We’d like to understand how you use our websites in order to improve them. Register your interest.

# 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 $(X,d)$ and $T:A→X$. A solution to the equation $Tx=x$ is called a fixed point of T. It is obvious that the condition $T(A)∩A≠∅$ is necessary for the existence of a fixed point for T. But there occur situations in which $d(x,Tx)>0$ for all $x∈A$. In such a situation, it is natural to find a point $x∈A$ 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 $T:A→X$ be a continuous function. Then there exists $x∈A$ such that $∥x−Tx∥=dist(Tx,A)=inf{∥Tx−a∥:a∈A}$.

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

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 $(X,d)$. A mapping $T:A∪B→A∪B$ is said to be a relatively nonexpansive mapping if

1. (i)

$T(A)⊆B$, $T(B)⊆A$;

2. (ii)

$d(Tx,Ty)≤d(x,y)$, for all $x∈A$, $y∈B$.

Theorem 3 

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

$∥x−Tx∥=∥Ty−y∥=dist(A,B).$

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 $T:A∪B→A∪B$ is said to be a relatively u-continuous mapping if it satisfies:

1. (i)

$T(A)⊆B$, $T(B)⊆A$;

2. (ii)

for each $ε>0$, there exists a $δ>0$ such that $d(Tx,Ty)<ε+dist(A,B)$, whenever $d(x,y)<δ+dist(A,B)$, for all $x∈A$, $y∈B$.

Theorem 6 

Let A, B be nonempty compact convex subsets of a strictly convex Banach space X and $T:A∪B→A∪B$ be a relatively u-continuous mapping. Then there exists $(x,y)∈A×B$ such that

$∥x−Tx∥=∥y−Ty∥=dist(A,B).$

Remark 7 

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

Example 8 

Let $(X= R 2 , ∥ ⋅ ∥ 2 )$ and consider $A={(0,t):0≤t≤1}$ and $B={(1,s):0≤s≤1}$. Define $T:A∪B→A∪B$ 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 $(X,d)$ 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 $c:[0,l]⊆R→X$ such that $c(0)=x$, $c(l)=y$, and $d(c(t),c( t ′ ))=|t− t ′ |$ for all t, $t ′ ∈[0,l]$. Moreover, X is called uniquely geodesic if there is exactly one geodesic joining x and y for each $x,y∈X$.

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

A point $z∈X$ belongs to the geodesic segment $[x,y]$ if and only if there exists $t∈[0,1]$ 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.

The metric $d:X×X→R$ in a geodesic space $(X,d)$ is convex if

$d ( z , ( 1 − t ) x + t y ) ≤(1−t)d(z,x)+td(z,y)$

for any $x,y,z∈X$ and $t∈[0,1]$.

Definition 10 

A geodesic metric space X is said to be strictly convex if for every $r>0$, a, x and $y∈X$ with $d(x,a)≤r$, $d(y,a)≤r$ and $x≠y$, it is the case that $d(a,p), where p is any point between x and y such that $p≠x$ and $p≠y$, 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 $CAT(0)$ spaces, is defined in  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 $x 0$, $x 1 ∈X$ there exists a point $y∈X$ such that for all points $z∈X$,

$d 2 (z,y)≤ 1 2 d 2 (z, x 0 )+ 1 2 d 2 (z, x 1 )− 1 4 d 2 ( x 0 , x 1 ).$

Proposition 13 

If $(X,d)$ is a global NPC space, then it is a geodesic space. Moreover, for any pair of points $x 0 , x 1 ∈X$ there exists a unique geodesic $γ:[0,1]→X$ connecting them. For $t∈[0,1]$ the intermediate points $γ t$ depend continuously on the endpoints $x 0$, $x 1$. Finally, for any $z∈X$,

$d 2 (z, γ t )≤(1−t) d 2 (z, x 0 )+t d 2 (z, x 1 )−t(1−t) d 2 ( x 0 , x 1 ).$

Corollary 14 

Let $(X,d)$ be a global NPC space, $γ,η:[0,1]→X$ be geodesics and $t∈[0,1]$. Then

