiyp<εMnfori=1,⋯,n−1,i.e.,
(32)b′ξ,yp<εn1M+λM+⋯+λn−1M<ε.
Thus,yn right converges to ξ with respect to the bipolar metric b′.
Similarly, sincexn right converges to ξ, one can show that xn right converges to ξ with respect to the bipolar metric b′.
In conclusion, the bisequencexn,yn is biconvergent in the bipolar metric space X,Y,b′, i.e., the bipolar metric space X,Y,b′ in B-complete.
This completes the proof.☐We conclude this section by an example, illustrating the importance of the requirement forT to be uniformly continuous.Example 22.
LetX=0,3 and let X×X be endowed with the partial metric b, defined as
(33)bx,y=x−yforallx,y∈0,32.
Observe that for0,3∪0,3=0,3, every mapping f:0,3∪0,3⟶0,3∪0,3 is as f:0,3⟶0,3.
DefineT:0,3⟶0,3 by
(34)Tx=1,if0≤x≤2,2,if2<x≤3.
The mappingT is covariant T:0,3,0,3,‖⇌0,3,0,3,‖, since T0,3⊂0,3. It is discontinuous at x=2, but T2x=1 for all x,y∈0,3, i.e., T2 is a contraction and the unique fixed point is x=1. Moreover, any real K∈0,1 can be used as contraction constant for T2. We can then apply Theorem 15 with any λ such that λ>1. From Theorem 15, the bipolar metric b′ is given by
(35)b′x,y=dx,y,ifx,y≤2orx,y>2,dx,y+λ,otherwise.
The bipolar metricb′ is not B-complete. Indeed, the bisequence xn,yn given by xn=2+1/n=yn is bi-Cauchy in X,b′ but is not biconvergent, as the only candidate for a limit is 2 but
(36)b′xn,yn=1n+λ>λforalln.


