# 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 $T:A\to X$. A solution to the equation $Tx=x$ is called a fixed point of T. It is obvious that the condition $T\left(A\right)\cap A\ne \mathrm{\varnothing }$ 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 $x\in A$. In such a situation, it is natural to find a point $x\in A$ such that x is closest to Tx in some sense. The following well-known best approximation theorem, due to Ky Fan , explores the existence of an approximate solution to the equation $Tx=x$.

Theorem 1 

Let A be a nonempty compact convex subset of a normed linear space X and $T:A\to X$ be a continuous function. Then there exists $x\in A$ such that $\parallel x-Tx\parallel =dist\left(Tx,A\right)=inf\left\{\parallel Tx-a\parallel :a\in A\right\}$.

The point $x\in A$ in Theorem 1 is called a best approximant of T in A. Let A, B be nonempty subsets of a metric space X and $T:A\to B$. A point ${x}_{0}\in A$ 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 .

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

Definition 2 

Let A, B be nonempty subsets of a metric space $\left(X,d\right)$. A mapping $T:A\cup B\to A\cup B$ is said to be a relatively nonexpansive mapping if

1. (i)

$T\left(A\right)\subseteq B$, $T\left(B\right)\subseteq A$;

2. (ii)

$d\left(Tx,Ty\right)\le d\left(x,y\right)$, for all $x\in A$, $y\in B$.

Theorem 3 

Let $\left(A,B\right)$ be a nonempty, weakly compact convex pair in a Banach space X. Let $T:A\cup B\to A\cup B$ be a relatively nonexpansive mapping and suppose $\left(A,B\right)$ has a proximal normal structure. Then there exists $\left(x,y\right)\in A×B$ such that

$\parallel x-Tx\parallel =\parallel Ty-y\parallel =dist\left(A,B\right).$

Remark 4 

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

In , 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  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 

Let A, B be nonempty subsets of a metric space X. A mapping $T:A\cup B\to A\cup B$ is said to be a relatively u-continuous mapping if it satisfies:

1. (i)

$T\left(A\right)\subseteq B$, $T\left(B\right)\subseteq A$;

2. (ii)

for each $\epsilon >0$, there exists a $\delta >0$ such that $d\left(Tx,Ty\right)<\epsilon +dist\left(A,B\right)$, whenever $d\left(x,y\right)<\delta +dist\left(A,B\right)$, for all $x\in A$, $y\in B$.

Theorem 6 

Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and $T:A\cup B\to A\cup B$ be a relatively u-continuous mapping. Then there exists $\left(x,y\right)\in A×B$ such that

$\parallel x-Tx\parallel =\parallel y-Ty\parallel =dist\left(A,B\right).$

Remark 7 

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

Example 8 

Let $\left(X={\mathbb{R}}^{2},{\parallel \cdot \parallel }_{2}\right)$ and consider $A=\left\{\left(0,t\right):0\le t\le 1\right\}$ and $B=\left\{\left(1,s\right):0\le s\le 1\right\}$. Define $T:A\cup B\to A\cup B$ by

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

Also, in , 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 

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 $c:\left[0,l\right]\subseteq \mathbb{R}\to X$ such that $c\left(0\right)=x$, $c\left(l\right)=y$, and $d\left(c\left(t\right),c\left({t}^{\mathrm{\prime }}\right)\right)=|t-{t}^{\mathrm{\prime }}|$ for all t, ${t}^{\mathrm{\prime }}\in \left[0,l\right]$. Moreover, X is called uniquely geodesic if there is exactly one geodesic joining x and y for each $x,y\in X$.

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 $z\in X$ belongs to the geodesic segment $\left[x,y\right]$ if and only if there exists $t\in \left[0,1\right]$ such that $d\left(z,x\right)=td\left(x,y\right)$ and $d\left(z,y\right)=\left(1-t\right)d\left(x,y\right)$. Hence, we write $z=\left(1-t\right)x+ty$.

