# Common Best Proximity Point Theorems for Generalized Proximal Weakly Contractive Mappings in b-Metric Space

Authors: Yohannes Gebru; Yitages Negede
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, common best proximity point theorems for weakly contractive mapping in b-metric spaces in the cases of nonself-mappings are proved; we introduced the notion of generalized proximal weakly contractive mappings in b-metric spaces and proved the existence and uniqueness of common best proximity point for these mappings in complete b-metric spaces. We also included some supporting examples that our finding is more generalized with the references we used.


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## Body

## 1. Introduction

The metric fixed point theory gained impetus due to its wide range of applicability to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, and so forth. The first fixed point theorem was given by Brouwer [1], but the credit of making concept useful and popular goes to polish mathematician, Banach [2] who proved the famous contraction mapping theorem in 1922 in the setting of metric space. This principle guarantees the existence and uniqueness of fixed point of certain self-maps of metric spaces and provides a constructive method to find those fixed points. This principle includes different directions in different spaces adopted by mathematicians for example metric spaces, G-metric spaces, partial metric spaces, and cone metric spaces.A classical best approximation theorem was introduced by Fan [3], which states that “if A is a non-empty compact convex subset of a Hausdorff locally convex topological vector space B and T:A⟶B is a continuous mapping, then there exists an element x∈A such that dx,Tx=dTx,A.” Afterwards, Prolla [4], Reich [5], and Sehgal and Singh [6] have derived extensions of Fan Theorem in many directions. The common fixed point theorem insists to the authors to investigation on common best proximity point theorem for nonself-mappings. The common best proximity point theorem assures a common optimal solution at which both the real valued multiobjective functions x⟶dx,Sx and x⟶dx,Tx attain the global minimal value dA,B. A number of authors have improved, generalized, and extended this basic result either by defining a new contractive mapping in the context of a complete metric space or extend best proximity results from fixed point theory (see [7–12]).Definition 1.
LetX be a nonempty set and T:X⟶X a self-map. A point x∈X is said to be fixed point of T if Tx=x.Example 2.
LetX=ℝ and T:X⟶X defined by Tx=x/2, for each x∈X.
Tx=x⇒x/2=x, and we get x=0∈X, which is a fixed point of T.Definition 3 (see [13]).
A functionψ:0,∞⟶0,∞ is called an altering distance function if the following properties are satisfied:
(i)
ψ is monotone increasing and continuous(ii)
ψt=0 if and only if t=0Example 4.
Defineψ:0,∞⟶0,∞ by ψt=t2/2.
ψ′t=2t/2=t≥0, which shows ψ is nondecreasing, satisfies that ψt=0⇐t=0, and ψ is continuous.Definition 5 (see [14]).
LetX be a nonempty set, and a mapping d:X×X⟶0,∞ is said to be metric if and only if, for all x,y,z∈X, the following conditions are satisfied:
(i)
dx,y=0 if and only if x=y and dx,y>0 if x≠y(ii)
dx,y=dy,x(iii)
dx,y≤dx,z+dz,yExample 6.
LetX=ℝ; then, X,. that means dx,y=x−y, for all x,y∈X which is a metric space.Definition 7 (see [2]).
LetX,d be a metric space and T:X⟶X be a self-map; then, T is said to be a contraction mapping if there exists a constant k∈0,1, such that dTx,Ty≤kdx,y, ∀x,y∈X.Example 8.
LetX=ℝ, dx,y=x−y, and a mapping T:ℝ⟶ℝ defined by Tx=x/3, ∀x,y∈ℝ.
Then,dTx,Ty=Tx−Ty=x/3−y/3=1/3x−y≤1/3x−y=1/3dx,y, which implies that dTx,Ty≤1/3dx,y and k=1/3∈0,1, for all x,y∈X. Therefore, T is contraction mapping.Definition 9 (see [15]).
LetX,d be a metric space. The mapping T:X⟶X is said to be contractive mapping if
(1)dTx,Ty<dx,yforallx,y∈Xwithx≠y.Example 10.
LetX=ℝ and dx,y=x−y, and a mapping T:ℝ⟶ℝ defined by Tx=x/2, ∀x,y∈ℝ.
Then,dTx,Ty=Tx−Ty=x/2−y/2=1/2x−y≤1/2x−y<x−y, which implies that dTx,Ty<dx,y with x≠y.