$d 2 ( γ t , η t )≤(1−t) d 2 ( γ 0 , η 0 )+t d 2 ( γ 1 , η 1 )−t(1−t) [ d ( γ 0 , γ 1 ) − d ( η 0 , η 1 ) ] 2$

and

$d( γ t , η t )≤(1−t)d( γ 0 , η 0 )+td( γ 1 , η 1 ).$

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

We need the following notations in the sequel. Let $(X,d)$ be a metric space and A, B be nonempty subsets of X. Define Given C a nonempty subset of X, the metric projection $P C :X→ 2 C$ 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 $T:A∪B→A∪B$ is said to be affine if

$T ( λ x + ( 1 − λ ) y ) =λTx+(1−λ)Ty,$

for all $x,y∈A$ or $x,y∈B$ and $λ∈(0,1)$.

Definition 19 

Let X be a metric space. A subset C of X is called approximatively compact if for any $y∈X$ and for any sequence ${ x n }$ in C such that $d( x n ,y)→dist(y,C)$ as $n→∞$, ${ x n }$ 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 $(X,d)$, there exists a unique map $P C :X→C$ (projection onto C) such that

$d ( P C ( x ) , x ) = inf y ∈ C d(x,y) for every x∈X;$
2. (ii)

$P C$ is orthogonal in the sense that

$d 2 (x,y)≥ d 2 ( x , P C ( x ) ) + d 2 ( P C ( x ) , y )$

for every $x∈X$, $y∈C$;

1. (iii)

$P C$ is nonexpansive,

$d ( P C ( x ) , P C ( z ) ) ≤d(x,z) for every x,z∈X.$

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 $(X,d)$, there exists a unique smallest convex set $co(A)= ⋃ n = 0 ∞ A n$, containing A and called convex hull of A. Where $A 0 =A$, and for $n∈N$, 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 $T:C→C$, 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 $A 0 ≠∅$ and $T:A∪B→A∪B$ be a relatively u-continuous mapping. Then $T( A 0 )⊆ B 0$ and $T( B 0 )⊆ A 0$.

Proof Choose $x∈ A 0$, then there exists $y∈B$ such that $d(x,y)=dist(A,B)$. But T is a relatively u-continuous mapping, then for each $ε>0$, there exists a $δ>0$ such that

$d(p,q)<δ+dist(A,B)impliesd(Tp,Tq)<ε+dist(A,B)$

for each $p∈A$, $q∈B$. Since $d(x,y)<δ+dist(A,B)$ for any $δ>0$, hence

$dist(A,B)≤d(Tx,Ty)<ε+dist(A,B)$

for each $ε>0$. Therefore, $d(Tx,Ty)=dist(A,B)$ and then $T(x)∈ B 0$. This shows that $T( A 0 )⊆ B 0$. Similarly, it can be seen that $T( B 0 )⊆ A 0$. □

Proposition 25 Let A, B be nonempty closed convex subsets of a global NPC space X with $A 0 ≠∅$, $T:A∪B→A∪B$ be a relatively u-continuous mapping, and $P:A∪B→A∪B$ be a mapping defined by

$P(x)={ P B ( x ) , if x ∈ A , P A ( x ) , if x ∈ B .$

Then $TP(x)=P(Tx)$ for all $x∈ A 0 ∪ B 0$, i.e., $P A (T(x))=T( P B (x))$ for $x∈ A 0$ and $T( P A (y))= P B (T(y))$ for $y∈ B 0$.

Proof Choose $x∈ A 0$, then there exists $y∈B$ 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 $T(x)∈ B 0$ and $T(y)∈ A 0$. Again, in view of the uniqueness of the projection operator, we have

$P A ( T ( x ) ) =T(y)=T ( P B ( x ) ) .$

So, $P A (T(x))=T( P B (x))$ for any $x∈ A 0$. Similarly, it can be shown that $T( P A (y))= P B (T(y))$ for any $y∈ B 0$. □

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

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

$d( x n m ,y)→dist(A,B)butd ( x n m , P A ( y ) ) ≥ε.$

Since $A 0$ is approximatively compact, there exists a subsequence ${ x n m ′ }$ of ${ x n m }$ which converges to a point $x∈A$. Hence,

