Open Access

Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and CAT ( 0 ) spaces

Fixed Point Theory and Applications20132013:57

https://doi.org/10.1186/1687-1812-2013-57

Received: 4 December 2012

Accepted: 21 February 2013

Published: 14 March 2013

Abstract

An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to CAT ( 0 ) spaces.

AMS Subject Classification:54E40, 54H25, 47H10, 37C25.

Keywords

α-nonexpansive mappingfixed pointIshihawa iteration algorithmuniformly convex Banach space CAT ( 0 ) spaces

1 Introduction

The purpose of this paper is to study fixed point theorems of α-nonexpansive mappings of CAT ( 0 ) spaces. A metric space X is a CAT ( 0 ) space if it is geodesically connected, and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane (see Section 4 for the precise definition). Our approach is to prove firstly weak and strong convergence theorems for Ishikawa iterations of α-nonexpansive mappings in uniformly convex Banach spaces. Then, we extend the results to CAT ( 0 ) spaces.

Here are the details. Let E be a (real) Banach space and let C be a nonempty subset of E. Let T : C E be a mapping. Denote by F ( T ) the set of fixed points of T, i.e., F ( T ) = { x C : T x = x } . We say that T is nonexpansive if T x T y x y for all x, y in C, and that T is quasi-nonexpansive if F ( T ) and T x y x y for all x in C and y in F ( T ) .

The concept of nonexpansivity of a map T from a convex set C into C plays an important role in the study of the Mann-type iteration given by
x n + 1 = β n T x n + ( 1 β n ) x n , x 1 C .
(1.1)
Here, { β n } is a real sequence in [ 0 , 1 ] satisfying some appropriate conditions, which is usually called a control sequence. A more general iteration scheme is the Ishikawa iteration given by
{ y n = β n T x n + ( 1 β n ) x n , x n + 1 = γ n T y n + ( 1 γ n ) x n ,
(1.2)

where the sequences { β n } and { γ n } satisfy some appropriate conditions. In particular, when all β n = 0 , the Ishikawa iteration (1.2) becomes the standard Mann iteration (1.1). Let T be nonexpansive and let C be a nonempty closed and convex subset of a uniformly convex Banach space E satisfying the Opial property. Takahashi and Kim [1] proved that, for any initial data x 1 in C, the sequence { x n } of iterations defined by the Ishikawa iteration (1.2) converges weakly to a fixed point of T, with appropriate choices of control sequences { β n } and { γ n } .

Following Aoyama and Kohsaka [2], a mapping T : C E is said to be α-nonexpansive for some real number α < 1 if
T x T y 2 α T x y 2 + α T y x 2 + ( 1 2 α ) x y 2 , x , y C .

Clearly, 0-nonexpansive maps are exactly nonexpansive maps. Moreover, T is Lipschitz continuous whenever α 0 . An example of a discontinuous α-nonexpansive mapping (with α > 0 ) has been given in [2]. See also Example 3.6(b).

An existence theorem for a fixed point of an α-nonexpansive mapping T of a nonempty bounded, closed and convex subset C of a uniformly convex Banach space E has been recently established by Aoyama and Kohsaka [2] with a non-constructive argument. In Section 3, we show that, under mild conditions on the control sequences { β n } and { γ n } , the fixed point set F ( T ) is nonempty if and only if the sequence { x n } obtained by the Ishikawa iteration (1.2) is bounded and lim inf n T x n x n = 0 . In this case, { x n } converges weakly or strongly to a fixed point of T.

In Section 5, we establish the existence result of an α-nonexpansive mapping in a CAT ( 0 ) -space in parallel to [2]. We then extend the convergence theorems obtained in Section 3 to the case of CAT ( 0 ) spaces, as we planned.

2 Preliminaries

Let E be a (real) Banach space with the norm and the dual space E . Denote by x n x the strong convergence of a sequence { x n } to x in E and by x n x the weak convergence. The modulus δ of the convexity of E is defined by
δ ( ϵ ) = inf { 1 x + y 2 : x 1 , y 1 , x y ϵ }
for every ϵ with 0 ϵ 2 . A Banach space E is said to be uniformly convex if δ ( ϵ ) > 0 for every 0 < ϵ 2 . Let S = { x E : x = 1 } . The norm of E is said to be Gâteaux differentiable if for each x, y in S, the limit
lim t 0 x + t y x t
(2.1)

exists. In this case, E is called smooth. If the limit (2.1) is attained uniformly in x, y in S, then E is called uniformly smooth. A Banach space E is said to be strictly convex if x + y 2 < 1 whenever x , y S and x y . It is well-known that E is uniformly convex if and only if E is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if E is smooth; for more details, see [3].

