# Best proximity point results in geodesic metric spaces

## Abstract

In this paper, the existence of a best proximity point for relatively u-continuous mappings is proved in geodesic metric spaces. As an application, we discuss the existence of common best proximity points for a family of not necessarily commuting relatively u-continuous mappings.

## 1 Introduction

Let A be a nonempty subset of a metric space $\left(X,d\right)$ and $T:Aâ†’X$. A solution to the equation $Tx=x$ is called a fixed point of T. It is obvious that the condition is necessary for the existence of a fixed point for T. But there occur situations in which $d\left(x,Tx\right)>0$ for all $xâˆˆA$. In such a situation, it is natural to find a point $xâˆˆA$ such that x is closest to Tx in some sense. The following well-known best approximation theorem, due to Ky Fan [1], explores the existence of an approximate solution to the equation $Tx=x$.

Theorem 1 [1]

Let A be a nonempty compact convex subset of a normed linear space X and $T:Aâ†’X$ be a continuous function. Then there exists $xâˆˆA$ such that $âˆ¥xâˆ’Txâˆ¥=dist\left(Tx,A\right)=inf\left\{âˆ¥Txâˆ’aâˆ¥:aâˆˆA\right\}$.

The point $xâˆˆA$ in Theorem 1 is called a best approximant of T in A. Let A, B be nonempty subsets of a metric space X and $T:Aâ†’B$. A point ${x}_{0}âˆˆA$ is called a best proximity point of T if $d\left({x}_{0},T{x}_{0}\right)=dist\left(A,B\right)$. Some interesting results in approximation theory can be found in [2â€“8].

Eldred et al. [2] defined relatively nonexpansive mappings and used the proximal normal structure to prove the existence of best proximity points for such mappings.

Definition 2 [2]

Let A, B be nonempty subsets of a metric space $\left(X,d\right)$. A mapping $T:AâˆªBâ†’AâˆªB$ is said to be a relatively nonexpansive mapping if

1. (i)

$T\left(A\right)âŠ†B$, $T\left(B\right)âŠ†A$;

2. (ii)

$d\left(Tx,Ty\right)â‰¤d\left(x,y\right)$, for all $xâˆˆA$, $yâˆˆB$.

Theorem 3 [2]

Let $\left(A,B\right)$ be a nonempty, weakly compact convex pair in a Banach space X. Let $T:AâˆªBâ†’AâˆªB$ be a relatively nonexpansive mapping and suppose $\left(A,B\right)$ has a proximal normal structure. Then there exists $\left(x,y\right)âˆˆAÃ—B$ such that

$âˆ¥xâˆ’Txâˆ¥=âˆ¥Tyâˆ’yâˆ¥=dist\left(A,B\right).$

Remark 4 [2]

Note that every nonexpansive self-map is a relatively nonexpansive map. Also, a relatively nonexpansive mapping need not be continuous.

In [8], Sankar Raj and Veeramani used a convergence theorem to prove the existence of best proximity points for relatively nonexpansive mappings in strictly convex Banach spaces.

Recently, Elderd, Sankar Raj and Veeramani [9] introduced a class of relatively u-continuous mappings and investigated the existence of best proximity points for such mappings in strictly convex Banach spaces.

Definition 5 [9]

Let A, B be nonempty subsets of a metric space X. A mapping $T:AâˆªBâ†’AâˆªB$ is said to be a relatively u-continuous mapping if it satisfies:

1. (i)

$T\left(A\right)âŠ†B$, $T\left(B\right)âŠ†A$;

2. (ii)

for each $\mathrm{Îµ}>0$, there exists a $\mathrm{Î´}>0$ such that $d\left(Tx,Ty\right)<\mathrm{Îµ}+dist\left(A,B\right)$, whenever $d\left(x,y\right)<\mathrm{Î´}+dist\left(A,B\right)$, for all $xâˆˆA$, $yâˆˆB$.

Theorem 6 [9]

Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and $T:AâˆªBâ†’AâˆªB$ be a relatively u-continuous mapping. Then there exists $\left(x,y\right)âˆˆAÃ—B$ such that

$âˆ¥xâˆ’Txâˆ¥=âˆ¥yâˆ’Tyâˆ¥=dist\left(A,B\right).$

Remark 7 [9]

Every relatively nonexpansive mapping is a relatively u-continuous mapping, but the converse is not true.

Example 8 [9]

Let $\left(X={\mathbb{R}}^{2},{âˆ¥â‹\dots âˆ¥}_{2}\right)$ and consider $A=\left\{\left(0,t\right):0â‰¤tâ‰¤1\right\}$ and $B=\left\{\left(1,s\right):0â‰¤sâ‰¤1\right\}$. Define $T:AâˆªBâ†’AâˆªB$ by

Then T is relatively u-continuous, but not relatively nonexpansive.