$d( x n m ′ ,y)→d(x,y).$

Also,

$d( x n m ′ ,y)→dist(A,B).$

Thus, $d(x,y)=dist(A,B)$. By Proposition 20, it follows that $x= P A (y)$. Finally, we obtain

$d ( x n m ′ , P A ( y ) ) →d ( x , P A ( y ) ) ≥ε,$

which implies that $x≠ P A (y)$. This leads to a contradiction and therefore $x n → P A (y)$. □

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 $T:A∪B→A∪B$ be a relatively u-continuous mapping. Then there exist $x 0 ∈A$, $y 0 ∈B$ such that

$d( x 0 ,T x 0 )=d( y 0 ,T y 0 )=dist(A,B).$

Proof By Proposition 24, since T is a relatively u-continuous mapping, we have $T( A 0 )⊆ B 0$ and $T( B 0 )⊆ A 0$. The result follows from Theorem 23 once we show that $P A ∘T: A 0 → A 0$ is a continuous mapping, where $P A :X→A$ is a metric projection operator.

To prove this, first notice that $P A ( B 0 )⊆ A 0$. Since X is a global NPC space, by Proposition 20, we obtain that $P A :X→A$ 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 $x n → x 0$ for some $x 0 ∈ A 0$. From Proposition 25, we have

$P B ( P A ( T x 0 ) ) = P B ( T ( P B x 0 ) ) =T ( P A ( P B x 0 ) ) =T x 0 .$

Notice that

$d ( x n , P B ( x 0 ) ) ≤ d ( x n , x 0 ) + d ( x 0 , P B ( x 0 ) ) = d ( x n , x 0 ) + dist ( A , B ) → dist ( A , B )$
(3.1)

as $n→∞$. Since T is relatively u-continuous, for each $ε>0$, there exists a $δ>0$ such that $d(x,y)<δ+dist(A,B)$ implies $d(Tx,Ty)<ε+dist(A,B)$ for all $x∈A$, $y∈B$. From (3.1), with this $δ>0$, it follows that there is $N∈N$ such that $d( x n , P B ( x 0 ))<δ+dist(A,B)$ for all $n≥N$. This implies

$d ( T ( x n ) , T ( P B ( x 0 ) ) ) <ε+dist(A,B)$

for all $n≥N$. Therefore,

$d ( T ( x n ) , P A ( T x 0 ) ) =d ( T ( x n ) , T ( P B ( x 0 ) ) ) →dist(A,B).$

This together with Theorem 26 implies that $T x n → P B ( P A (T x 0 ))=T x 0$. Thus, T is continuous on $A 0$.

Now, since $P A ∘T$ is a continuous mapping of $A 0$, by the Schauder fixed point theorem for a global NPC space, Theorem 23, $P A ∘T$ has a fixed point $x 0 ∈ A 0$. From $P A (T x 0 )= x 0$, we find that $d( x 0 ,T x 0 )=dist(T x 0 ,A)$. But since $T x 0 ∈ B 0$, there is $x ′ ∈ A 0$ such that $d( x ′ ,T x 0 )=dist(A,B)$. Consequently,

$dist(A,B)≤dist(T x 0 ,A)≤d ( T x 0 , x ′ ) =dist(A,B),$

which gives

$dist(T x 0 ,A)=dist(A,B).$

Thus, $d( x 0 ,T x 0 )=dist(A,B)$. 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 $T:A∪B→A∪B$ 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 $x n → x 0$ for some $x 0 ∈ A 0$. By the continuity of T on $A 0$, we have $x 0 ∈ F A (T)$. Therefore, $F A (T)$ is closed and then compact. Now we claim that $F A (T)$ is convex. In fact, let $λ∈[0,1]$, $x 1$, $x 2 ∈ F A (T)$, and $z=(1−λ) x 1 +λ x 2$. Since the distance function d is convex with respect to both variables, by Corollary 14, we have