A Banach space E is said to satisfy the Opial property [4] if, for every weakly convergent sequence x n x in E, we have
lim sup n x n x < lim sup n x n y

for all y in E with y x . It is well known that all Hilbert spaces, all finite dimensional Banach spaces and the Banach spaces l p ( 1 p < ) satisfy the Opial property, while the uniformly convex spaces L p [ 0 , 2 π ] ( p 2 ) do not; see, for example, [46].

Let { x n } be a bounded sequence in a Banach space E. For any x in E, we set
r ( x , { x n } ) = lim sup n x x n .
The asymptotic radius of { x n } relative to a nonempty closed and convex subset C of E is defined by
r ( C , { x n } ) = inf { r ( x , { x n } ) : x C } .
The asymptotic center of { x n } relative to C is the set
A ( C , { x n } ) = { x C : r ( x , { x n } ) = r ( C , { x n } ) } .

It is well known that if E is uniformly convex, then A ( C , { x n } ) consists of exactly one point; see [7, 8].

Lemma 2.1 Let C be a nonempty subset of a Banach space E. Let T : C E be an α-nonexpansive mapping for some α < 1 such that F ( T ) . Then T is quasi-nonexpansive. Moreover, F ( T ) is norm closed.

Proof Let x C and z F ( T ) . Then we have
T x z 2 = T x T z 2 α T x z 2 + α T z x 2 + ( 1 2 α ) x z 2 = α T x z 2 + α z x 2 + ( 1 2 α ) x z 2 = α T x z 2 + ( 1 α ) x z 2 .
Therefore,
T x z x z .

This inequality ensures the closedness of F ( T ) . □

Lemma 2.2 Let C be a nonempty subset of a Banach space E. Let T : C E be an α-nonexpansive mapping for some α < 1 . Then the following assertions hold.
  1. (i)
    If 0 α < 1 , then
    x T y 2 1 + α 1 α x T x 2 + 2 1 α ( α x y + T x T y ) x T x + x y 2 , x , y C .
     
  2. (ii)
    If α < 0 , then
    x T y 2 x T x 2 + 2 1 α [ ( α ) T x y + T x T y ] x T x + x y 2 , x , y C .
     
Proof
  1. (i)
    Observe
    x T y 2 = x T x + T x T y 2 ( x T x + T x T y ) 2 = x T x 2 + T x T y 2 + 2 x T x T x T y x T x 2 + α T x y 2 + α x T y 2 + ( 1 2 α ) x y 2 + 2 x T x T x T y x T x 2 + α ( T x x + x y ) 2 + α x T y 2 + ( 1 2 α ) x y 2 + 2 x T x T x T y x T x 2 + α T x x 2 + α x y 2 + 2 α T x x x y + α x T y 2 + ( 1 2 α ) x y 2 + 2 x T x T x T y = ( 1 + α ) x T x 2 + 2 α T x x x y + α x T y 2 + ( 1 α ) x y 2 + 2 x T x T x T y .
     
This implies that
x T y 2 1 + α 1 α x T x 2 + 2 1 α ( α x y + T x T y ) x T x + x y 2 .
(ii) Observe
x T y 2 = x T x + T x T y 2 ( x T x + T x T y ) 2 = x T x 2 + T x T y 2 + 2 x T x T x T y x T x 2 + α T x y 2 + α x T y 2 + ( 1 2 α ) x y 2 + 2 x T x T x T y = x T x 2 + α T x y 2 + α x T y 2 + ( 1 α ) x y 2 α x y 2 + 2 x T x T x T y x T x 2 + α T x y 2 + α x T y 2 + ( 1 α ) x y 2 α [ x T x 2 + T x y 2 + 2 x T x T x y ] + 2 x T x T x T y = ( 1 α ) x T x 2 + α x T y 2 + ( 1 α ) x y 2 2 α x T x T x y + 2 x T x T x T y = ( 1 α ) x T x 2 + α x T y 2 + ( 1 α ) x y 2 + 2 [ ( α ) T x y + T x T y ] x T x .
This implies that
x T y 2 x T x 2 + 2 1 α [ ( α ) T x y + T x T y ] x T x + x y 2 .

