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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:AX. 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 xA. In such a situation, it is natural to find a point xA 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:AX be a continuous function. Then there exists xA such that xTx=dist(Tx,A)=inf{Txa:aA}.

The point xA 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:AB. 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 [28].

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 (X,d). A mapping T:ABAB 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 xA, yB.

Theorem 3 [2]

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

xTx=Tyy=dist(A,B).

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:ABAB 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 xA, yB.

Theorem 6 [9]

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

xTx=yTy=dist(A,B).

Remark 7 [9]

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

Example 8 [9]

Let (X= R 2 , 2 ) and consider A={(0,t):0t1} and B={(1,s):0s1}. Define T:ABAB by

T(x,y)={ ( 1 , y ) if  x = 0 , ( 0 , y ) if  x = 1 .

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]RX 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,yX.

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 zX 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)=(1t)d(x,y). Hence, we write z=(1t)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×XR in a geodesic space (X,d) is convex if

d ( z , ( 1 t ) x + t y ) (1t)d(z,x)+td(z,y)

for any x,y,zX and t[0,1].

Definition 10 [10]

A geodesic metric space X is said to be strictly convex if for every r>0, a, x and yX with d(x,a)r, d(y,a)r and xy, it is the case that d(a,p)<r, where p is any point between x and y such that px and py, 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 [1113].

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 X there exists a point yX such that for all points zX,

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 [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 zX,

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

Corollary 14 [13]

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

and

d( γ t , η t )(1t)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 [1316].

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

P C (x)= { y C : d ( x , y ) = dist ( x , C ) } ,for every xX,

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:ABAB is said to be affine if

T ( λ x + ( 1 λ ) y ) =λTx+(1λ)Ty,

for all x,yA or x,yB and λ(0,1).

Definition 19 [8]

Let X be a metric space. A subset C of X is called approximatively compact if for any yX 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 [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 (X,d), there exists a unique map P C :XC (projection onto C) such that

    d ( P C ( x ) , x ) = inf y C d(x,y) for every xX;
  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 xX, yC;

  1. (iii)

    P C is nonexpansive,

    d ( P C ( x ) , P C ( z ) ) d(x,z) for every x,zX.

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 co(A)= n = 0 A n , containing A and called convex hull of A. Where A 0 =A, and for nN, 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:CC, 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:ABAB 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 yB 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 pA, qB. 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:ABAB be a relatively u-continuous mapping, and P:ABAB 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 yB 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 [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 yB 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 xA. 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:ABAB 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 :XA 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 :XA 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 xA, yB. From (3.1), with this δ>0, it follows that there is NN such that d( x n , P B ( x 0 ))<δ+dist(A,B) for all nN. This implies

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

for all nN. 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:ABAB 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=(1λ) 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:ABAB 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:ABAB 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 [18])

Let T,S:ABAB 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:ABAB 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 :iN} 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:ABAB. 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,jN. 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 :iN}. 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 ):iN} 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. □

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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