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 $d:X×X\to \mathbb{R}$ in a geodesic space $\left(X,d\right)$ is convex if

$d\left(z,\left(1-t\right)x+ty\right)\le \left(1-t\right)d\left(z,x\right)+td\left(z,y\right)$

for any $x,y,z\in X$ and $t\in \left[0,1\right]$.

Definition 10 

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

Remark 11 

Every strictly convex metric space is uniquely geodesic.

In , Fernández-León proved the existence and uniqueness of best proximity points in strictly convex metric spaces. For more details about geodesic spaces, one may check .

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  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 ${x}_{0}$, ${x}_{1}\in X$ there exists a point $y\in X$ such that for all points $z\in X$,

${d}^{2}\left(z,y\right)\le \frac{1}{2}{d}^{2}\left(z,{x}_{0}\right)+\frac{1}{2}{d}^{2}\left(z,{x}_{1}\right)-\frac{1}{4}{d}^{2}\left({x}_{0},{x}_{1}\right).$

Proposition 13 

If $\left(X,d\right)$ is a global NPC space, then it is a geodesic space. Moreover, for any pair of points ${x}_{0},{x}_{1}\in X$ there exists a unique geodesic $\gamma :\left[0,1\right]\to X$ connecting them. For $t\in \left[0,1\right]$ the intermediate points ${\gamma }_{t}$ depend continuously on the endpoints ${x}_{0}$, ${x}_{1}$. Finally, for any $z\in X$,

${d}^{2}\left(z,{\gamma }_{t}\right)\le \left(1-t\right){d}^{2}\left(z,{x}_{0}\right)+t{d}^{2}\left(z,{x}_{1}\right)-t\left(1-t\right){d}^{2}\left({x}_{0},{x}_{1}\right).$

Corollary 14 

Let $\left(X,d\right)$ be a global NPC space, $\gamma ,\eta :\left[0,1\right]\to X$ be geodesics and $t\in \left[0,1\right]$. Then

${d}^{2}\left({\gamma }_{t},{\eta }_{t}\right)\le \left(1-t\right){d}^{2}\left({\gamma }_{0},{\eta }_{0}\right)+t{d}^{2}\left({\gamma }_{1},{\eta }_{1}\right)-t\left(1-t\right){\left[d\left({\gamma }_{0},{\gamma }_{1}\right)-d\left({\eta }_{0},{\eta }_{1}\right)\right]}^{2}$

and

$d\left({\gamma }_{t},{\eta }_{t}\right)\le \left(1-t\right)d\left({\gamma }_{0},{\eta }_{0}\right)+td\left({\gamma }_{1},{\eta }_{1}\right).$

Corollary 14 shows that the distance function $\left(x,y\right)↦d\left(x,y\right)$ 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 .

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 ${P}_{C}:X\to {2}^{C}$ is the mapping

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

Definition 18 

Let A, B be nonempty convex subsets of a geodesic metric space. A mapping $T:A\cup B\to A\cup B$ is said to be affine if

$T\left(\lambda x+\left(1-\lambda \right)y\right)=\lambda Tx+\left(1-\lambda \right)Ty,$

for all $x,y\in A$ or $x,y\in B$ and $\lambda \in \left(0,1\right)$.

Definition 19 

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

In , 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 ${P}_{C}:X\to C$ (projection onto C) such that

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

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

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

for every $x\in X$, $y\in C$;

1. (iii)

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

$d\left({P}_{C}\left(x\right),{P}_{C}\left(z\right)\right)\le d\left(x,z\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}\phantom{\rule{0.1em}{0ex}}x,z\in X.$

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

Remark 22 

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

Based on Proposition 20, Niculescu and Roventa  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 $T:C\to C$, 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 ${A}_{0}\ne \mathrm{\varnothing }$ and $T:A\cup B\to A\cup B$ be a relatively u-continuous mapping. Then $T\left({A}_{0}\right)\subseteq {B}_{0}$ and $T\left({B}_{0}\right)\subseteq {A}_{0}$.