Also, in [9], the authors proved the existence of common best proximity points for a family of commuting relatively u-continuous mappings.

The aim of this paper is to discuss the existence of a best proximity point for relatively u-continuous mappings in the frameworks of geodesic metric spaces. As an application, we investigate the existence of common best proximity points for a family of not necessarily commuting relatively u-continuous mappings.

## 2 Preliminaries

In this section, we give some preliminaries.

Definition 9 [10]

A metric space $\left(X,d\right)$ is said to be a geodesic space if every two points x and y of X are joined by a geodesic, i.e., a map $c:\left[0,l\right]âŠ†\mathbb{R}â†’X$ such that $c\left(0\right)=x$, $c\left(l\right)=y$, and $d\left(c\left(t\right),c\left({t}^{\mathrm{â€²}}\right)\right)=|tâˆ’{t}^{\mathrm{â€²}}|$ for all t, ${t}^{\mathrm{â€²}}âˆˆ\left[0,l\right]$. Moreover, X is called uniquely geodesic if there is exactly one geodesic joining x and y for each $x,yâˆˆX$.

The midpoint m between two points x and y in a uniquely geodesic metric space has the property $d\left(x,m\right)=d\left(y,m\right)=\frac{1}{2}d\left(x,y\right)$. A trivial example of a geodesic space is a Banach space with usual segments as geodesic segments.

A point $zâˆˆX$ belongs to the geodesic segment $\left[x,y\right]$ if and only if there exists $tâˆˆ\left[0,1\right]$ such that $d\left(z,x\right)=td\left(x,y\right)$ and $d\left(z,y\right)=\left(1âˆ’t\right)d\left(x,y\right)$. Hence, we write $z=\left(1âˆ’t\right)x+ty$.

A subset A of a geodesic metric space X is said to be convex if it contains any geodesic segment that joins each pair of points of A.

The metric $d:XÃ—Xâ†’\mathbb{R}$ in a geodesic space $\left(X,d\right)$ is convex if

$d\left(z,\left(1âˆ’t\right)x+ty\right)â‰¤\left(1âˆ’t\right)d\left(z,x\right)+td\left(z,y\right)$

for any $x,y,zâˆˆX$ and $tâˆˆ\left[0,1\right]$.

Definition 10 [10]

A geodesic metric space X is said to be strictly convex if for every $r>0$, a, x and $yâˆˆX$ with $d\left(x,a\right)â‰¤r$, $d\left(y,a\right)â‰¤r$ and , it is the case that $d\left(a,p\right), where p is any point between x and y such that and , i.e., p is any point in the interior of a geodesic segment that joins x and y.

Remark 11 [10]

Every strictly convex metric space is uniquely geodesic.

In [10], FernÃ¡ndez-LeÃ³n proved the existence and uniqueness of best proximity points in strictly convex metric spaces. For more details about geodesic spaces, one may check [11â€“13].

In the particular framework of geodesic metric spaces, the concept of global nonpositive curvature (global NPC spaces), also known as the $CAT\left(0\right)$ spaces, is defined in [13] as follows.

Definition 12 A global NPC space is a complete metric space $\left(X,d\right)$ for which the following inequality holds true: for each pair of points ${x}_{0}$, ${x}_{1}âˆˆX$ there exists a point $yâˆˆX$ such that for all points $zâˆˆX$,

${d}^{2}\left(z,y\right)â‰¤\frac{1}{2}{d}^{2}\left(z,{x}_{0}\right)+\frac{1}{2}{d}^{2}\left(z,{x}_{1}\right)âˆ’\frac{1}{4}{d}^{2}\left({x}_{0},{x}_{1}\right).$

Proposition 13 [13]

If $\left(X,d\right)$ is a global NPC space, then it is a geodesic space. Moreover, for any pair of points ${x}_{0},{x}_{1}âˆˆX$ there exists a unique geodesic $\mathrm{Î³}:\left[0,1\right]â†’X$ connecting them. For $tâˆˆ\left[0,1\right]$ the intermediate points ${\mathrm{Î³}}_{t}$ depend continuously on the endpoints ${x}_{0}$, ${x}_{1}$. Finally, for any $zâˆˆX$,

${d}^{2}\left(z,{\mathrm{Î³}}_{t}\right)â‰¤\left(1âˆ’t\right){d}^{2}\left(z,{x}_{0}\right)+t{d}^{2}\left(z,{x}_{1}\right)âˆ’t\left(1âˆ’t\right){d}^{2}\left({x}_{0},{x}_{1}\right).$

Corollary 14 [13]

Let $\left(X,d\right)$ be a global NPC space, $\mathrm{Î³},\mathrm{Î·}:\left[0,1\right]â†’X$ be geodesics and $tâˆˆ\left[0,1\right]$. Then