 □

Proposition 2.3 (Demiclosedness principle)

Let C be a subset of a Banach space E with the Opial property. Let T : C C be an α-nonexpansive mapping for some α < 1 . If { x n } converges weakly to z and lim n T x n x n = 0 , then T z = z . That is, I T is demiclosed at zero, where I is the identity mapping on E.

Proof Since { x n } converges weakly to z and lim n T x n x n = 0 , both { x n } and { T x n } are bounded. Let M 1 = sup { x n , T x n , z , T z : n N } < . If 0 α < 1 , then in view of Lemma 2.2(i),
x n T z 2 1 + α 1 α x n T x n 2 + 2 1 α ( α x n z + T x n T z ) x n T x n + x n z 2 1 + α 1 α x n T x n 2 + 4 M 1 ( 1 + α ) 1 α x n T x n + x n z 2 .
If α < 0 , then in view of Lemma 2.2(ii),
x n T z 2 x n T x n 2 + 2 1 α [ ( α ) T x n z + T x n T z ] x n T x n + x n z 2 x n T x n 2 + 4 M 1 x n T x n + x n z 2 .
These relations imply
lim sup n x n T z lim sup n x n z .

From the Opial property, we obtain T z = z . □

The following result has been proved in [9].

Lemma 2.4 Let r > 0 be a fixed real number. If E is a uniformly convex Banach space, then there exists a continuous strictly increasing convex function g : [ 0 , + ) [ 0 , + ) with g ( 0 ) = 0 such that
λ x + ( 1 λ ) y 2 λ x 2 + ( 1 λ ) y 2 λ ( 1 λ ) g ( x y )

for all x, y in B r ( 0 ) = { u E : u r } and λ [ 0 , 1 ] .

Recently, Aoyama and Kohsaka [2] proved the following fixed point theorem for α-nonexpansive mappings of Banach spaces.

Lemma 2.5 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let T : C C be an α-nonexpansive mapping for some α < 1 . Then the following conditions are equivalent.
  1. (i)

    There exists x in C such that { T n x } n = 1 is bounded.

     
  2. (ii)

    F ( T ) .

     

3 Fixed point and convergence theorems in Banach spaces

Lemma 3.1 Let C be a nonempty closed and convex subset of a Banach space E. Let T : C C be an α-nonexpansive mapping for some α < 1 . Let a sequence { x n } with x 1 in C be defined by the Ishikawa iteration (1.2) such that { β n } and { γ n } are arbitrary sequences in [ 0 , 1 ] . Suppose that the fixed point set F ( T ) contains an element z. Then the following assertions hold.
  1. (1)

    max { x n + 1 z , y n z } x n z for all n = 1 , 2 ,  .

     
  2. (2)

    lim n x n z exists.

     
  3. (3)

    lim n d ( x n , F ( T ) ) exists, where d ( x , F ( T ) ) denotes the distance from x to F ( T ) .

     

Proof

In view of Lemma 2.1, we conclude that
y n z = β n T x n + ( 1 β n ) x n z β n T x n z + ( 1 β n ) x n z β n x n z + ( 1 β n ) x n z = x n z .
Consequently,
x n + 1 z = γ n T y n + ( 1 γ n ) x n z γ n T y n z + ( 1 γ n ) x n z γ n y n z + ( 1 γ n ) x n z γ n x n z + ( 1 γ n ) x n z = x n z .

This implies that { x n z } is a bounded and nonincreasing sequence. Thus, lim n x n z exists.

In the same manner, we see that { d ( x n , F ( T ) ) } is also a bounded nonincreasing real sequence, and thus converges. □

Theorem 3.2 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] and let { x n } be a sequence with x 1 in C defined by the Ishikawa iteration (1.2).
  1. 1.

    If { x n } is bounded and lim inf n T x n x n = 0 , then the fixed point set F ( T ) .

     
  2. 2.

    Assume F ( T ) . Then { x n } is bounded, and the following hold.

     
Case 1: 0 < α < 1 .
  1. (a)

    lim inf n T x n x n = 0 when lim sup n γ n ( 1 γ n ) > 0 .

     
  2. (b)

    lim n T x n x n = 0 when lim inf n γ n ( 1 γ n ) > 0 .

     
Case 2: α 0 .
  1. (a)
    lim inf n T x n x n = 0 when
    { lim inf n γ n ( 1 γ n ) > 0 , lim inf n β n < 1 , or { lim sup n γ n ( 1 γ n ) > 0 , lim sup n β n < 1 .
     
  2. (b)

    lim n T x n x n = 0 when lim inf n γ n ( 1 γ n ) > 0 and lim sup n β n < 1 .