## 3. Generalizations of Banach Fixed Point Theorem

The aim of this section is to propose a generalization of the Banach fixed point result via a Caristi-type argument. We begin by this first technical lemma.Lemma 23.
LetX,Y,d be a bipolar metric space and consider a bisequence xn,yn. If there exists one constant γ∈0,1 such that
(37)dxn,yn+1≤γdxn−1,yn;dxn+1,yn+1≤γdxn,yn;dxn+1,yn≤γdxn,yn−1,∀n≥0,then xn,yn is a Cauchy bisequence.Proof.
The following proof is inspired by the first part of the proof of Theorem 5.1 of [1].
Sincedxn+1,yn+1≤γdxn,yn, it is obvious that
(38)dxn+1,yn+1≤γndx0,y0.
Sincedxn,yn+1≤γdxn−1,yn+1, it is obvious that
(39)dxn+1,yn+1≤γndx0,y1.
Now let us setM=dx0,y0+dx0,y1. For n,p∈ℕ, we have
(40)dxn+p,yn≤dxn+p,yn+1+dxn,yn+1+dxn,yn≤dxn+p,yn+1+γnM≤dxn+p,yn+2+dxn+1,yn+2+dxn+1,yn+1+γnM≤dxn+p,yn+2+γn+1+γnM⋮≤dxn+p,yn+p+γn+p−1+⋯+γn+1+γnM≤γn+p+⋯+γn+1+γnM≤γnM∑i=0∞γi=γnM1−γ.
Hence,(41)dxn+p,yn≤γnM1−γ.
Similarly, one can establish that(42)dxn,yn+p≤γnM1−γ.
Since0≤γ<1, for any ε>0, there exists n0∈ℕ such that
(43)γn0M1−γ<ε3.
Then,(44)dxn,ym≤dxn,yn0+dxn0,yn0+dxn0,ym≤3γn0M1−γ<ε,and xn,yn is a Cauchy bisequence.☐We are now ready to give the first Banach contraction principle (BCP) generalization.Theorem 24.
LetX,Y,d be a B-complete bipolar metric space with X∩Y≠∅ and T:X,Y,d⇉X,Y,d be a covariant map. Suppose that there exists a function ϕ:X∪Y⟶ℝ satisfying
(i)
ϕ is bounded from below(ii)
dTx,Ty>0impliesdTx,Ty≤ϕx−ϕTxdx,y, whenever x,y∈X×Y
Then,T has at least one fixed point in X∪Y.Proof.
Letx0∈X,y0∈Y. Define the bisequence xn,yn by xn+1=Txn,yn+1=Tyn.
We prove that the bisequencexn,yn is Cauchy.
(1)
If there existn0,m0 such that dxn0,ym0=0, then xn0=ym0 and(45)Txn0=xn0+1=ym0+1=Tym0,i.e.,dxn0+1,ym0+1=0.
Therefore,(46)limn⟶∞,m⟶∞dxn,ym=0.
And the bisequencexn,yn is Cauchy.
(2)
Now, assume that for alln≥0,m≥0,dxn,ym>0. Owing to (ii), we derive that(47)dxn+1,yn+1=dTxn,Tyn≤ϕxn−ϕTxndxn,yn≤ϕxn−ϕxn+1dxn,yn.
So we have(48)0<dxn+1,yn+1dxn,yn≤ϕxn−ϕxn+1,whenevern∈ℕ.
Thus, the sequenceϕxn is positive and nonincreasing. Hence, it converges to some l≥0. Observe that for each n∈ℕ, we have
(49)∑i=1ndxi+1,yi+1dxi,yi≤∑k=1nϕxi−ϕxi+1=ϕx0−ϕxn⟶ϕx0−l<∞asn⟶∞.
The above entails that(50)∑i=1∞dxi+1,yi+1dxi,yi<∞,limn⟶∞dxn+1,yn+1dxn,yn=0.
It follows, that forδ∈0,1, there exists n0∈ℕ such that
(51)dxn+1,yn+1≤δdxn,yn∀n≥n0.
Similarly, one can prove that forδ∈0,1, there exists n1∈ℕ such that
(52)dxn,yn+1≤δdxn−1,yn∀n≥n1,dxn+1,yn≤δdxn,yn−1∀n≥n2.
Now using Lemma23, we obtain that the bisequence xn,yn is Cauchy.
Since the bisequencexn,yn is Cauchy, it is biconvergent to some ω∈X∩Y. We claim that ω is the fixed point of T. Assuming that dω,Tω>0 and employing assumption (ii) of the theorem, we find that
(53)dω,Tω≤dω,yn+dxn+1,yn+dxn+1,Tω≤dω,yn+dxn+1,yn+ϕxn−ϕTωdxn,ω⟶0asn⟶∞.
Consequently, we obtaindω,Tω=0, that is, Tω=ω.☐Example 25.
LetX=0,1,2,Y=0,2 and let us endow X×Y with the following bipolar metric:
(54)d0,1=1=d1,0,d0,2=1=d2,0,d1,2=32=d2,1,da,a=0,foralla,a∈X×Y.
It is easy to see thatX,Y,d is a B-complete bipolar metric space because any Cauchy bisequence xn,yn is eventually constant, i.e., there exists n0∈ℕ such that for n≥n0, xn,yn=a,a∈X×Y for some a∈X∩Y. Hence, it is biconvergent.
LetT:X,Y,d⇉X,Y,d defined by T0=0,T1=2,T2=0. Define ϕ:X⟶0,∞ as ϕ2=2,ϕ0=0,ϕ1=4.T is a covariant map, and for all x,y∈X×Y,
(55)dTx,Ty>0⇐x,y∈1,0,1,2.
We have(56)dT1,T0=d2,0=1≤2=ϕ1−ϕT1d1,0,dT1,T2=d2,0=1≤3=ϕ1−ϕT1d1,2.
Thus, the mappingT satisfies our conditions and also has a fixed point.
Note thatdT1,T0=d2,0=1=d1,0; thus, T does not satisfy the Banach contraction principle.Remark 26.
In a few words, Example25 tells us that Theorem 24 is not a consequence of the Banach contraction principle on complete bipolar metric spaces. It is therefore very natural to ask if the reverse holds, i.e., if the Banach contraction principle is a consequence of Theorem 24 on complete bipolar metric spaces. The authors plan to look at that question in another study.Here goes the second Banach contraction principle (BCP) generalization.Theorem 27.