${d}^{2}\left({\mathrm{Î³}}_{t},{\mathrm{Î·}}_{t}\right)â‰¤\left(1âˆ’t\right){d}^{2}\left({\mathrm{Î³}}_{0},{\mathrm{Î·}}_{0}\right)+t{d}^{2}\left({\mathrm{Î³}}_{1},{\mathrm{Î·}}_{1}\right)âˆ’t\left(1âˆ’t\right){\left[d\left({\mathrm{Î³}}_{0},{\mathrm{Î³}}_{1}\right)âˆ’d\left({\mathrm{Î·}}_{0},{\mathrm{Î·}}_{1}\right)\right]}^{2}$

and

$d\left({\mathrm{Î³}}_{t},{\mathrm{Î·}}_{t}\right)â‰¤\left(1âˆ’t\right)d\left({\mathrm{Î³}}_{0},{\mathrm{Î·}}_{0}\right)+td\left({\mathrm{Î³}}_{1},{\mathrm{Î·}}_{1}\right).$

Corollary 14 shows that the distance function $\left(x,y\right)â†¦d\left(x,y\right)$ in a global NPC space is convex with respect to both variables. Consequently, all balls in a global NPC space are convex.

Example 15 Every Hilbert space is a global NPC space.

Example 16 Every metric tree is a global NPC space.

Example 17 A Riemannian manifold is a global NPC space if and only if it is complete, simply connected, and of nonpositive curvature.

More details about global NPC spaces can be found in [13â€“16].

We need the following notations in the sequel. Let $\left(X,d\right)$ be a metric space and A, B be nonempty subsets of X. Define

Given C a nonempty subset of X, the metric projection ${P}_{C}:Xâ†’{2}^{C}$ is the mapping

where ${2}^{C}$ denotes the set of all subsets of C.

Definition 18 [9]

Let A, B be nonempty convex subsets of a geodesic metric space. A mapping $T:AâˆªBâ†’AâˆªB$ is said to be affine if

$T\left(\mathrm{Î»}x+\left(1âˆ’\mathrm{Î»}\right)y\right)=\mathrm{Î»}Tx+\left(1âˆ’\mathrm{Î»}\right)Ty,$

for all $x,yâˆˆA$ or $x,yâˆˆB$ and $\mathrm{Î»}âˆˆ\left(0,1\right)$.

Definition 19 [8]

Let X be a metric space. A subset C of X is called approximatively compact if for any $yâˆˆX$ and for any sequence $\left\{{x}_{n}\right\}$ in C such that $d\left({x}_{n},y\right)â†’dist\left(y,C\right)$ as $nâ†’\mathrm{âˆž}$, $\left\{{x}_{n}\right\}$ has a subsequence which converges to a point in C.

In [13], Sturm presented the following result which ensures the existence and uniqueness of the metric projection on a global NPC space.

Proposition 20

1. (i)

For each closed convex set C in a global NPC space $\left(X,d\right)$, there exists a unique map ${P}_{C}:Xâ†’C$ (projection onto C) such that

$d\left({P}_{C}\left(x\right),x\right)=\underset{yâˆˆC}{inf}d\left(x,y\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}\phantom{\rule{0.1em}{0ex}}xâˆˆX;$
2. (ii)

${P}_{C}$ is orthogonal in the sense that

${d}^{2}\left(x,y\right)â‰¥{d}^{2}\left(x,{P}_{C}\left(x\right)\right)+{d}^{2}\left({P}_{C}\left(x\right),y\right)$

for every $xâˆˆX$, $yâˆˆC$;

1. (iii)

${P}_{C}$ is nonexpansive,

$d\left({P}_{C}\left(x\right),{P}_{C}\left(z\right)\right)â‰¤d\left(x,z\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}\phantom{\rule{0.1em}{0ex}}x,zâˆˆX.$

Remark 21 Note that the existence of a unique metric projection does not need the compactness of C.

Remark 22 [13]

For any subset A of a global NPC space $\left(X,d\right)$, there exists a unique smallest convex set $\mathit{co}\left(A\right)={â‹ƒ}_{n=0}^{\mathrm{âˆž}}{A}_{n}$, containing A and called convex hull of A. Where ${A}_{0}=A$, and for $nâˆˆ\mathbb{N}$, the set ${A}_{n}$ consists of all points in global NPC space X which lie on geodesics which start and end in ${A}_{nâˆ’1}$.

Based on Proposition 20, Niculescu and Roventa [17] proved the Schauder fixed point theorem in the setting of a global NPC space.

Theorem 23 Let C be a closed convex subset of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Then every continuous map $T:Câ†’C$, whose image $T\left(C\right)$ is relatively compact, has a fixed point.

## 3 Main results

In this section, we will prove the existence of best proximity points for a relatively u-continuous mapping. Also, we obtain a result on the existence of common best proximity points for a family of not necessarily commuting relatively u-continuous mappings.