     
Proof Assume that { x n } is bounded and lim inf n T x n x n = 0 . There is a bounded subsequence { T x n k } of { T x n } such that lim k T x n k x n k = 0 . Suppose A ( C , { x n k } ) = { z } . Let M 1 = sup { x n k , T x n k , z , T z : k N } < . If 0 α < 1 , then, by Lemma 2.2(i), we have
x n k T z 2 1 + α 1 α x n k T x n k 2 + 2 1 α ( α x n k z + T x n k T z ) x n k T x n k + x n k z 2 1 + α 1 α x n k T x n k 2 + 4 M 1 ( 1 + α ) 1 α T x n k x n k + x n k z 2 .
This implies that
lim sup k x n k T z 2 1 + α 1 α lim sup k x n k T x n k 2 + 4 M 1 ( 1 + α ) 1 α lim sup k T x n k x n k + lim sup k x n k z 2 = lim sup k x n k z 2 .
If α < 0 , then, by Lemma 2.2(ii), we have
x n k T z 2 x n k T x n k 2 + 2 1 α ( ( α ) T x n k z + T x n k T z ) x n k T x n k + x n k z 2 1 + α 1 α x n k T x n k 2 + 4 M 1 ( 1 + α ) 1 α T x n k x n k + x n k z 2 .
This implies again that
lim sup k x n k T z 2 1 + α 1 α lim sup k x n k T x n k 2 + 4 M 1 ( 1 + α ) 1 α lim sup k T x n k x n k + lim sup k x n k z 2 = lim sup k x n k z 2 .
Thus, we have in all cases
r ( T z , { x n k } ) = lim sup n x n k T z lim sup n x n k z = r ( z , { x n k } ) .

This means that T z A ( C , { x n k } ) . By the uniform convexity of E, we conclude that T z = z .

Conversely, let F ( T ) and let z F ( T ) . It follows from Lemma 3.1 that lim n x n z exists and hence { x n } is bounded. In view of Lemmas 2.1 and 2.4, we obtain a continuous strictly increasing convex function g : [ 0 , + ) [ 0 , + ) with g ( 0 ) = 0 such that
x n + 1 z 2 = γ n T y n + ( 1 γ n ) x n z 2 γ n T y n z 2 + ( 1 γ n ) x n z 2 γ n ( 1 γ n ) g ( T y n x n ) γ n y n z 2 + ( 1 γ n ) x n z 2 γ n ( 1 γ n ) g ( T y n x n ) γ n x n z 2 + ( 1 γ n ) x n z 2 γ n ( 1 γ n ) g ( T y n x n ) = x n z 2 γ n ( 1 γ n ) g ( T y n x n ) .
(3.1)
In view of (3.1), we conclude by applying Lemma 3.1 that
γ n ( 1 γ n ) g ( T y n x n ) x n z 2 x n + 1 z 2 0 , as  n .
It follows that
lim inf n g ( T y n x n ) = 0 whenever  lim sup n γ n ( 1 γ n ) > 0 .
From the property of g, we deduce that
lim inf n T y n x n = 0 in case  lim sup n γ n ( 1 γ n ) > 0 .
(3.2)
In the same manner, we also obtain that
lim n T y n x n = 0 in case  lim inf n γ n ( 1 γ n ) > 0 .
(3.3)
On the other hand, from (1.2) we get
T x n y n = ( 1 β n ) ( T x n x n ) , x n y n = β n ( x n T x n ) .
(3.4)

Observing (3.4), we see that the assertions about the case α 0 follow from (3.2) and (3.3).