Therefore,T is contractive mapping.Definition 11 (see[16]).
LetX,d be a metric space and T:X⟶X, a mapping T is said to be weakly contractive if
(2)dTx,Ty≤dx,y−ϕdx,y,forallx,y∈X,where ϕ:0,∞⟶0,∞ is altering function.Remark 12.
Ifϕt=1−kt with k∈0,1 and t∈0,∞, a weak contraction reduces to a contraction.Example 13.
LetX=0,∞ be endowed by dx,y=x−y, and let T:X⟶X define by Tx=x/1+x for each, x∈X.
Defineϕ:0,∞⟶0,∞ by ϕt=t2/1+t.
Claim:T is weakly contractive.
ϕ′t=2t+t2/1+t2≥0 which shows ϕ is nondecreasing and satisfies that ϕt=0⇐t=0 and ϕ is continuous.
Then,dTx,Ty=x/1+x−y/1+y=x−y/1+x1+y≤x−y/1+x−y=x−y−x−y2/1+x−y=dx,y−ϕdx,y, for all x,y∈X.
SoT is weakly contractive.Definition 14 (see [17]).
LetX be a nonempty set and s≥1 be a given real number. A mapping d:X×X⟶0,∞ is said to be a b-metric if and only if, for all x,y,z∈X, the following conditions are satisfied:
(i)
dx,y=0 if and only if x=y and dx,y>0 if x≠y(ii)
dx,y=dy,x(iii)
dx,y≤sdx,z+sdz,yRemark 15 (see [18]).
We should note that a b-metric space withs=1 is a metric space. We can find several examples of b-metric spaces which are not metric spaces.Example 16 (see [19]).
LetX,ρ be a metric space, and dx,y=ρx,yp, where p>1 is a real number. Then, dx,y is a b-metric space with
(3)s=2p−1.Definition 17 (see [20]).
LetX,d be a b-metric space with parameter s≥1. Then, a sequence {xn} in X is said to be
(i)
b-convergent if and only if there existsx∈X such that dxn,x⟶0 as n⟶∞(ii)
a b-Cauchy sequence if and only ifdxn,xm⟶0 as n,m⟶∞, for all n,m∈ℕ
In addition, a b-metric space is called complete if and only if each Cauchy sequence in this space is b-convergent.Example 18.
LetX=0,∞ and dx,y=x−y2; then, the space X,d is a complete b-metric space.Definition 19 (see [21]).
Letf and g be two self-mappings on a nonempty set X. If w=fx=gx, for some x∈X, then x is said to be the coincidence point of f and g, where w is called the point of coincidence of f and g. Let Cf,g denote the set of all coincidence points of f and g.Definition 20 (see [21]).
Letf and g be two self-mappings defined on a nonempty set X. Then, f and g are said to be weakly compatible if they commute at every coincidence point, that is, fx=gx⇒fgx=gfx, for every x∈Cf,g.Example 21.
(i)
f,g:R⟶R defined by fx=x/3 and gx=x2, x∈ℝ. In this example, f and g have coincidence point at x=0, and x=1/3 but f and g are not weakly compatible(ii)
X=0,3 equipped with the usual metric space dx,y=x−y
Definef,g:X⟶X by the following:
(4)fx=x,ifx∈0,1,3,ifx∈1,3,gx=3−x,ifx∈0,1,3,ifx∈1,3.
This example shows, for anyx∈1,3, fgx=gfx. Therefore, f and g are weakly compatible maps on 0,3.
In this study, motivated and inspired by Yan Hao and Hongyan Guan [22], we introduce the notion of generalized proximal weakly contractive mappings in b-metric spaces and prove a common best proximity point theorem for generalized proximal weakly contractive mapping defined on complete b-metric spaces.


## 2. Preliminaries

Definition 22 (see [23]).
LetA and B be nonempty subsets of a metric space X,d. We denote by A0 and B0 the following sets:
(5)A0=x∈A:dx,y=dA,B,forsomey∈B,(6)B0=y∈B:dx,y=dA,B,forsomex∈A,
wheredA,B=infdx,y:x∈A,y∈B is the distance between A and B.Definition 23 (see [24]).
LetA,B be nonempty subset of metric space X,d. Given a nonself-mapping T:A⟶B, then an element x∗∈A is called best proximity point of the mapping if
(7)dx∗,Tx∗=dA,B.Definition 24 (see [25]).