Proposition 24 Let A, B be nonempty subsets of a metric space X with and $T:AâˆªBâ†’AâˆªB$ be a relatively u-continuous mapping. Then $T\left({A}_{0}\right)âŠ†{B}_{0}$ and $T\left({B}_{0}\right)âŠ†{A}_{0}$.

Proof Choose $xâˆˆ{A}_{0}$, then there exists $yâˆˆB$ such that $d\left(x,y\right)=dist\left(A,B\right)$. But T is a relatively u-continuous mapping, then for each $\mathrm{Îµ}>0$, there exists a $\mathrm{Î´}>0$ such that

$d\left(p,q\right)<\mathrm{Î´}+dist\left(A,B\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(Tp,Tq\right)<\mathrm{Îµ}+dist\left(A,B\right)$

for each $pâˆˆA$, $qâˆˆB$. Since $d\left(x,y\right)<\mathrm{Î´}+dist\left(A,B\right)$ for any $\mathrm{Î´}>0$, hence

$dist\left(A,B\right)â‰¤d\left(Tx,Ty\right)<\mathrm{Îµ}+dist\left(A,B\right)$

for each $\mathrm{Îµ}>0$. Therefore, $d\left(Tx,Ty\right)=dist\left(A,B\right)$ and then $T\left(x\right)âˆˆ{B}_{0}$. This shows that $T\left({A}_{0}\right)âŠ†{B}_{0}$. Similarly, it can be seen that $T\left({B}_{0}\right)âŠ†{A}_{0}$.â€ƒâ–¡

Proposition 25 Let A, B be nonempty closed convex subsets of a global NPC space X with , $T:AâˆªBâ†’AâˆªB$ be a relatively u-continuous mapping, and $P:AâˆªBâ†’AâˆªB$ be a mapping defined by

$P\left(x\right)=\left\{\begin{array}{cc}{P}_{B}\left(x\right),\hfill & \mathit{\text{if}}\phantom{\rule{0.1em}{0ex}}xâˆˆA,\hfill \\ {P}_{A}\left(x\right),\hfill & \mathit{\text{if}}\phantom{\rule{0.1em}{0ex}}xâˆˆB.\hfill \end{array}$

Then $TP\left(x\right)=P\left(Tx\right)$ for all $xâˆˆ{A}_{0}âˆª{B}_{0}$, i.e., ${P}_{A}\left(T\left(x\right)\right)=T\left({P}_{B}\left(x\right)\right)$ for $xâˆˆ{A}_{0}$ and $T\left({P}_{A}\left(y\right)\right)={P}_{B}\left(T\left(y\right)\right)$ for $yâˆˆ{B}_{0}$.

Proof Choose $xâˆˆ{A}_{0}$, then there exists $yâˆˆB$ such that $d\left(x,y\right)=dist\left(A,B\right)$. According to Proposition 20, since the metric projection is unique, we have $y={P}_{B}\left(x\right)$ and $x={P}_{A}\left(y\right)$. Recalling that T is relatively u-continuous, therefore, as in the proof of Proposition 24, $d\left(Tx,Ty\right)=dist\left(A,B\right)$. Thus, it follows that $T\left(x\right)âˆˆ{B}_{0}$ and $T\left(y\right)âˆˆ{A}_{0}$. Again, in view of the uniqueness of the projection operator, we have

${P}_{A}\left(T\left(x\right)\right)=T\left(y\right)=T\left({P}_{B}\left(x\right)\right).$

So, ${P}_{A}\left(T\left(x\right)\right)=T\left({P}_{B}\left(x\right)\right)$ for any $xâˆˆ{A}_{0}$. Similarly, it can be shown that $T\left({P}_{A}\left(y\right)\right)={P}_{B}\left(T\left(y\right)\right)$ for any $yâˆˆ{B}_{0}$.â€ƒâ–¡

By an analogous argument to the proof of Theorem 3.1 [8], we can prove the following theorem.

Theorem 26 Let A, B be two nonempty subsets of a global NPC space X such that A is closed convex and B is closed. If ${A}_{0}$ is approximatively compact and $\left\{{x}_{n}\right\}$ is a sequence in ${A}_{0}$, and $yâˆˆB$ such that $d\left({x}_{n},y\right)â†’dist\left(A,B\right)$, then ${x}_{n}â†’{P}_{A}\left(y\right)$.