In what follows, we discuss the case 0 < α < 1 . Assume first lim inf n γ n ( 1 γ n ) > 0 . By Lemma 2.1 and (3.3), we see that M 2 : = sup { T x n , T y n : n N } < . Since T is α-nonexpansive, in view of (3.4), we obtain
T x n x n 2 = T x n T y n + T y n x n 2 ( T x n T y n + T y n x n ) 2 = T x n T y n 2 + T y n x n 2 + 2 T x n T y n T y n x n α T x n y n 2 + α T y n x n 2 + ( 1 2 α ) x n y n 2 + T y n x n 2 + 4 M 2 T y n x n α ( 1 β n ) ( T x n x n ) 2 + ( α + 1 ) T y n x n 2 + ( 1 2 α ) β n ( x n T x n ) 2 + 4 M 2 T y n x n [ α ( 1 β n ) 2 + ( 1 2 α ) β n 2 ] T x n x n 2 + ( α + 1 ) T y n x n 2 + 4 M 2 T y n x n .
(3.5)
Case (i): If 0 < α < 1 2 , then (3.5) becomes
T x n x n 2 [ α ( 1 β n ) 2 + ( 1 2 α ) β n 2 ] T x n x n 2 + ( α + 1 ) T y n x n 2 + 4 M 2 T y n x n = ( 1 α ) T x n x n 2 + ( α + 1 ) T y n x n 2 + 4 M 2 T y n x n ,
since all β n are in [ 0 , 1 ] . We then derive from (3.3) that
T x n x n 2 1 + α α T y n x n 2 + 4 M 2 α T y n x n 0 , as  n .
(3.6)
Case (ii): If 1 2 α < 1 , then (3.5) becomes
T x n x n 2 [ α ( 1 β n ) 2 + ( 1 2 α ) β n 2 ] T x n x n 2 + ( α + 1 ) T y n x n 2 + 4 M 2 T y n x n α T x n x n 2 + ( α + 1 ) T y n x n 2 + 4 M 2 T y n x n .
We then derive from (3.3) again that
T x n x n 2 1 + α 1 α T y n x n 2 + 4 M 2 1 α T y n x n 0 , as  n .
(3.7)
Finally, we assume lim sup n γ n ( 1 γ n ) > 0 instead. By (3.2) we have subsequences { x n k } and { y n k } of { x n } and { y n } , respectively, such that
lim k T y n k x n k = 0 .

Replacing M 2 by the number sup { T x n k , T y n k : k N } < and dealing with the subsequences { x n k } and { y n k } in (3.6) and (3.7), we will arrive at the desired conclusion that lim k T x n k x n k = 0 . This gives lim inf n T x n x n = 0 . □

Theorem 3.3 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E with the Opial property. Let T : C C be an α-nonexpansive mapping with a nonempty fixed point set F ( T ) for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] and let { x n } be a sequence with x 1 in C defined by the Ishikawa iteration (1.2).

Assume that lim inf n γ n ( 1 γ n ) > 0 , and assume, in addition, lim sup n β n < 1 if α 0 . Then { x n } converges weakly to a fixed point of T.

Proof It follows from Theorem 3.2 that { x n } is bounded and lim n T x n x n = 0 . The uniform convexity of E implies that E is reflexive; see, for example, [3]. Then there exists a subsequence { x n i } of { x n } such that x n i p C as i . In view of Proposition 2.3, we conclude that p F ( T ) . We claim that x n p as n . Suppose on the contrary that there exists a subsequence { x n j } of { x n } converging weakly to some q in C with p q . By Proposition 2.3, we see that q F ( T ) . Lemma 3.1 says that lim n x n z exists for all z in F ( T ) . The Opial property then implies
lim n x n p = lim i x n i p < lim i x n i q = lim n x n q = lim j x n j q < lim j x n j p = lim n x n p .

This is a contradiction. Thus p = q , and the desired assertion follows. □

Theorem 3.4 Let C be a nonempty compact and convex subset of a uniformly convex Banach space E. Let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] .

When 0 < α < 1 , we assume lim sup n γ n ( 1 γ n ) > 0 . When α 0 , we assume either
{ lim inf n γ n ( 1 γ n ) > 0 , lim inf n β n < 1 , or { lim sup n γ n ( 1 γ n ) > 0 , lim sup n β n < 1 .

Let { x n } be a sequence with x 1 in C defined by the Ishikawa iteration (1.2). Then { x n } converges strongly to a fixed point z of T.

Proof Since C is bounded, it follows from Lemma 2.5 that the fixed point set F ( T ) of T is nonempty. In view of Theorem 3.2, the sequence { x n } is bounded and lim inf n T x n x n = 0 . By the compactness of C, there exists a subsequence { x n k } of { x n } converging strongly to some z in C, and lim k T x n k x n k = 0 . In particular, { T x n k } is bounded. Let M 3 = sup { x n k , T x n k , z , T z : k N } < . If 0 α < 1 , then, in view of Lemma 2.2(i), we obtain
x n k T z 2 1 + α 1 α x n k T x n k 2 + 2 1 α ( α x n k z + T x n k T z ) x n k T x n k + x n k z 2 1 + α 1 α x n k T x n k 2 + 4 M 3 ( 1 + α ) 1 α T x n k x n k + x n k z 2 .
Therefore,
lim sup k x n k T z 2 1 + α 1 α lim sup k x n k T x n k 2 + 4 M 3 ( 1 + α ) 1 α lim sup k T x n k x n k + lim sup k x n k z 2 .
If α < 0 , then, in view of Lemma 2.2(ii), we obtain
x n k T z 2 x n k T x n k 2 + 2 1 α [ ( α ) T x n k z + T x n k T z ] x n k T x n k + x n k z 2 x n k T x n k 2 + 4 M 3 ( 1 α ) 1 α T x n k x n k + x n k z 2 .
Therefore,
lim sup k x n k T z 2 lim sup k x n k T x n k 2 + 4 M 3 lim sup k T x n k x n k + lim sup k x n k z 2 .