Letf,g:A⟶B be nonself-mappings. An element x⋆∈A is said to be a common best proximity point of the pair f,g if this condition is satisfied:
(8)dx⋆,fx⋆=dA,B=dx⋆,gx⋆.Definition 25 (see [26]).
Letf,g:A⟶B be mappings. A pair f,g is said to commute proximally if for each x,u,v∈A,
(9)du,fx=dv,gx=dA,B⇒fv=gu.Lemma 26 (see [19]).
LetX,d be a b-metric space with parameter s≥1. Assume that xn and yn are b-convergent to x and y, respectively. Then, we have the following:
(10)1s2dx,y≤liminfn⟶∞dxn,yn≤limsupn⟶∞dxn,yn≤s2dx,y.
In particular, ifx=y, then we have limn⟶∞dxn,yn=0.
Moreover, for eachz∈X, we have the following:
(11)1sdx,z≤liminfn⟶∞dxn,z≤limsupn⟶∞dxn,z≤sdx,z.Definition 27 (see [22]).
A functionf:X⟶0,∞, where X,d is a b-metric space, is called lower semicontinuous if for all x∈X, and a sequence xn is b-convergent to x, and we have
(12)fx≤liminfn⟶∞fxn.
Consider the following:
Ψ=ψ:0,∞⟶0,∞ such that ψ is continuous and nondecreasing function}.
Also, we denoteΦ=ϕ:0,∞⟶0,∞ such that ϕ is nondecreasing and lower semicontinuous, and
(13)ϕt=0⇔t=0.
Hao and Guan [22] proved the following common fixed point results for generalized weakly contractive mapping in b-metric spaces:Theorem 28 (see [22]).
LetX,d be a complete b-metric space with parameter s≥1, and let f,g:X⟶X be given self-mappings satisfying g as injective and fX⊂gX where gX is closed. Suppose φ:X⟶0,∞ is a lower semicontinuous function and p≥2 is a constant. If there are functions ψ∈Ψ,ϕ∈Φ such that
(14)ψspdfx,fy+φfx+φfy≤ψmx,y,d,f,g,φ−ϕlx,y,d,f,g,φ,where
(15)mx,y,d,f,g,φ=maxdgx,gy+φgx+φgy,12dfx,gx+φfx+φgx+dfy,gy+φfy+φgy,12sdfx,gy+φfx+φgy+dfy,gx+φfy+φgx,(16)lx,y,d,f,g,φ=maxdgx,gy+φgx+φgy,dfy,gy+φfy+φgy,
thenf and g have a unique coincidence point in X. Moreover, f and g have a unique common fixed point provided that f and g are weakly compatible.


## 3. Result and Discussion

Definition 29.
LetX,d be a b-metric space and A and B be two nonempty subset of a b-metric space X,d with parameter s≥1 and p≥2 is a constant. A pair of map f,g:A⟶B is said to be a generalized proximal weakly contractive mapping, if for all x,y,h,t,r,m∈A,
(17)dh,fx=dA,B,(18)dt,fy=dA,B,(19)dr,gx=dA,B,(20)dm,gy=dA,B,
then(21)ψspdh,t+φh+φt≤ψmdx,y,h,t,r,m,d,φ−ϕldx,y,h,t,r,m,d,φ,where
(22)mdx,y,h,t,r,m,d,φ=maxdr,m+φr+φm,12dh,r+φh+φr+dt,m+φt+φm,12sdh,m+φh+φm+dt,r+φt+φr.
ldx,y,h,t,r,m,d,φ=maxdr,m+φr+φm,dt,m+φt+φm,ψ∈Ψ,ϕ∈Φ, and φ:X⟶0,∞ is a lower semicontinuous function.Theorem 30.
LetA,B be a pair of nonempty subsets of a complete b-metric space X,d, and assume that A0 and B0 are nonempty such that A0 is closed. Define a pair of mapping f,g:A⟶B satisfying the following conditions:
(i)
f and g are generalized proximal weakly contractive mapping(ii)
fA0⊆B0, and fA0⊂gA0(iii)
f and g are continuous mapping(iv)
f and g are commute proximity
Then,f and g have a unique common best proximity point.Proof.
We prove the existence of common best proximity point.
Letx0∈A0. Since fA0⊂gA0, there exists x1∈A0 such that
(23)fx0=gx1.