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

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

for each $p\in A$, $q\in B$. Since $d\left(x,y\right)<\delta +dist\left(A,B\right)$ for any $\delta >0$, hence

$dist\left(A,B\right)\le d\left(Tx,Ty\right)<\epsilon +dist\left(A,B\right)$

for each $\epsilon >0$. Therefore, $d\left(Tx,Ty\right)=dist\left(A,B\right)$ and then $T\left(x\right)\in {B}_{0}$. This shows that $T\left({A}_{0}\right)\subseteq {B}_{0}$. Similarly, it can be seen that $T\left({B}_{0}\right)\subseteq {A}_{0}$. □

Proposition 25 Let A, B be nonempty closed convex subsets of a global NPC space X with ${A}_{0}\ne \mathrm{\varnothing }$, $T:A\cup B\to A\cup B$ be a relatively u-continuous mapping, and $P:A\cup B\to A\cup B$ be a mapping defined by

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

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

Proof Choose $x\in {A}_{0}$, then there exists $y\in B$ 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 $T\left(x\right)\in {B}_{0}$ and $T\left(y\right)\in {A}_{0}$. 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 $x\in {A}_{0}$. Similarly, it can be shown that $T\left({P}_{A}\left(y\right)\right)={P}_{B}\left(T\left(y\right)\right)$ for any $y\in {B}_{0}$. □

By an analogous argument to the proof of Theorem 3.1 , 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 $y\in B$ such that $d\left({x}_{n},y\right)\to dist\left(A,B\right)$, then ${x}_{n}\to {P}_{A}\left(y\right)$.

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

$d\left({x}_{{n}_{m}},y\right)\to dist\left(A,B\right)\phantom{\rule{1em}{0ex}}\text{but}\phantom{\rule{1em}{0ex}}d\left({x}_{{n}_{m}},{P}_{A}\left(y\right)\right)\ge \epsilon .$

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

$d\left({x}_{{n}_{m}^{\mathrm{\prime }}},y\right)\to d\left(x,y\right).$

Also,

$d\left({x}_{{n}_{m}^{\mathrm{\prime }}},y\right)\to dist\left(A,B\right).$

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

$d\left({x}_{{n}_{m}^{\mathrm{\prime }}},{P}_{A}\left(y\right)\right)\to d\left(x,{P}_{A}\left(y\right)\right)\ge \epsilon ,$

which implies that $x\ne {P}_{A}\left(y\right)$. This leads to a contradiction and therefore ${x}_{n}\to {P}_{A}\left(y\right)$. □

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 $T:A\cup B\to A\cup B$ be a relatively u-continuous mapping. Then there exist ${x}_{0}\in A$, ${y}_{0}\in B$ 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 $T\left({A}_{0}\right)\subseteq {B}_{0}$ and $T\left({B}_{0}\right)\subseteq {A}_{0}$. The result follows from Theorem 23 once we show that ${P}_{A}\circ T:{A}_{0}\to {A}_{0}$ is a continuous mapping, where ${P}_{A}:X\to A$ is a metric projection operator.

To prove this, first notice that ${P}_{A}\left({B}_{0}\right)\subseteq {A}_{0}$. Since X is a global NPC space, by Proposition 20, we obtain that ${P}_{A}:X\to A$ 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 ${x}_{n}\to {x}_{0}$ for some ${x}_{0}\in {A}_{0}$. 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

$\begin{array}{rcl}d\left({x}_{n},{P}_{B}\left({x}_{0}\right)\right)& \le & d\left({x}_{n},{x}_{0}\right)+d\left({x}_{0},{P}_{B}\left({x}_{0}\right)\right)\\ =& d\left({x}_{n},{x}_{0}\right)+dist\left(A,B\right)\to dist\left(A,B\right)\end{array}$
(3.1)

as $n\to \mathrm{\infty }$. Since T is relatively u-continuous, for each $\epsilon >0$, there exists a $\delta >0$ such that $d\left(x,y\right)<\delta +dist\left(A,B\right)$ implies $d\left(Tx,Ty\right)<\epsilon +dist\left(A,B\right)$ for all $x\in A$, $y\in B$. From (3.1), with this $\delta >0$, it follows that there is $N\in \mathbb{N}$ such that $d\left({x}_{n},{P}_{B}\left({x}_{0}\right)\right)<\delta +dist\left(A,B\right)$ for all $n\ge N$. This implies

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

for all $n\ge N$. Therefore,

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

This together with Theorem 26 implies that $T{x}_{n}\to {P}_{B}\left({P}_{A}\left(T{x}_{0}\right)\right)=T{x}_{0}$. Thus, T is continuous on ${A}_{0}$.