Proof Assume the contrary, then there exists $\mathrm{Îµ}>0$ and a subsequence $\left\{{x}_{{n}_{m}}\right\}$ of $\left\{{x}_{n}\right\}$ such that

$d\left({x}_{{n}_{m}},y\right)â†’dist\left(A,B\right)\phantom{\rule{1em}{0ex}}\text{but}\phantom{\rule{1em}{0ex}}d\left({x}_{{n}_{m}},{P}_{A}\left(y\right)\right)â‰¥\mathrm{Îµ}.$

Since ${A}_{0}$ is approximatively compact, there exists a subsequence $\left\{{x}_{{n}_{m}^{\mathrm{â€²}}}\right\}$ of $\left\{{x}_{{n}_{m}}\right\}$ which converges to a point $xâˆˆA$. Hence,

$d\left({x}_{{n}_{m}^{\mathrm{â€²}}},y\right)â†’d\left(x,y\right).$

Also,

$d\left({x}_{{n}_{m}^{\mathrm{â€²}}},y\right)â†’dist\left(A,B\right).$

Thus, $d\left(x,y\right)=dist\left(A,B\right)$. By Proposition 20, it follows that $x={P}_{A}\left(y\right)$. Finally, we obtain

$d\left({x}_{{n}_{m}^{\mathrm{â€²}}},{P}_{A}\left(y\right)\right)â†’d\left(x,{P}_{A}\left(y\right)\right)â‰¥\mathrm{Îµ},$

which implies that . This leads to a contradiction and therefore ${x}_{n}â†’{P}_{A}\left(y\right)$.â€ƒâ–¡

The following theorem guarantees the existence of best proximity points for a relatively u-continuous mapping in a global NPC space.

Theorem 27 Let A, B be nonempty closed convex subsets of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Let ${A}_{0}$, ${B}_{0}$ be nonempty compact convex and $T:AâˆªBâ†’AâˆªB$ be a relatively u-continuous mapping. Then there exist ${x}_{0}âˆˆA$, ${y}_{0}âˆˆB$ such that

$d\left({x}_{0},T{x}_{0}\right)=d\left({y}_{0},T{y}_{0}\right)=dist\left(A,B\right).$

Proof By Proposition 24, since T is a relatively u-continuous mapping, we have $T\left({A}_{0}\right)âŠ†{B}_{0}$ and $T\left({B}_{0}\right)âŠ†{A}_{0}$. The result follows from Theorem 23 once we show that ${P}_{A}âˆ˜T:{A}_{0}â†’{A}_{0}$ is a continuous mapping, where ${P}_{A}:Xâ†’A$ is a metric projection operator.

To prove this, first notice that ${P}_{A}\left({B}_{0}\right)âŠ†{A}_{0}$. Since X is a global NPC space, by Proposition 20, we obtain that ${P}_{A}:Xâ†’A$ is a continuous mapping. In what follows, we see that the mapping T is continuous on ${A}_{0}$, In fact, let $\left\{{x}_{n}\right\}$ be a sequence in ${A}_{0}$ such that ${x}_{n}â†’{x}_{0}$ for some ${x}_{0}âˆˆ{A}_{0}$. From Proposition 25, we have

${P}_{B}\left({P}_{A}\left(T{x}_{0}\right)\right)={P}_{B}\left(T\left({P}_{B}{x}_{0}\right)\right)=T\left({P}_{A}\left({P}_{B}{x}_{0}\right)\right)=T{x}_{0}.$

Notice that

$\begin{array}{rcl}d\left({x}_{n},{P}_{B}\left({x}_{0}\right)\right)& â‰¤& d\left({x}_{n},{x}_{0}\right)+d\left({x}_{0},{P}_{B}\left({x}_{0}\right)\right)\\ =& d\left({x}_{n},{x}_{0}\right)+dist\left(A,B\right)â†’dist\left(A,B\right)\end{array}$
(3.1)

as $nâ†’\mathrm{âˆž}$. Since T is relatively u-continuous, for each $\mathrm{Îµ}>0$, there exists a $\mathrm{Î´}>0$ such that $d\left(x,y\right)<\mathrm{Î´}+dist\left(A,B\right)$ implies $d\left(Tx,Ty\right)<\mathrm{Îµ}+dist\left(A,B\right)$ for all $xâˆˆA$, $yâˆˆB$. From (3.1), with this $\mathrm{Î´}>0$, it follows that there is $Nâˆˆ\mathbb{N}$ such that $d\left({x}_{n},{P}_{B}\left({x}_{0}\right)\right)<\mathrm{Î´}+dist\left(A,B\right)$ for all $nâ‰¥N$. This implies

$d\left(T\left({x}_{n}\right),T\left({P}_{B}\left({x}_{0}\right)\right)\right)<\mathrm{Îµ}+dist\left(A,B\right)$

for all $nâ‰¥N$. Therefore,

$d\left(T\left({x}_{n}\right),{P}_{A}\left(T{x}_{0}\right)\right)=d\left(T\left({x}_{n}\right),T\left({P}_{B}\left({x}_{0}\right)\right)\right)â†’dist\left(A,B\right).$

This together with Theorem 26 implies that $T{x}_{n}â†’{P}_{B}\left({P}_{A}\left(T{x}_{0}\right)\right)=T{x}_{0}$. Thus, T is continuous on ${A}_{0}$.