Also,x1∈A0. Since fA0⊂gA0, there exists x2∈A0 such that
(24)fx1=gx2.
Continuing this process in a similar fashion, obtain the sequencexn and xn+1 in A0 such that
(25)fxn=gxn+1,
for eachn≥0.
SincefA0⊆B0 and A0 is nonempty set, there exists un∈A0 such that
(26)dun,fxn=dA,B,
for alln≥0.
Further, we obtain that(27)dA,B=dun,fxn=dun,gxn+1,
for alln≥0.
Our first goal is to show thatfu=gu, for some u∈A0.
Suppose thatun=un+1, for some n≥0, by (2) and (3), we get that
(28)dun+1,fxn+1=dA,B=dun,fxn=dun,gxn+1.
Sincef and g commute proximally, fun=un+1=gun, and so we are done.
Assume thatun≠un+1, for all n≥0. From (3), note that
(29)dun,fxn=dun+1,fxn+1=dA,B=dun−1,gxn=dun,gxn+1,
for alln≥1. Since a pair f,g is generalized proximal weakly contractive map with x=xn, y=xn+1, we have that
(30)ψdun,un+1+φun+φun+1≤ψspdun,un+1+φun+φun+1≤ψmdxn,xn+1,un−1,un,un,un+1,d,φ−ϕldxn,xn+1,un−1,un,un,un+1,d,φ,where
(31)ψmdxn,xn+1,un−1,un,un,un+1,d,φ=maxdun−1,un+φun−1+φun,12dun,un−1+φun+φun−1+dun+1,un+φun+1+φun,12sdun,un+φun+φun+dun+1,un−1+φun+1+φun−1≤maxdun,un−1+φun+φun−1,dun,un+1+φun+φun+1,ϕldxn,xn+1,un−1,un,un,un+1,d,φ=maxdun−1,un+φun−1+φun,dun,un+1+φun+φun+1.
Ifdun,un+1+φun+φun+1>dun−1,un+φun−1+φun, for some n∈ℕ, in view of (5)–(8), we have
(32)ψdun,un+1+φun+φun+1≤ψmdxn,xn+1,un−1,un,un,un+1,d,φ−ϕldxn,xn+1,un−1,un,un,un+1,d,φ≤ψdun,un+1+φun+φun+1−ϕdun,un+1+φun+φun+1,
which impliesϕdun,un+1+φun+φun+1=0.
Hence,un=un+1, a contradiction.
Thus, we have(33)dun,un+1+φun+φun+1≤dun−1,un+φun−1+φun,mdxn,xn+1,un−1,un,un,un+1,d,φ≤dun−1,un+φun−1+φun,ldxn,xn+1,un−1,un,un,un+1,d,φ=dun−1,un+φun−1+φun.
It follows from (10) that {dun,un+1+φun+φun+1} is a nonincreasing sequence, and so there exists r≥0 such that
(34)limn⟶∞dun,un+1+φun+φun+1=r.
By (5), (11), and (12), we can obtain
(35)ψdun,un+1+φun+φun+1≤ψmdxn,xn+1,un−1,un,un,un+1,d,φ−ϕldxn,xn+1,un−1,un,un,un+1,d,φ≤ψdun−1,un+φun−1+φun−ϕdun−1,un+φun−1+φun.
Now assume thatr>0. Taking the upper limit as n⟶∞ in (15), we have
(36)limsupn⟶∞ψdun,un+1+φun+φun+1≤limsupn⟶∞ψmdxn,xn+1,un−1,un,un,un+1,d,φ−limsupn⟶∞ϕldxn,xn+1,un−1,un,un,un+1,d,φ≤limsupn⟶∞ψdun−1,un+φun−1+φun−liminfn⟶∞ϕdun−1,un+φun−1+φun,
which implies thatψr≤ψr−ϕr, a contradiction. Thus, we have
(37)limn⟶∞dun,un+1+φun+φun+1=r=0.
It follows that(38)limn⟶∞dun,un+1=0,limn⟶∞φun=0.
Now, we claim that {un} is a Cauchy sequence.
Suppose contradiction, that is, {un}, is not a Cauchy sequence. Then, there exists ε>0 such that there are subsequences {umk} and {unk} of {un} so that for all k∈ℕ with nk>mk>k, we obtain
(39)ε≤dumk,unk,dumk,unk−1<ε.
