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# Best proximity point results in geodesic metric spaces

- Maryam A Alghamdi
^{1}, - Mohammed A Alghamdi
^{2}and - Naseer Shahzad
^{2}Email author

**2012**:234

https://doi.org/10.1186/1687-1812-2012-234

© Alghamdi et al.; licensee Springer 2012

**Received:**17 May 2012**Accepted:**7 November 2012**Published:**28 December 2012

## 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.

## Keywords

- Convex Subset
- Nonexpansive Mapping
- Nonempty Subset
- Finite Subset
- Nonempty Closed Convex Subset

## 1 Introduction

Let *A* be a nonempty subset of a metric space $(X,d)$ 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 $T(A)\cap A\ne \mathrm{\varnothing}$ is necessary for the existence of a fixed point for *T*. But there occur situations in which $d(x,Tx)>0$ 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 [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* $T:A\to X$ *be a continuous function*. *Then there exists* $x\in A$ *such that* $\parallel x-Tx\parallel =dist(Tx,A)=inf\{\parallel Tx-a\parallel :a\in A\}$.

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 $d({x}_{0},T{x}_{0})=dist(A,B)$. Some interesting results in approximation theory can be found in [2–8].

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]

*A*,

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

- (i)
$T(A)\subseteq B$, $T(B)\subseteq A$;

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

**Theorem 3** [2]

*Let*$(A,B)$

*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*$(A,B)$

*has a proximal normal structure*.

*Then there exists*$(x,y)\in A\times B$

*such that*

**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]

*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:

- (i)
$T(A)\subseteq B$, $T(B)\subseteq A$;

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

**Theorem 6** [9]

*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*$(x,y)\in A\times B$

*such that*

**Remark 7** [9]

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

**Example 8** [9]

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 $(X,d)$ 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 $c:[0,l]\subseteq \mathbb{R}\to X$ such that $c(0)=x$, $c(l)=y$, and $d(c(t),c({t}^{\mathrm{\prime}}))=|t-{t}^{\mathrm{\prime}}|$ for all *t*, ${t}^{\mathrm{\prime}}\in [0,l]$. 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 $d(x,m)=d(y,m)=\frac{1}{2}d(x,y)$. 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 $[x,y]$ if and only if there exists $t\in [0,1]$ such that $d(z,x)=td(x,y)$ and $d(z,y)=(1-t)d(x,y)$. Hence, we write $z=(1-t)x+ty$.

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*.

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

**Definition 10** [10]

A geodesic metric space *X* is said to be strictly convex if for every $r>0$, *a*, *x* and $y\in X$ with $d(x,a)\le r$, $d(y,a)\le r$ and $x\ne y$, it is the case that $d(a,p)<r$, 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** [10]

Every strictly convex metric space is uniquely geodesic.

In [10], 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 [11–13].

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

**Definition 12**A global NPC space is a complete metric space $(X,d)$ 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 $z\in X$,

**Proposition 13** [13]

*If*$(X,d)$

*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*$\gamma :[0,1]\to X$

*connecting them*.

*For*$t\in [0,1]$

*the intermediate points*${\gamma}_{t}$

*depend continuously on the endpoints*${x}_{0}$, ${x}_{1}$.

*Finally*,

*for any*$z\in X$,

**Corollary 14** [13]

*Let*$(X,d)$

*be a global NPC space*, $\gamma ,\eta :[0,1]\to X$

*be geodesics and*$t\in [0,1]$.

*Then*

*and*

Corollary 14 shows that the distance function $(x,y)\mapsto d(x,y)$ 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 [13–16].

*A*,

*B*be nonempty subsets of

*X*. Define

*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** [9]

*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

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

**Definition 19** [8]

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 $\{{x}_{n}\}$ in *C* such that $d({x}_{n},y)\to dist(y,C)$ as $n\to \mathrm{\infty}$, $\{{x}_{n}\}$ 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**

- (i)
*For each closed convex set**C**in a global NPC space*$(X,d)$,*there exists a unique map*${P}_{C}:X\to C$ (*projection onto**C*)*such that*$d({P}_{C}(x),x)=\underset{y\in C}{inf}d(x,y)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}\phantom{\rule{0.1em}{0ex}}x\in X;$ - (ii)${P}_{C}$
*is orthogonal in the sense that*${d}^{2}(x,y)\ge {d}^{2}(x,{P}_{C}(x))+{d}^{2}({P}_{C}(x),y)$

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

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

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

**Remark 22** [13]

For any subset *A* of a global NPC space $(X,d)$, there exists a unique smallest convex set $\mathit{co}(A)={\bigcup}_{n=0}^{\mathrm{\infty}}{A}_{n}$, 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 [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* $T:C\to C$, *whose image* $T(C)$ *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* $T({A}_{0})\subseteq {B}_{0}$ *and* $T({B}_{0})\subseteq {A}_{0}$.