LetX,Y,d be a B-complete bipolar metric space with X∩Y≠∅ and T:X,Y,d⇉X,Y,d be a continuous covariant map. If there exists a function θ:0,∞⟶0,∞ such that limt⟶0+θt=0, and θ0=0 satisfying the additional condition:
(57)dTx,Ty≤θdx,y−θdTx,Ty,whenever x,y∈X×Y, then T has a unique fixed point in X∪Y.Proof.
We follow similar steps as in the proof of Theorem24. Let x0∈X,y0∈Y. Define the bisequence xn,yn by xn+1=Txn,yn+1=Tyn. For any n∈ℕ,(58)0≤dxn+1,yn+1=dTxn,Tyn≤θdxn,yn−θdxn+1,yn+1.
Hence, the sequenceθdxn,xn+1 is nonincreasing of nonnegative numbers. Since θ is bounded below, there exists K≥0 such that limn⟶∞θdxn,xn+1=K.
We can then write that(59)0≤limn⟶∞dxn+1,yn+1≤K−K=0,i.e.,
(60)limn⟶∞dxn+1,yn+1=0.
Similarly, one can see that(61)limn⟶∞dxn,yn+1=0=limn⟶∞dxn+1,yn.
So the bisequencexn,yn is Cauchy and then is biconvergent to some u∈X∩Y, and Tyn=yn+1⟶u∈X∩Y guarantees that Tyn has a unique limit. Since the map T is continuous, xn⟶u implies that Txn⟶Tu and combining this with xn⟶u gives Tu=u. Hence, u is a fixed point of T.
Ifv is any fixed point of T, then Tv=v implies that v∈X∩Y and we have, by (57),
(62)du,v=dTu,Tv≤θdu,v−θdTu,Tv=0⇒du,v=0,
and sou=v.☐Remark 28.
Note that Theorem27 is indeed a generalization of the Banach contraction principle (see [1], Theorem 5.1.) over B-complete bipolar metric spaces, as we demonstrate it in the coming lines.IfT:X,Y,d⇉X,Y,d is a Banach contraction, there exists k∈0,1 such that
(63)dTx,Ty≤kdx,y,forallx,y∈X×Y.Hence,(64)dTx,Ty≤kdx,y≤k1+k−kdx,y,forallx,y∈X×Y.Consequently,(65)1−kdTx,Ty≤kdx,y−kdTx,Ty,forallx,y∈X×Y.This yields(66)dTx,Ty≤k1−kdx,y−k1−kdTx,Tyforallx,y∈X×Y.By choosing the linear function(67)ξt=k1−kt,we can easily rewrite the above as
(68)dTx,Ty≤ξdx,y−ξdTx,Tyforallx,y∈X×Y,which is of the form (57).Moreover, settingθt=texpt, we deduce the following corollary.Corollary 29.
LetX,Y,d be a B-complete bipolar metric space with X∩Y≠∅ and T:X,Y,d⇉X,Y,d be a continuous covariant map. If there exists a function θ:0,∞⟶0,∞ such that limt⟶0+θt=0 and θ0=0 satisfying the additional condition:
(69)dTx,Ty1+expdTx,Tydx,yexpdx,y≤1,whenever x,y∈X×Y,x≠y. Then, T has a unique fixed point in X∪Y.Example 30.
Let(70)X=xi=ii+12,i=1,2,⋯,Y=xi=ii+12,i=2,3,⋯,and let us endow X×Y with the following bipolar metric:
(71)dxi,yj=∣xi−yj∣∀xi,yj∈X×Y.
LetT:X,Y,dÃX,Y,d be defined by
(72)Tx1=x1,Txi=xi−1fori≥2,Tyj=yi−1forj≥2.We need only check the following two cases.Case 1.
x=x1 and y=yj,j≥2, so
(73)dx,y=1−yj,dTx,Ty=1−yj−1.
We have(74)dTx,Ty1+expdTx,Tydx,yexpdx,y=yj−1−11+expyj−1−1yj−1expyj−1=jj−1/2−11+expjj−1/2−1jj+1/2−1expjj+1/2−1≤2expjj−1/2−1expjj+1/2−1<2exp−j≤1.Case 2.
x=xi,i≥2, and y=yj,j≥2, and so
(75)dx,y=xi−yj,dTx,Ty=xi−1−yj−1.
One could observe that fori≥2,j≥2,xi=yi and without loss of generality, we may assume that j≥i. So we get
(76)dTx,Ty1+expdTx,Tydx,yexpdx,y=yj−1−xi−11+expyj−1−xi−1yj−xiexpyj−xi=jj−1/2−ii−1/21+expjj−1/2−ii−1/2jj+1/2−ii+1/2expjj+1/2−ii+1/2≤2j+i−1j+i+1expjj−1/2−ii−1/2expjj+1/2−ii+1/2<2expi−j≤2exp−1≤1.So, by Corollary29, T has a unique fixed point. Here, Tx1=x1.We point out that the mappingT is not a Banach contraction. Indeed, just observe that
(77)supdTx1,Tyjdx1,yj=supyj−1−1yj−1=supjj−1/2−1jj+1/2−1=1.


## 4. Conclusion and Going Further

In this paper, we introduced new generalizations/variants of the Banach contraction principle using Caristi-type arguments in the setting of bipolar metric spaces. We plan to investigate how these generalizations can be powerful tools in finding solutions of integral differential equations, for instance. Another interesting line of research could be to think of the multivalued version of the new contraction considered in the present manuscript, as this could provide a powerful tool for the existence solution of Volterra-integral inclusions. Furthermore, it is our opinion that the conclusions of Theorem21 and the subsequent results remain valid if we consider contravariant mappings. The authors plan to take up this investigation in another manuscript. The contravariant case appears to be more easy but contains, however, some tricky steps than need to be taken care of wisely.



---

*Source: 4846877-2021-08-13.xml*