By triangular inequality in b-metric space and (19) and (20), we have
(40)ε≤dumk,unk≤sdumk,unk−1+sdunk−1,unk<sε+sdunk−1,unk.
Taking the upper limit ask⟶∞ in the above inequality, we have
(41)ε≤limsupn⟶∞dumk,unk≤sε,εs≤limsupn⟶∞dumk,unk−1≤ε.
Also, we have(42)ε≤dumk,unk≤sdumk,umk−1+sdumk−1,unk≤sdumk,umk−1+s2dumk−1,umk+s2dumk,unk≤sdumk,umk−1+s2dumk−1,umk+s3ε.
Then, by taking the upper limit ask⟶∞ in (42), we have
(43)ε≤limsupn⟶∞sdumk−1,unk≤s3ε,
which implies(44)εs≤limsupn⟶∞dumk−1,unk≤s2ε.
It is from(45)ε≤dumk,unk≤sdumk,umk−1+sdumk−1,unk≤sdumk,umk−1+s2dumk−1,unk−1+s2dunk−1,unk≤sdumk,umk−1+s3dumk−1,umk+s3dumk,unk−1≤sdumk,umk−1+s3dumk−1,umk+s3ε.
By taking the upper limit ask⟶∞ in (43), we have
(46)εs2≤limsupn⟶∞dumk−1,unk−1≤sε.
In similar fashion by taking the lower limit, we can obtain(47)ε≤liminfn⟶∞dumk,unk≤sε,εs≤liminfn⟶∞dumk,unk−1≤ε,εs≤liminfn⟶∞dumk−1,unk≤s2ε,εs2≤liminfn⟶∞dumk−1,unk−1≤sε.
Sinceumk and unk satisfy equations (26) and (27), we obtain that
(48)dunk,fxnk=dA,B=dunk−1,gxnk,dumk,fxmk=dA,B=dumk−1,gxmk,
for eachk∈ℕ. Since f and g are generalized proximal weakly contractive mapping with x=xnk and y=xmk, we have
(49)ψdunk,umk+φunk+φumk≤ψmdxnk,xmk,unk,umk,unk−1,umk−1,d,φ−ϕldxnk,xmk,unk,umk,unk−1,umk−1,d,φ.
From the definition, we have(50)mdxnk,xmk,unk,umk,unk−1,umk−1,d,φ≤maxdumk−1,unk−1+φumk−1+φunk−1,12dunk,unk−1+φunk+φunk−1+dumk,umk−1+φumk+φumk−1,12sdunk,umk−1+φunk+φumk−1+dunk−1,umk+φunk−1+φumk−1.
Taking the upper limit ask⟶∞, we obtain
(51)limsupn⟶∞mdxnk,xmk,unk,umk,unk−1,umk−1,d,φ≤maxsε,0,ε+s2ε2s=sε.
Also, we have(52)ldxnk,xmk,unk,umk,unk−1,umk−1,d,φ=maxdumk−1,unk−1+φumk−1+φunk−1,dumk,umk−1+φumk+φumk−1.
By taking the lower limit ask⟶∞, we have
(53)sε≥liminfn⟶∞ldxnk,xmk,unk,umk,unk−1,umk−1,d,φ≥εs2.
By applying generalized proximal weakly contractive mapping withx=xnk and y=xmk, we have
(54)ψsε≤ψspε≤ψsplimsupn⟶∞dumk,unk+φumk+φunk≤ψlimsupn⟶∞mdxnk,xmk,unk,umk,unk−1,umk−1,d,φ−liminfn⟶∞ϕldxnk,xmk,unk,umk,unk−1,umk−1,d,φ≤ψsε−ϕliminfn⟶∞ldxnk,xmk,unk,umk,unk−1,umk−1,d,φ,
which implies that(55)liminfn⟶∞ldxnk,xmk,unk,umk,unk−1,umk−1,d,φ=0,
a contradiction to (53). Hence, the sequence un is Cauchy. Since A0 be a closed subset of the complete b-metric space X, there exists u∈A0 such that
(56)limn⟶∞un=u.
By the definition ofφ, we have
(57)φu≤liminfn⟶∞φun=0⇒φu=0.
Consider, by (2) and (3), that
(58)dun,fxn=dun−1,gxn=dA,B.
Sincef and g are commute proximally,
(59)fun−1=gun,
for alln∈ℕ. By continuity of f and g,
(60)fu=limn⟶∞fun−1=limn⟶∞gun=gu.