$dist ( A , B ) ≤ d ( z , T z ) = d ( ( 1 − λ ) x 1 + λ x 2 , T ( ( 1 − λ ) x 1 + λ x 2 ) ) = d ( ( 1 − λ ) x 1 + λ x 2 , ( 1 − λ ) T x 1 + λ T x 2 ) ≤ ( 1 − λ ) d ( x 1 , T x 1 ) + λ d ( x 2 , T x 1 ) = dist ( A , B ) .$

This implies that $d(z,Tz)=dist(A,B)$, i.e., $z∈ F A (T)$. 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 $T,S:A∪B→A∪B$ be relatively u-continuous mappings such that S and T are commuting on $F A (T)∪ F B (T)$. Then $S( F A (T))⊆ F B (T)$ and $S( F B (T))⊆ F A (T)$.

Proof For each $x∈ F A (T)$, we have $d(x,Tx)=dist(A,B)$. Since S is a relatively u-continuous mapping, then for $δ>0$,

$d(x,Tx)<δ+dist(A,B)impliesd ( S ( x ) , S ( T x ) ) <ε+dist(A,B)$

for each $ε>0$. 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 $Sx∈ F B (T)$. This shows that $S( F A (T))⊆ F B (T)$. Also, we can prove that $S( F B (T))⊆ F A (T)$. □

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 $T,S:A∪B→A∪B$ be mappings. The pair ${S,T}$ is called a cyclic Banach pair if $S( F A (T))⊆ F B (T)$ and $S( F B (T))⊆ F A (T)$.

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

Example 31 Let $X= R 2$ with the Euclidean metric and consider (as in ) Let $T,S:A∪B→A∪B$ be defined as Then T and S are relatively u-continuous mappings. Since

$TS(0,x)≠ST(0,x),$

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

$F A (T)= { ( 0 , 0 ) } and F B (T)= { ( 1 , 0 ) }$

and

$S ( F A ( T ) ) ⊆ F B (T),S ( F B ( T ) ) ⊆ F A (T).$

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 $T,S:A∪B→A∪B$ be affine relatively u-continuous mappings. If ${S,T}$ is a cyclic Banach pair, then $F A (T)∩ F A (S)≠∅$.

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 $x∈ F A (T)$, we have

$dist(A,B)≤dist ( F A ( T ) , F B ( T ) ) ≤d(x,Tx)=dist(A,B),$

which implies that $dist( F A (T), F B (T))=dist(A,B)$. By the definition of cyclic Banach pairs $S: F A (T)∪ F B (T)→ F A (T)∪ F B (T)$. Since ${S,T}$ is a cyclic Banach pair and since for each $ε>0$ there exists a $δ>0$ such that

$d(x,y)<δ+dist(A,B)impliesd ( S ( x ) , S ( y ) ) <ε+dist(A,B)$

for all $x∈ F A (T)$, $y∈ F B (T)$, hence S is a relatively u-continuous mapping on $F A (T)∪ F B (T)$. The conditions of Theorem 27 are satisfied, so there exists $x 0 ∈ F A (T)$ such that

$d( x 0 ,S x 0 )=dist ( F A ( T ) , F B ( T ) ) =dist(A,B).$

Thus, $x 0 ∈ F A (S)$. This implies that $F A (T)∩ F A (S)≠∅$. □

Next, we will extend Theorem 32 to the case of a countable family of not necessarily commuting relatively u-continuous mappings. Let $Ω={ T i :i∈N}$ be a family of relatively u-continuous mappings. Define for each $i=1,…,n$.

Definition 33 Let A, B be nonempty subsets of a metric space $(X,d)$ and let $T,S:A∪B→A∪B$. The pair ${S,T}$ is called a symmetric cyclic Banach pair if ${S,T}$ and ${T,S}$ are cyclic Banach pairs, that is, $S( F A (T))⊆ F B (T)$, $S( F B (T))⊆ F A (T)$, $T( F A (S))⊆ F B (S)$ and $T( F B (S))⊆ F A (S)$.

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 $i,j∈N$. Then Ω has a common best proximity in A.