*Proof*Choose $x\in {A}_{0}$, then there exists $y\in B$ such that $d(x,y)=dist(A,B)$. But

*T*is a relatively

*u*-continuous mapping, then for each $\epsilon >0$, there exists a $\delta >0$ such that

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

**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*

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

*Proof*Choose $x\in {A}_{0}$, then there exists $y\in B$ such that $d(x,y)=dist(A,B)$. According to Proposition 20, since the metric projection is unique, we have $y={P}_{B}(x)$ and $x={P}_{A}(y)$. Recalling that

*T*is relatively

*u*-continuous, therefore, as in the proof of Proposition 24, $d(Tx,Ty)=dist(A,B)$. Thus, it follows that $T(x)\in {B}_{0}$ and $T(y)\in {A}_{0}$. Again, in view of the uniqueness of the projection operator, we have

So, ${P}_{A}(T(x))=T({P}_{B}(x))$ for any $x\in {A}_{0}$. Similarly, it can be shown that $T({P}_{A}(y))={P}_{B}(T(y))$ for any $y\in {B}_{0}$. □

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* $\{{x}_{n}\}$ *is a sequence in *${A}_{0}$, *and* $y\in B$ *such that* $d({x}_{n},y)\to dist(A,B)$, *then* ${x}_{n}\to {P}_{A}(y)$.

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

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

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*

*Proof* By Proposition 24, since *T* is a relatively *u*-continuous mapping, we have $T({A}_{0})\subseteq {B}_{0}$ and $T({B}_{0})\subseteq {A}_{0}$. 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.

*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 $\{{x}_{n}\}$ 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

*T*is relatively

*u*-continuous, for each $\epsilon >0$, there exists a $\delta >0$ such that $d(x,y)<\delta +dist(A,B)$ implies $d(Tx,Ty)<\epsilon +dist(A,B)$ 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 $d({x}_{n},{P}_{B}({x}_{0}))<\delta +dist(A,B)$ for all $n\ge N$. This implies

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

Thus, $d({x}_{0},T{x}_{0})=dist(A,B)$. This completes the proof. □

*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* ${F}_{A}(T)$ *is a nonempty compact convex subset of* ${A}_{0}$ *and* ${F}_{B}(T)$ *is a nonempty compact convex subset of* ${B}_{0}$.

*Proof*It is obvious that ${F}_{A}(T)$ is a nonempty subset of ${A}_{0}$ by Theorem 27. Assume that $\{{x}_{n}\}$ is a sequence in ${F}_{A}(T)$ such that ${x}_{n}\to {x}_{0}$ for some ${x}_{0}\in {A}_{0}$. By the continuity of

*T*on ${A}_{0}$, we have ${x}_{0}\in {F}_{A}(T)$. Therefore, ${F}_{A}(T)$ is closed and then compact. Now we claim that ${F}_{A}(T)$ is convex. In fact, let $\lambda \in [0,1]$, ${x}_{1}$, ${x}_{2}\in {F}_{A}(T)$, and $z=(1-\lambda ){x}_{1}+\lambda {x}_{2}$. Since the distance function

*d*is convex with respect to both variables, by Corollary 14, we have

This implies that $d(z,Tz)=dist(A,B)$, *i.e.*, $z\in {F}_{A}(T)$. Therefore, ${F}_{A}(T)$ is convex. Similarly, it can be shown that ${F}_{B}(T)$ 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* ${F}_{A}(T)\cup {F}_{B}(T)$. *Then* $S({F}_{A}(T))\subseteq {F}_{B}(T)$ *and* $S({F}_{B}(T))\subseteq {F}_{A}(T)$.