Now, we claim the existence of common best proximity point off and g. Since fA0⊆B0, there exists x⋆∈A0 such that
(61)dx⋆,fu=dx⋆,gu=dA,B.
By the assumption thatf and g commute proximally, fx⋆=gx⋆.
According to the assumption thatfA0⊆B0, there exists z⋆∈A0 such that
(62)dz⋆,fx⋆=dz⋆,gx⋆=dA,B.
Next, we claim thatx⋆=z⋆. Suppose that x⋆≠z⋆, that is, dx⋆,z⋆>0. By applying generalized proximal weakly contractive mapping with x=u and y=x⋆, we observe that
(63)ψdx⋆,z⋆+φx⋆+φz⋆≤ψspdx⋆,z⋆+φx⋆+φz⋆≤ψmdu,x⋆,x⋆,z⋆,x⋆,z⋆,d,φ−ϕldu,x⋆,x⋆,z⋆,x⋆,z⋆,d,φ,where
(64)mdu,x⋆,x⋆,z⋆,x⋆,z⋆,d,φ=maxdx⋆,z⋆+φx⋆+φz⋆,12dx⋆,x⋆+φx⋆+φx⋆+dz⋆,z⋆+φz⋆+φz⋆,12sdx⋆,z⋆+φx⋆+φz⋆+dz⋆,x⋆+φz⋆+φx⋆≤dx⋆,z⋆+φx⋆+φz⋆,(65)ldu,x⋆,x⋆,z⋆,x⋆,z⋆,d,φ=maxdx⋆,z⋆+φx⋆+φz⋆,dz⋆,z⋆+φz⋆+φz⋆=dx⋆,z⋆+φx⋆+φz⋆.
From (63)–(65), we have
(66)ψdx⋆,z⋆+φx⋆+φz⋆≤ψdx⋆,z⋆+φx⋆+φz⋆−ϕdx⋆,z⋆+φx⋆+φz⋆,
which implies(67)dx⋆,z⋆+φx⋆+φz⋆=0,⇒dx⋆,z⋆=0,φx⋆=0.
This contradicts the assumptionx⋆≠z⋆. Thus, x⋆=z⋆. Hence,
(68)dx⋆,fx⋆=dA,B=dx⋆,gx⋆.
That is, the elementx⋆∈A is a common best proximity point of f and g.
Finally, we have to show that the pointx⋆ is unique.
Lety⋆∈A be another common best proximity point of f and g. Then,
(69)dx⋆,fx⋆=dy⋆,fy⋆=dA,B=dx⋆,gx⋆=dy⋆,gy⋆.
Sincef and g are generalized proximal weakly contractive mapping, we obtain that
(70)ψdx⋆,y⋆+φx⋆+φy⋆≤ψspdx⋆,y⋆+φx⋆+φy⋆≤ψmdx⋆,y⋆,x⋆,y⋆,x⋆,y⋆,d,φ−ϕldx⋆,y⋆,x⋆,y⋆,x⋆,y⋆,d,φ,where
(71)mdx⋆,y⋆,x⋆,y⋆,x⋆,y⋆,d,φ=maxdx⋆,y⋆+φx⋆+φy⋆,12dx⋆,x⋆+φx⋆+φx⋆+dy⋆,y⋆+φy⋆+φy⋆,12sdx⋆,y⋆+φx⋆+φy⋆+dy⋆,x⋆+φy⋆+φx⋆≤dx⋆,y⋆+φx⋆+φy⋆,(72)ldx⋆,y⋆,x⋆,y⋆,x⋆,y⋆,d,φ=maxdx⋆,y⋆+φx⋆+φy⋆,dy⋆,y⋆+φy⋆+φy⋆=dx⋆,y⋆+φx⋆+φy⋆.
Now, from (71) and (72), we have
(73)ψdx⋆,y⋆+φx⋆+φy⋆≤ψdx⋆,y⋆+φx⋆+φy⋆−ϕdx⋆,y⋆+φx⋆+φy⋆.
By the properties ofϕ and from (73), we have
(74)dx⋆,y⋆+φx⋆+φy⋆=0,⇒dx⋆,y⋆=0,φx⋆=0,
which contradict the supposition thatx⋆≠y⋆. Thus, x⋆=y⋆.
Therefore,f and g have a unique common best proximity point.