It follows that lim k x n k T z = 0 . Thus we have T z = z . By Lemma 3.1, lim n x n z exists. Therefore, z is the strong limit of the sequence { x n } . □

Let C be a nonempty closed and convex subset of a Banach space E. A mapping T : C C is said to satisfy condition (I) [10] if

there exists a nondecreasing function f : [ 0 , ) [ 0 , ) with f ( 0 ) = 0 and f ( r ) > 0 for all r > 0 such that
d ( x , T x ) f ( d ( x , F ( T ) ) ) , x C .

Using Theorem 3.2, we can prove the following result.

Theorem 3.5 Let C be a nonempty closed and convex subset of a uniformly convex Banach space E. Let T : C C be an α-nonexpansive mapping with a nonempty fixed point set F ( T ) for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] . When 0 < α < 1 , we assume lim sup n γ n ( 1 γ n ) > 0 . When α 0 , we assume either
{ lim inf n γ n ( 1 γ n ) > 0 , lim inf n β n < 1 , or { lim sup n γ n ( 1 γ n ) > 0 , lim sup n β n < 1 .

Let { x n } be a sequence with x 1 in C defined by the Ishikawa iteration (1.2). If T satisfies condition (I), then { x n } converges strongly to a fixed point z of T.

Proof

It follows from Theorem 3.2 that
lim inf n T x n x n = 0 .
Therefore, there is a subsequence { x n k } of { x n } such that
lim k T x n k x n k = 0 .
Since T satisfies condition (I), with respect to the sequence { x n k } , we obtain
lim k d ( x n k , F ( T ) ) = 0 .
This implies that, there exist a subsequence of { x n k } , denoted also by { x n k } , and a sequence { z k } in F ( T ) such that
d ( x n k , z k ) < 1 2 k , k N .
(3.8)
In view of Lemma 3.1, we have
x n k + 1 z k x n k z k < 1 2 k , k N .
This implies
z k + 1 z k z k + 1 x n k + 1 + x n k + 1 z k 1 2 ( k + 1 ) + 1 2 k < 1 2 ( k 1 ) , k = 1 , 2 , .

Consequently, { z k } is a Cauchy sequence in F ( T ) . Due to the closedness of F ( T ) in E (see Lemma 2.1), we deduce that lim k z k = z for some z in F ( T ) . It follows from (3.8) that lim k x n k = z . By Lemma 3.1, we see that lim n x n z exists. This forces lim n x n z = 0 . □

The following examples explain why we need to impose some conditions on the control sequences in previous theorems.

Examples 3.6 (a) Let T : [ 1 , 1 ] [ 1 , 1 ] be defined by T x = x . Then T is a 0-nonexpansive (i.e., nonexpansive) mapping. Setting all β n = 1 , the Ishikawa iteration (1.2) provides a sequence
x n + 1 = γ n T 2 x n + ( 1 γ n ) x n = x n , n = 1 , 2 , ,

no matter how we choose { γ n } . Unless x 1 = 0 , we can never reach the unique fixed point 0 of T via { x n } .

(b) Let T : [ 0 , 4 ] [ 0 , 4 ] be defined by
T x = { 0 if  x 4 , 2 if  x = 4 .
Then T is a 1 2 -nonexpansive mapping. Indeed, for any x in [ 0 , 4 ) and y = 4 , we have
| T x T y | 2 = 4 8 + 1 2 | x 2 | 2 = 1 2 | T x y | 2 + 1 2 | x T y | 2 .
The other cases can be verified similarly. It is worth mentioning that T is neither nonexpansive nor continuous. Setting all β n = 1 , the Ishikawa iteration (1.2) provides a sequence
x n + 1 = γ n T 2 x n + ( 1 γ n ) x n , n = 1 , 2 , .
For any arbitrary starting point x 1 in [ 0 , 4 ] , we have T 2 x n = 0 and
x n + 1 = ( 1 γ n ) x n = ( 1 γ 1 ) ( 1 γ 2 ) ( 1 γ n ) x 1 = k = 1 n ( 1 γ k ) x 1 , n = 1 , 2 , .