*Proof*For each $x\in {F}_{A}(T)$, we have $d(x,Tx)=dist(A,B)$. Since

*S*is a relatively

*u*-continuous mapping, then for $\delta >0$,

for each $\epsilon >0$. Therefore, $d(S(x),S(Tx))=dist(A,B)$. The commutativity for *S* and *T* on ${F}_{A}(T)$ implies that $d(S(x),T(Sx))=dist(A,B)$. Thus, we deduce that $Sx\in {F}_{B}(T)$. This shows that $S({F}_{A}(T))\subseteq {F}_{B}(T)$. Also, we can prove that $S({F}_{B}(T))\subseteq {F}_{A}(T)$. □

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

**Definition 30** Let *A*, *B* be nonempty subsets of a metric space $(X,d)$ and let $T,S:A\cup B\to A\cup B$ be mappings. The pair $\{S,T\}$ is called a cyclic Banach pair if $S({F}_{A}(T))\subseteq {F}_{B}(T)$ and $S({F}_{B}(T))\subseteq {F}_{A}(T)$.

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

*T*and

*S*are relatively

*u*-continuous mappings. Since

*T*and

*S*are non-commuting mappings. Also, $dist(A,B)=1$. It is easy to verify that

Therefore, $\{S,T\}$ 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* $\{S,T\}$ *is a cyclic Banach pair*, *then* ${F}_{A}(T)\cap {F}_{A}(S)\ne \mathrm{\varnothing}$.

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

*S*is a relatively

*u*-continuous mapping on ${F}_{A}(T)\cup {F}_{B}(T)$. The conditions of Theorem 27 are satisfied, so there exists ${x}_{0}\in {F}_{A}(T)$ such that

Thus, ${x}_{0}\in {F}_{A}(S)$. This implies that ${F}_{A}(T)\cap {F}_{A}(S)\ne \mathrm{\varnothing}$. □

*u*-continuous mappings. Let $\mathrm{\Omega}=\{{T}_{i}:i\in \mathbb{N}\}$ 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 $(X,d)$ and let $T,S:A\cup B\to A\cup B$. The pair $\{S,T\}$ is called a symmetric cyclic Banach pair if $\{S,T\}$ and $\{T,S\}$ are cyclic Banach pairs, that is, $S({F}_{A}(T))\subseteq {F}_{B}(T)$, $S({F}_{B}(T))\subseteq {F}_{A}(T)$, $T({F}_{A}(S))\subseteq {F}_{B}(S)$ and $T({F}_{B}(S))\subseteq {F}_{A}(S)$.

**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* $\{{T}_{i},{T}_{j}\}$ *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 ${F}_{A}({T}_{1})\cap {F}_{A}({T}_{2})\cap {F}_{A}({T}_{3})\ne \mathrm{\varnothing}$. By an analogous argument to the proof of Theorem 32, ${F}_{A}({T}_{i})$ is a nonempty compact convex subset of ${A}_{0}$, ${F}_{B}({T}_{i})$ is a nonempty compact convex subset of ${B}_{0}$ and $dist({F}_{A}({T}_{i}),{F}_{B}({T}_{i}))=dist(A,B)$, for $i=1,2,3$. So, we have ${F}_{A}({T}_{1})\cap {F}_{A}({T}_{2})$ and ${F}_{B}({T}_{1})\cap {F}_{B}({T}_{2})$ are nonempty compact convex with

Suppose that ${T}_{3}$ is a mapping on $({F}_{A}({T}_{1})\cap {F}_{A}({T}_{2}))\cup ({F}_{B}({T}_{1})\cap {F}_{B}({T}_{2}))$. Since both of $\{{T}_{3},{T}_{1}\}$ and $\{{T}_{3},{T}_{2}\}$ are cyclic Banach pairs, ${T}_{3}$ is a relatively *u*-continuous mapping on $({F}_{A}({T}_{1})\cap {F}_{A}({T}_{2}))\cup ({F}_{B}({T}_{1})\cap {F}_{B}({T}_{2}))$. From Theorem 27, ${T}_{3}$ has a best proximity point $z\in {F}_{A}({T}_{1})\cap {F}_{A}({T}_{2})$. This shows that ${F}_{A}({T}_{1})\cap {F}_{A}({T}_{2})\cap {F}_{A}({T}_{3})\ne \mathrm{\varnothing}$.

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

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

## Declarations

### 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.

## Authors’ Affiliations

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