The proof is completed.Example 31.
LetX=ℝ2 and d:X×X⟶0,∞ be defined by dx1,x2,y1,y2=x1−y12+x2−y22, for all x1,x2,y1,y2∈X and X,d is a complete b-metric space with parameter s=2.
Suppose(75)A=x,0:0≤x≤1,B=x,1:0≤x≤1.
Letf,g:A⟶B be the mapping defined by
(76)fx,0=x8,1,gx,0=7x8,1.
φ:X⟶0,∞, defined by φx,0=x2, and define a mapping ψ,ϕ:0,∞⟶0,∞ with ψt=t, and ϕt=35t/98. Clearly, φ is lower semicontinuous function, and ψ is continuous and nondecreasing function. Further, ϕ is nondecreasing and lower semicontinuous, and ϕt=0⇔t=0.
dA,B=infdx,0,x,1:x,0∈A,x,1∈B, which implies that dA,B=infx−x2+0−12=1 and implies that dA,B=1.
Notice thatf and g are continuous. Now, we check that f and g are generalized proximal weakly contractive mapping.
In fact, for allx,0,y,0,h,0,t,0,r,0,m,0∈A, we have
dh,0,fx,0=dA,B implies that dh,0,x/8,1=1, which implies that h=x/8
dt,0,fy,0=dA,B; this implies dt,0,y/8,1=1, which implies that t=y/8
dr,0,gx,0=dA,B implies that dr,0,7x/8,1=1, implies that r=7x/8
dm,0,gy,0=dA,B implies that dm,0,7y/8,1=1, implies that m=7y/8
Then, by applying the generalized proximal weakly contractive mapping definition withs=p=2, we have the following:
(77)ψspdh,0,t,0+φh,0+φt,0≤ψmdx,0,y,0,h,0,t,0,r,0,m,0,d,φ−ϕldx,0,y,0,h,0,t,0,r,0,m,0,d,φ.
Now, we have(78)ψspdh,0,t,0+φh,0+φt,0=22.dx8,0,y8,0+φx8,0+φy8,0=4.dx8,0,y8,0+x82+y82=4.x8−y82+0−02+x82+y82=4.x8−y82+x82+y82≤4.164.2x2+y2=18x2+y2,ψmdx,0,y,0,h,0,t,0,r,0,m,0,d,φ=maxdr,0,m,0+φr,0+φm,0,12dh,0,r,0+φh,0+dt,0,m,0+φt,0+φm,0,12sdh,0,m,0+φh,0+φm,0+dt,0,r,0+φt,0+φr,0≥ψ12dh,0,r,0+φh,0+φr,0+dt,0,m,0+φt,0+φm,0=12dx8,0,7x8,0+φx8,0+φ7x8,0+dy8,0,7y8,0+φy8,0+φ7y8,0=12x8−7x82+0−02+x82+7x82+y8−7y82+0−02+y82+7y82=12.16436x2+x2+49x2+36y2+y2+49y2=12.16486x2+86y2=4364x2+y2,ϕldx,0,y,0,h,0,t,0,r,0,m,0,d,φ=ϕmaxdr,0,m,0+φr,0+φm,0,dt,0,m,0+φt,0+φm,0=3598maxd7x8,0,7y8,0+φ7x8,0+φ7y8,0,dy8,0,7y8,0+φy8,0+φ7y8,0=3598max7x8−7y82+0−02+7x82+7y82,y8−7y82+0−02+y82+7y82=3598max4932x2+y2,4332y2≤3564x2+y2.
According to above inequalities, we get that(79)ψspdh,0,t,0+φh,0+φt,0≤18x2+y2=4364x2+y2−3564x2+y2≤ψmdx,0,y,0,h,0,t,0,r,0,m,0,d,φ−ϕldx,0,y,0,h,0,t,0,r,0,m,0,d,φ.
Hence,f and g are generalized proximal weakly contractive mappings.
Next, consider, by the definition ofA0,B0, that A0=A and B0=B; thus, fA0 and gA0⊆B0. Additionally, fA0=x,1:0≤x≤1/8⊂x,1:0≤x≤7/8=gA0.
Now, it remains to show thatf and g commute proximally. Let x,u,v∈A such that
(80)du,fx=dv,gx=dA,B.