Consider two possible choices of the values of γ n :

Case 1. If we set γ n = 1 2 , n = 1 , 2 ,  , then lim n γ n ( 1 γ n ) = 1 / 4 > 0 and x n 0 , the unique fixed point of T.

Case 2. If we set γ n = 1 ( n + 1 ) 2 , n = 1 , 2 ,  , then lim n γ n ( 1 γ n ) = 0 and x n = n + 2 2 n + 2 x 1 x 1 / 2 . Unless x 1 = 0 , we can never reach the unique fixed point 0 of T via x n .

4 An existence result in CAT ( 0 ) spaces

Let ( X , d ) be a metric space. A geodesic path joining x to y in X (or briefly, a geodesic from x to y) is a map c from a closed interval [ 0 , l ] R into X such that c ( 0 ) = x , c ( l ) = y , and d ( c ( t ) , c ( t ) ) = | t t | for all t, t in [ 0 , l ] . In particular, c is an isometry and d ( x , y ) = l . The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [ x , y ] . The space ( X , d ) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be a uniquely geodesic if there exists exactly one geodesic joining x and y for each x, y in X. A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle Δ ( x 1 , x 2 , x 3 ) in a geodesic space ( X , d ) consists of three points x 1 , x 2 , x 3 in X (the vertices of Δ), together with a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for a geodesic triangle Δ ( x 1 , x 2 , x 3 ) in a geodesic space ( X , d ) is a triangle Δ ¯ ( x 1 , x 2 , x 3 ) : = Δ ( x ¯ 1 , x ¯ 2 , x ¯ 3 ) in the Euclidean plane E 2 together with a one-to-one correspondence x x ¯ from Δ onto Δ ¯ such that it is an isometry on each of the three segments. A geodesic space X is said to be a CAT ( 0 ) space if all geodesic triangles Δ satisfy the CAT ( 0 ) inequality:
d ( x , y ) d E 2 ( x ¯ , y ¯ ) , x , y Δ .

It is easy to see that a CAT ( 0 ) space is uniquely geodesic.

It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT ( 0 ) space. Other examples include inner product spaces, -trees (see, for example, [11]), Euclidean building (see, for example, [12]), and the complex Hilbert ball with a hyperbolic metric (see, for example, [8]). For a thorough discussion on other spaces and on the fundamental role they play in geometry, see, for example, [1214].

We collect some properties of CAT ( 0 ) spaces. For more details, we refer the readers to [1517].

Lemma 4.1 [16]

Let ( X , d ) be a CAT ( 0 ) space. Then the following assertions hold.
  1. (i)
    For x, y in X and t in [ 0 , 1 ] , there exists a unique point z in [ x , y ] such that
    d ( x , z ) = t d ( x , y ) and d ( y , z ) = ( 1 t ) d ( x , y ) .
    (4.1)

    We use the notation ( 1 t ) x t y for the unique point z satisfying (4.1).

     
  2. (ii)
    For x, y in X and t in [ 0 , 1 ] , we have
    d ( ( 1 t ) x t y , z ) ( 1 t ) d ( x , z ) + t d ( y , z ) .
     

The notion of asymptotic centers in a Banach space can be extended to a CAT ( 0 ) space as well by simply replacing the distance defined by with the one defined by the metric d ( , ) . In particular, in a CAT ( 0 ) space, A ( C , { x n } ) consists of exactly one point whenever C is a closed and convex set and { x n } is a bounded sequence; see [[18], Proposition 7].

Definition 4.2 [19, 20]

A sequence { x n } in a CAT ( 0 ) space X is said to Δ-converge to x in X if x is the unique asymptotic center of { x n k } for every subsequence { x n k } of { x n } . In this case, we write Δ - lim n x n = x , and we call x the Δ-limit of { x n } .

Lemma 4.3 [19]

Every bounded sequence in a complete CAT ( 0 ) space X has a Δ-convergent subsequence.

Lemma 4.4 [21]

Let C be a closed and convex subset of a complete CAT ( 0 ) space X. If { x n } is a bounded sequence in C, then the asymptotic center of { x n } is in C.