Now, since ${P}_{A}\circ T$ is a continuous mapping of ${A}_{0}$, by the Schauder fixed point theorem for a global NPC space, Theorem 23, ${P}_{A}\circ T$ has a fixed point ${x}_{0}\in {A}_{0}$. 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 $T{x}_{0}\in {B}_{0}$, there is ${x}^{\mathrm{\prime }}\in {A}_{0}$ such that $d\left({x}^{\mathrm{\prime }},T{x}_{0}\right)=dist\left(A,B\right)$. Consequently,

$dist\left(A,B\right)\le dist\left(T{x}_{0},A\right)\le d\left(T{x}_{0},{x}^{\mathrm{\prime }}\right)=dist\left(A,B\right),$

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. □

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 $T:A\cup B\to A\cup B$ 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 ${x}_{n}\to {x}_{0}$ for some ${x}_{0}\in {A}_{0}$. By the continuity of T on ${A}_{0}$, we have ${x}_{0}\in {F}_{A}\left(T\right)$. 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 $\lambda \in \left[0,1\right]$, ${x}_{1}$, ${x}_{2}\in {F}_{A}\left(T\right)$, and $z=\left(1-\lambda \right){x}_{1}+\lambda {x}_{2}$. Since the distance function d is convex with respect to both variables, by Corollary 14, we have

$\begin{array}{rcl}dist\left(A,B\right)& \le & d\left(z,Tz\right)\\ =& d\left(\left(1-\lambda \right){x}_{1}+\lambda {x}_{2},T\left(\left(1-\lambda \right){x}_{1}+\lambda {x}_{2}\right)\right)\\ =& d\left(\left(1-\lambda \right){x}_{1}+\lambda {x}_{2},\left(1-\lambda \right)T{x}_{1}+\lambda T{x}_{2}\right)\\ \le & \left(1-\lambda \right)d\left({x}_{1},T{x}_{1}\right)+\lambda d\left({x}_{2},T{x}_{1}\right)\\ =& dist\left(A,B\right).\end{array}$

This implies that $d\left(z,Tz\right)=dist\left(A,B\right)$, i.e., $z\in {F}_{A}\left(T\right)$. 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 ${B}_{0}$. □

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 $T,S:A\cup B\to A\cup B$ be relatively u-continuous mappings such that S and T are commuting on ${F}_{A}\left(T\right)\cup {F}_{B}\left(T\right)$. Then $S\left({F}_{A}\left(T\right)\right)\subseteq {F}_{B}\left(T\right)$ and $S\left({F}_{B}\left(T\right)\right)\subseteq {F}_{A}\left(T\right)$.

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

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

for each $\epsilon >0$. 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 $Sx\in {F}_{B}\left(T\right)$. This shows that $S\left({F}_{A}\left(T\right)\right)\subseteq {F}_{B}\left(T\right)$. Also, we can prove that $S\left({F}_{B}\left(T\right)\right)\subseteq {F}_{A}\left(T\right)$. □

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 $T,S:A\cup B\to A\cup B$ be mappings. The pair $\left\{S,T\right\}$ is called a cyclic Banach pair if $S\left({F}_{A}\left(T\right)\right)\subseteq {F}_{B}\left(T\right)$ and $S\left({F}_{B}\left(T\right)\right)\subseteq {F}_{A}\left(T\right)$.

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 ) Let $T,S:A\cup B\to A\cup B$ be defined as Then T and S are relatively u-continuous mappings. Since

$TS\left(0,x\right)\ne ST\left(0,x\right),$

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

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

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 $T,S:A\cup B\to A\cup B$ be affine relatively u-continuous mappings. If $\left\{S,T\right\}$ is a cyclic Banach pair, then ${F}_{A}\left(T\right)\cap {F}_{A}\left(S\right)\ne \mathrm{\varnothing }$.