Now, since ${P}_{A}âˆ˜T$ is a continuous mapping of ${A}_{0}$, by the Schauder fixed point theorem for a global NPC space, Theorem 23, ${P}_{A}âˆ˜T$ has a fixed point ${x}_{0}âˆˆ{A}_{0}$. From ${P}_{A}\left(T{x}_{0}\right)={x}_{0}$, we find that $d\left({x}_{0},T{x}_{0}\right)=dist\left(T{x}_{0},A\right)$. But since $T{x}_{0}âˆˆ{B}_{0}$, there is ${x}^{\mathrm{â€²}}âˆˆ{A}_{0}$ such that $d\left({x}^{\mathrm{â€²}},T{x}_{0}\right)=dist\left(A,B\right)$. Consequently,

$dist\left(A,B\right)â‰¤dist\left(T{x}_{0},A\right)â‰¤d\left(T{x}_{0},{x}^{\mathrm{â€²}}\right)=dist\left(A,B\right),$

which gives

$dist\left(T{x}_{0},A\right)=dist\left(A,B\right).$

Thus, $d\left({x}_{0},T{x}_{0}\right)=dist\left(A,B\right)$. This completes the proof.â€ƒâ–¡

Next, we will show that Theorem 27 is also true for an appropriate family of relatively u-continuous mappings. The following notations define the set of all best proximity points of a relatively u-continuous mapping:

Theorem 28 Let A, B be nonempty closed convex subsets of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Let ${A}_{0}$, ${B}_{0}$ be nonempty compact convex and $T:AâˆªBâ†’AâˆªB$ be a relatively u-continuous mapping. Let T be affine. Then ${F}_{A}\left(T\right)$ is a nonempty compact convex subset of ${A}_{0}$ and ${F}_{B}\left(T\right)$ is a nonempty compact convex subset of ${B}_{0}$.

Proof It is obvious that ${F}_{A}\left(T\right)$ is a nonempty subset of ${A}_{0}$ by Theorem 27. Assume that $\left\{{x}_{n}\right\}$ is a sequence in ${F}_{A}\left(T\right)$ such that ${x}_{n}â†’{x}_{0}$ for some ${x}_{0}âˆˆ{A}_{0}$. By the continuity of T on ${A}_{0}$, we have ${x}_{0}âˆˆ{F}_{A}\left(T\right)$. Therefore, ${F}_{A}\left(T\right)$ is closed and then compact. Now we claim that ${F}_{A}\left(T\right)$ is convex. In fact, let $\mathrm{Î»}âˆˆ\left[0,1\right]$, ${x}_{1}$, ${x}_{2}âˆˆ{F}_{A}\left(T\right)$, and $z=\left(1âˆ’\mathrm{Î»}\right){x}_{1}+\mathrm{Î»}{x}_{2}$. Since the distance function d is convex with respect to both variables, by Corollary 14, we have

$\begin{array}{rcl}dist\left(A,B\right)& â‰¤& d\left(z,Tz\right)\\ =& d\left(\left(1âˆ’\mathrm{Î»}\right){x}_{1}+\mathrm{Î»}{x}_{2},T\left(\left(1âˆ’\mathrm{Î»}\right){x}_{1}+\mathrm{Î»}{x}_{2}\right)\right)\\ =& d\left(\left(1âˆ’\mathrm{Î»}\right){x}_{1}+\mathrm{Î»}{x}_{2},\left(1âˆ’\mathrm{Î»}\right)T{x}_{1}+\mathrm{Î»}T{x}_{2}\right)\\ â‰¤& \left(1âˆ’\mathrm{Î»}\right)d\left({x}_{1},T{x}_{1}\right)+\mathrm{Î»}d\left({x}_{2},T{x}_{1}\right)\\ =& dist\left(A,B\right).\end{array}$

This implies that $d\left(z,Tz\right)=dist\left(A,B\right)$, i.e., $zâˆˆ{F}_{A}\left(T\right)$. Therefore, ${F}_{A}\left(T\right)$ is convex. Similarly, it can be shown that ${F}_{B}\left(T\right)$ is a nonempty compact convex subset of ${B}_{0}$.â€ƒâ–¡

Lemma 29 Let A, B be nonempty closed convex subsets of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Let ${A}_{0}$, ${B}_{0}$ be nonempty compact convex and $T,S:AâˆªBâ†’AâˆªB$ be relatively u-continuous mappings such that S and T are commuting on ${F}_{A}\left(T\right)âˆª{F}_{B}\left(T\right)$. Then $S\left({F}_{A}\left(T\right)\right)âŠ†{F}_{B}\left(T\right)$ and $S\left({F}_{B}\left(T\right)\right)âŠ†{F}_{A}\left(T\right)$.