Proof First, we prove that $F A ( T 1 )∩ F A ( T 2 )∩ F A ( T 3 )≠∅$. 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 $F A ( T 1 )∩ F A ( T 2 )$ and $F B ( T 1 )∩ F B ( T 2 )$ are nonempty compact convex with

$dist ( F A ( T 1 ) ∩ F A ( T 2 ) , F B ( T 1 ) ∩ F B ( T 2 ) ) =dist(A,B).$

Suppose that $T 3$ is a mapping on $( F A ( T 1 )∩ F A ( T 2 ))∪( F B ( T 1 )∩ F B ( T 2 ))$. 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 $( F A ( T 1 )∩ F A ( T 2 ))∪( F B ( T 1 )∩ F B ( T 2 ))$. From Theorem 27, $T 3$ has a best proximity point $z∈ F A ( T 1 )∩ F A ( T 2 )$. This shows that $F A ( T 1 )∩ F A ( T 2 )∩ F A ( T 3 )≠∅$.

By induction, for a finite symmetric cyclic Banach family $Ω ′ ={ T 1 , T 2 ,…, T n }$ of affine relatively u-continuous mappings, there exists $x 0 ∈ ⋂ i = 1 n F A ( T i )$.

Now, let $Ω={ T i :i∈N}$. For each $T i$, $F A ( T i )$ is a nonempty compact convex of $A 0$, and for $i=1,…,n$, we have

$⋂ i = 1 n F A ( T i )≠∅.$

This shows that the set ${ F A ( T i ):i∈N}$ has a finite intersection property. Thus, we have

$⋂ i = 1 ∞ F A ( T i )≠∅,$

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

## References

1. 1.

Fan K: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 1969, 122: 234–240.

2. 2.

Eldred AA, Kirk WA, Veeramani P: Proximal normal structure and relatively nonexpansive mappings. Stud. Math. 2005, 171(3):283–293. 10.4064/sm171-3-5

3. 3.

Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323(2):1001–1006. 10.1016/j.jmaa.2005.10.081

4. 4.

Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156(18):2942–2948. 10.1016/j.topol.2009.01.017

5. 5.

Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022

6. 6.

Markin J, Shahzad N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal. 2009, 70(6):2435–2441. 10.1016/j.na.2008.03.045

7. 7.

Vetro C: Best proximity points: convergence and existence theorems for p -cyclic mappings. Nonlinear Anal. 2010, 73: 2283–2291. 10.1016/j.na.2010.06.008

8. 8.

Sankar Raj V, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10(1):21–28.

9. 9.

Eldred AA, Sankar Raj V, Veeramani P: On best proximity pair theorems for relatively u-continuous mappings. Nonlinear Anal. 2011, 74: 3870–3875. 10.1016/j.na.2011.02.021

10. 10.

Fernández-León A: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Anal. 2010, 73: 915–921. 10.1016/j.na.2010.04.005

11. 11.

Bridson MR, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.

12. 12.

Papadopoulus A: Metric Spaces, Convexity and Nonpositive Curvature. European Math. Soc., Zurich; 2005.

13. 13.

Sturm, KT: Probability measures on metric spaces of nonpositive curvature. In: Auscher, P, Coulhon, T, Grigor’yan, A (eds.) Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces. Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs. Paris, France. Contemporary Mathematics, vol. 338, pp. 357-390 (2003)

14. 14.

Ballmann W DMV Seminar 25. In Lectures in Spaces of Nonpositive Curvature. Birkhäuser, Basel; 2005.

15. 15.

Jost J Lectures in Mathematics ETH Zurich. In Nonpositive Curvature: Geometric and Analytic Aspects. Birkhäuser, Basel; 1997.

16. 16.

Niculescu CP, Roventa I: Fan’s inequality in geodesic spaces. Appl. Math. Lett. 2009, 22: 1529–1533. 10.1016/j.aml.2009.03.020

17. 17.

Niculescu CP, Roventa I: Schauder fixed point theorem in spaces with global nonpositive curvature. Fixed Point Theory Appl. 2009., 2009: Article ID 906727

18. 18.

Sankar Raj V: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052

## 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.

## Author information

Authors

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

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 