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 $x\in {F}_{A}\left(T\right)$, we have

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

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 $S:{F}_{A}\left(T\right)\cup {F}_{B}\left(T\right)\to {F}_{A}\left(T\right)\cup {F}_{B}\left(T\right)$. Since $\left\{S,T\right\}$ is a cyclic Banach pair and since for each $\epsilon >0$ there exists a $\delta >0$ such that

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

for all $x\in {F}_{A}\left(T\right)$, $y\in {F}_{B}\left(T\right)$, hence S is a relatively u-continuous mapping on ${F}_{A}\left(T\right)\cup {F}_{B}\left(T\right)$. The conditions of Theorem 27 are satisfied, so there exists ${x}_{0}\in {F}_{A}\left(T\right)$ 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, ${x}_{0}\in {F}_{A}\left(S\right)$. This implies that ${F}_{A}\left(T\right)\cap {F}_{A}\left(S\right)\ne \mathrm{\varnothing }$. □

Next, we will extend Theorem 32 to the case of a countable family of not necessarily commuting relatively u-continuous mappings. Let $\mathrm{\Omega }=\left\{{T}_{i}:i\in \mathbb{N}\right\}$ be a family of relatively u-continuous mappings. Define for each $i=1,\dots ,n$.

Definition 33 Let A, B be nonempty subsets of a metric space $\left(X,d\right)$ and let $T,S:A\cup B\to A\cup B$. 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, $S\left({F}_{A}\left(T\right)\right)\subseteq {F}_{B}\left(T\right)$, $S\left({F}_{B}\left(T\right)\right)\subseteq {F}_{A}\left(T\right)$, $T\left({F}_{A}\left(S\right)\right)\subseteq {F}_{B}\left(S\right)$ and $T\left({F}_{B}\left(S\right)\right)\subseteq {F}_{A}\left(S\right)$.

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 Ω 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 $i,j\in \mathbb{N}$. Then Ω has a common best proximity in A.

Proof First, we prove that ${F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)\cap {F}_{A}\left({T}_{3}\right)\ne \mathrm{\varnothing }$. 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 ${F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)$ and ${F}_{B}\left({T}_{1}\right)\cap {F}_{B}\left({T}_{2}\right)$ are nonempty compact convex with

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

Suppose that ${T}_{3}$ is a mapping on $\left({F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)\right)\cup \left({F}_{B}\left({T}_{1}\right)\cap {F}_{B}\left({T}_{2}\right)\right)$. 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 $\left({F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)\right)\cup \left({F}_{B}\left({T}_{1}\right)\cap {F}_{B}\left({T}_{2}\right)\right)$. From Theorem 27, ${T}_{3}$ has a best proximity point $z\in {F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)$. This shows that ${F}_{A}\left({T}_{1}\right)\cap {F}_{A}\left({T}_{2}\right)\cap {F}_{A}\left({T}_{3}\right)\ne \mathrm{\varnothing }$.

By induction, for a finite symmetric cyclic Banach family ${\mathrm{\Omega }}^{\mathrm{\prime }}=\left\{{T}_{1},{T}_{2},\dots ,{T}_{n}\right\}$ of affine relatively u-continuous mappings, there exists ${x}_{0}\in {\bigcap }_{i=1}^{n}{F}_{A}\left({T}_{i}\right)$.

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

$\bigcap _{i=1}^{n}{F}_{A}\left({T}_{i}\right)\ne \mathrm{\varnothing }.$

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

$\bigcap _{i=1}^{\mathrm{\infty }}{F}_{A}\left({T}_{i}\right)\ne \mathrm{\varnothing },$

i.e., Γ has a common best proximity point in A. □

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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

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### Keywords

• Convex Subset
• Nonexpansive Mapping
• Nonempty Subset
• Finite Subset
• Nonempty Closed Convex Subset 