Proof For each $xâˆˆ{F}_{A}\left(T\right)$, we have $d\left(x,Tx\right)=dist\left(A,B\right)$. Since S is a relatively u-continuous mapping, then for $\mathrm{Î´}>0$,

$d\left(x,Tx\right)<\mathrm{Î´}+dist\left(A,B\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(S\left(x\right),S\left(Tx\right)\right)<\mathrm{Îµ}+dist\left(A,B\right)$

for each $\mathrm{Îµ}>0$. Therefore, $d\left(S\left(x\right),S\left(Tx\right)\right)=dist\left(A,B\right)$. The commutativity for S and T on ${F}_{A}\left(T\right)$ implies that $d\left(S\left(x\right),T\left(Sx\right)\right)=dist\left(A,B\right)$. Thus, we deduce that $Sxâˆˆ{F}_{B}\left(T\right)$. This shows that $S\left({F}_{A}\left(T\right)\right)âŠ†{F}_{B}\left(T\right)$. Also, we can prove that $S\left({F}_{B}\left(T\right)\right)âŠ†{F}_{A}\left(T\right)$.â€ƒâ–¡

Now, we define a new class of mappings called cyclic Banach pairs.

Definition 30 Let A, B be nonempty subsets of a metric space $\left(X,d\right)$ and let $T,S:AâˆªBâ†’AâˆªB$ be mappings. The pair $\left\{S,T\right\}$ is called a cyclic Banach pair if $S\left({F}_{A}\left(T\right)\right)âŠ†{F}_{B}\left(T\right)$ and $S\left({F}_{B}\left(T\right)\right)âŠ†{F}_{A}\left(T\right)$.

The following is an example of a pair of non-commuting mappings that are relatively u-continuous and that are a cyclic Banach pair.

Example 31 Let $X={\mathbb{R}}^{2}$ with the Euclidean metric and consider (as in [18])

Let $T,S:AâˆªBâ†’AâˆªB$ be defined as

Then T and S are relatively u-continuous mappings. Since

T and S are non-commuting mappings. Also, $dist\left(A,B\right)=1$. It is easy to verify that

${F}_{A}\left(T\right)=\left\{\left(0,0\right)\right\}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{F}_{B}\left(T\right)=\left\{\left(1,0\right)\right\}$

and

$S\left({F}_{A}\left(T\right)\right)âŠ†{F}_{B}\left(T\right),\phantom{\rule{2em}{0ex}}S\left({F}_{B}\left(T\right)\right)âŠ†{F}_{A}\left(T\right).$

Therefore, $\left\{S,T\right\}$ is a cyclic Banach pair.

The following theorem proves that two relatively u-continuous mappings which are not necessarily commuting have common best proximity points.

Theorem 32 Let A, B be nonempty closed convex subsets of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Let ${A}_{0}$, ${B}_{0}$ be nonempty compact convex and $T,S:AâˆªBâ†’AâˆªB$ be affine relatively u-continuous mappings. If $\left\{S,T\right\}$ is a cyclic Banach pair, then .

Proof By Theorem 28, ${F}_{A}\left(T\right)$ is a nonempty compact convex subset of ${A}_{0}$ and ${F}_{B}\left(T\right)$ is a nonempty compact convex subset of ${B}_{0}$. For each $xâˆˆ{F}_{A}\left(T\right)$, we have

$dist\left(A,B\right)â‰¤dist\left({F}_{A}\left(T\right),{F}_{B}\left(T\right)\right)â‰¤d\left(x,Tx\right)=dist\left(A,B\right),$

which implies that $dist\left({F}_{A}\left(T\right),{F}_{B}\left(T\right)\right)=dist\left(A,B\right)$. By the definition of cyclic Banach pairs $S:{F}_{A}\left(T\right)âˆª{F}_{B}\left(T\right)â†’{F}_{A}\left(T\right)âˆª{F}_{B}\left(T\right)$. Since $\left\{S,T\right\}$ is a cyclic Banach pair and since for each $\mathrm{Îµ}>0$ there exists a $\mathrm{Î´}>0$ such that

$d\left(x,y\right)<\mathrm{Î´}+dist\left(A,B\right)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}d\left(S\left(x\right),S\left(y\right)\right)<\mathrm{Îµ}+dist\left(A,B\right)$

for all $xâˆˆ{F}_{A}\left(T\right)$, $yâˆˆ{F}_{B}\left(T\right)$, hence S is a relatively u-continuous mapping on ${F}_{A}\left(T\right)âˆª{F}_{B}\left(T\right)$. The conditions of Theorem 27 are satisfied, so there exists ${x}_{0}âˆˆ{F}_{A}\left(T\right)$ such that

$d\left({x}_{0},S{x}_{0}\right)=dist\left({F}_{A}\left(T\right),{F}_{B}\left(T\right)\right)=dist\left(A,B\right).$

Thus, ${x}_{0}âˆˆ{F}_{A}\left(S\right)$. This implies that .â€ƒâ–¡

Next, we will extend Theorem 32 to the case of a countable family of not necessarily commuting relatively u-continuous mappings. Let $\mathrm{Î©}=\left\{{T}_{i}:iâˆˆ\mathbb{N}\right\}$ be a family of relatively u-continuous mappings. Define

for each $i=1,â€¦,n$.