Consequently,x=x^,0, u=u^,0, and v=v^,0, where u^=x/8 and v^=7x^/8. Thus,
(81)fx=fx^,0=x^8,1,gx=gx^,0=7x^8,1,du,fx=du^,0,fx^,0=dx^8,0,x^8,1=1=dA,B,dv,gx=dv^,0,gx^,0=d7x^8,0,7x^8,1=1=dA,B.
Therefore,(82)du,fx=dv,gx=dA,B.
Now, we claim that(83)fv=gu,gu=gu^,0=gx^8,0=7x^64,1,fv=fv^,0=f7x^8,0=7x^64,1,
which impliesfv=gu.
Hence,du,fx=dv,gx=dA,B⇒fv=gu.
Therefore,f and g are commute proximally.
Finally, by Theorem30, we can conclude that there is a unique common best proximity point of the pair f,g. In fact, the point 0,0 is the unique common best proximity point of f,g.
To show this, there existsx⋆,0∈A such that
(84)dx⋆,0,fx⋆,0=dx⋆,0,gx⋆,0=dA,B=1,where x⋆,0 is common best proximity point of f and g. Now find x⋆(85)dx⋆,0,fx⋆,0=dA,B=1,
and this implies that(86)dx⋆,0,x⋆8,1=1,
implying that(87)x⋆−x⋆82+0−12=1.
From this, we get(88)x⋆−x⋆82=0.
Hence,x⋆=0, and also, from
(89)dx⋆,0,gx⋆,0=dA,B=1,
we have(90)dx⋆,0,7x⋆8,1=1,
implying that(91)x⋆−7x⋆82+0−12=1.
From this, we get(92)x⋆−7x⋆82=0.
Hence,x⋆=0.
Therefore, the pointx⋆,0=0,0∈A is a unique common best proximity point of f and g.
Ifφ=0 in Theorem 30, we can get the following result:Corollary 32.
LetA,B be a pair of nonempty subsets of a complete b-metric space X,d, and assume that A0 and B0 are nonempty such that A0 is closed. Define a pair of mapping f,g:A⟶B satisfying the following conditions:
(i)
For allx,y,h,t,r,m∈A,(93)dh,fx=dA,B,dt,fy=dA,B,dr,gx=dA,B,dm,gy=dA,B,
then(94)ψspdh,t≤ψmdx,y,h,t,r,m,d−ϕldx,y,h,t,r,m,d,where
(95)mdx,y,h,t,r,m,d=maxdr,m,12dh,r+dt,m,12sdh,m+dt,r,ldx,y,h,t,r,m,d=maxdr,m,dt,m,ψ∈Ψ,ϕ∈Φ,(ii)
fA0⊆B0 and fA0⊂gA0.(iii)
f and g are continuous mapping(iv)
f and g are commute proximity
Then,f and g have a unique common best proximity point.
If we consider the corresponding problem in the setting of metric space, that is,s=1 in Theorem 30, we can obtain the following:Corollary 33.
LetA,B be a pair of nonempty subsets of a complete b-metric space X,d and assume that A0 and B0 are nonempty such that A0 is closed. Define a pair of mapping f,g:A⟶B satisfying the following conditions:
(i) For allx,y,h,t,r,m∈A,
(96)dh,fx=dA,B,dt,fy=dA,B,dr,gx=dA,B,dm,gy=dA,B,
then(97)ψdh,t+φh+φt≤ψmdx,y,h,t,r,m,d,φ−ϕldx,y,h,t,r,m,d,φ,where
(98)mdx,y,h,t,r,m,d,φ=maxdr,m+φr+φm,12dh,r+φh+φr+dt,m+φt+φm,12dh,m+φh+φm+dt,r+φt+φr,ldx,y,h,t,r,m,d,φ=maxdr,m+φr+φm,dt,m+φt+φm,
is the same as Theorem30, ψ∈Ψ, ϕ∈Φ, and φ:X⟶0,∞ is a lower semicontinuous function.
(99)fA0⊆B0,andfA0⊂gA0
(ii)f and g are continuous mapping
(iii)f and g are commute proximity
Then,f and g have a unique common best proximity point.


## 4. Conclusion

This study is concerned with the existence and uniqueness of common best proximity point for generalized proximal weakly contractive mapping in complete b-metric spaces, and in this study, we have defined the notion of generalized proximal weakly contractive mapping in b-metric spaces.



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*Source: 5982401-2022-08-24.xml*