Definition 33 Let A, B be nonempty subsets of a metric space $\left(X,d\right)$ and let $T,S:AâˆªBâ†’AâˆªB$. The pair $\left\{S,T\right\}$ is called a symmetric cyclic Banach pair if $\left\{S,T\right\}$ and $\left\{T,S\right\}$ are cyclic Banach pairs, that is, $S\left({F}_{A}\left(T\right)\right)âŠ†{F}_{B}\left(T\right)$, $S\left({F}_{B}\left(T\right)\right)âŠ†{F}_{A}\left(T\right)$, $T\left({F}_{A}\left(S\right)\right)âŠ†{F}_{B}\left(S\right)$ and $T\left({F}_{B}\left(S\right)\right)âŠ†{F}_{A}\left(S\right)$.

Theorem 34 Let A, B be nonempty closed convex subsets of a global NPC space X with the property that the closed convex hull of every finite subset of X is compact. Let ${A}_{0}$, ${B}_{0}$ be nonempty compact convex and Î© a countable family of affine relatively u-continuous mappings such that $\left\{{T}_{i},{T}_{j}\right\}$ is a symmetric cyclic Banach pair for each $i,jâˆˆ\mathbb{N}$. Then Î© has a common best proximity in A.

Proof First, we prove that . By an analogous argument to the proof of Theorem 32, ${F}_{A}\left({T}_{i}\right)$ is a nonempty compact convex subset of ${A}_{0}$, ${F}_{B}\left({T}_{i}\right)$ is a nonempty compact convex subset of ${B}_{0}$ and $dist\left({F}_{A}\left({T}_{i}\right),{F}_{B}\left({T}_{i}\right)\right)=dist\left(A,B\right)$, for $i=1,2,3$. So, we have ${F}_{A}\left({T}_{1}\right)âˆ©{F}_{A}\left({T}_{2}\right)$ and ${F}_{B}\left({T}_{1}\right)âˆ©{F}_{B}\left({T}_{2}\right)$ are nonempty compact convex with

$dist\left({F}_{A}\left({T}_{1}\right)âˆ©{F}_{A}\left({T}_{2}\right),{F}_{B}\left({T}_{1}\right)âˆ©{F}_{B}\left({T}_{2}\right)\right)=dist\left(A,B\right).$

Suppose that ${T}_{3}$ is a mapping on $\left({F}_{A}\left({T}_{1}\right)âˆ©{F}_{A}\left({T}_{2}\right)\right)âˆª\left({F}_{B}\left({T}_{1}\right)âˆ©{F}_{B}\left({T}_{2}\right)\right)$. Since both of $\left\{{T}_{3},{T}_{1}\right\}$ and $\left\{{T}_{3},{T}_{2}\right\}$ are cyclic Banach pairs, ${T}_{3}$ is a relatively u-continuous mapping on $\left({F}_{A}\left({T}_{1}\right)âˆ©{F}_{A}\left({T}_{2}\right)\right)âˆª\left({F}_{B}\left({T}_{1}\right)âˆ©{F}_{B}\left({T}_{2}\right)\right)$. From Theorem 27, ${T}_{3}$ has a best proximity point $zâˆˆ{F}_{A}\left({T}_{1}\right)âˆ©{F}_{A}\left({T}_{2}\right)$. This shows that .

By induction, for a finite symmetric cyclic Banach family ${\mathrm{Î©}}^{\mathrm{â€²}}=\left\{{T}_{1},{T}_{2},â€¦,{T}_{n}\right\}$ of affine relatively u-continuous mappings, there exists ${x}_{0}âˆˆ{â‹‚}_{i=1}^{n}{F}_{A}\left({T}_{i}\right)$.

Now, let $\mathrm{Î©}=\left\{{T}_{i}:iâˆˆ\mathbb{N}\right\}$. For each ${T}_{i}$, ${F}_{A}\left({T}_{i}\right)$ is a nonempty compact convex of ${A}_{0}$, and for $i=1,â€¦,n$, we have

This shows that the set $\left\{{F}_{A}\left({T}_{i}\right):iâˆˆ\mathbb{N}\right\}$ has a finite intersection property. Thus, we have

i.e., Î“ has a common best proximity point in A.â€ƒâ–¡

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## Acknowledgements

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (3-843-D1432). The authors, therefore, acknowledge DSR with thanks for technical and financial support. The authors are in debt to the anonymous reviewers whose comments helped improve the quality of the paper.

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Alghamdi, M.A., Alghamdi, M.A. & Shahzad, N. Best proximity point results in geodesic metric spaces. Fixed Point Theory Appl 2012, 234 (2012). https://doi.org/10.1186/1687-1812-2012-234