Lemma 4.5 [22]

Let X be a complete CAT ( 0 ) space and let x X . Suppose that 0 < b t n c < 1 and x n , y n X for n = 1 , 2 ,  . If for some r 0 we have
lim sup n d ( x n , x ) r , lim sup n d ( y n , x ) r , and lim n d ( t n x n ( 1 t n ) y n , x ) = r ,

then lim n d ( x n , y n ) = 0 .

Recall that the Ishikawa iteration in CAT ( 0 ) spaces is described as follows: For any initial point x 1 in C, we define the iterates { x n } by
{ y n = β n T x n ( 1 β n ) x n , x n + 1 = γ n T y n ( 1 γ n ) x n ,
(4.2)

where the sequences { β n } and { γ n } satisfy some appropriate conditions.

We introduce the notion of α-nonexpansive mappings of CAT ( 0 ) spaces.

Definition 4.6 Let C be a nonempty subset of a CAT ( 0 ) space X and let α < 1 . A mapping T : C X is said to be α-nonexpansive if
d ( T x , T y ) 2 α d ( T x , y ) 2 + α d ( x , T y ) 2 + ( 1 2 α ) d ( x , y ) 2 , x , y C .

The following is the CAT ( 0 ) counterpart to Lemma 2.5. However, we do not know if the compactness assumption can be removed from the negative α case.

Lemma 4.7 Let C be a nonempty closed and convex subset of a complete CAT ( 0 ) space X. Let T : C C be an α-nonexpansive mapping for some α < 1 . In the case 0 α < 1 , we have F ( T ) if and only if { T n x } n = 1 is bounded for some x in C. If C is compact, we always have F ( T ) .

Proof Assume first that 0 α < 1 . The necessity is obvious. We verify the sufficiency. Suppose that { T n x } n = 1 is bounded for some x in C. Set x n : = T n x for n = 1 , 2 ,  . By the boundedness of { x n } n = 1 , there exists z in X such that A ( C , { x n } ) = { z } . It follows from Lemma 4.4 that z C . Furthermore, we have
d ( x n , T z ) 2 α d ( x n , z ) 2 + α d ( x n 1 , T z ) 2 + ( 1 2 α ) d ( x n 1 , z ) 2 , n = 1 , 2 , .
This implies
lim sup n d ( x n , T z ) 2 α lim sup n d ( x n , z ) 2 + α lim sup n d ( x n 1 , T z ) 2 + ( 1 2 α ) lim sup n d ( x n 1 , z ) 2 .
Thus,
lim sup n d ( x n , T z ) lim sup n d ( x n , z ) .

Consequently, T z A ( { x n } ) = { z } , ensuring that F ( T ) .

Next, we assume α < 0 and C is compact. In particular, T is continuous and the sequence of x n : = T n x for any x in C is bounded. In what follows, we adapt the arguments in [2] with slight modifications.

Let μ be a Banach limit, i.e., μ is a bounded unital positive linear functional of such that μ s = μ . Here, s is the left shift operator on . We write μ n a n for the value of μ ( a ) with a = ( a n ) in as usual. In particular, μ n a n + 1 = μ ( s ( a ) ) = μ ( a ) = μ n a n . As showed in [[2], Lemmas 3.1 and 3.2], we have
μ n d ( x n , T y ) 2 μ n d ( x n , y ) 2 , y C ,
(4.3)
and
g ( y ) : = μ n d ( x n , y ) 2

defines a continuous function from C into .

By compactness, there exists y in C such that g ( y ) = inf g ( C ) . Suppose that there is another z in C such that g ( z ) = g ( y ) . Let m be the midpoint in the geodesic segment joining y to z. In view of Lemma 4.1, we see that g is convex. Thus, g ( m ) = g ( y ) too. Observing the comparison triangles in E 2 , we have
d ( x n , y ) 2 + d ( x n , z ) 2 2 d ( x n , m ) 2 + 1 2 d ( y , z ) 2 , n = 1 , 2 , .
Consequently,
μ n d ( x n , y ) 2 + μ n d ( x n , z ) 2 2 μ n d ( x n , m ) 2 + 1 2 μ n d ( y , z ) 2 .
This amounts to say
g ( y ) + g ( z ) 2 g ( m ) + 1 2 d ( y , z ) 2 .

Since g ( y ) = g ( z ) = g ( m ) , we have y = z . Finally, it follows from (4.3) that g ( T y ) g ( y ) = inf g ( C ) . By uniqueness, we have T y = y F ( T ) . □

The proofs of the following results are similar to those in Sections 2 and 3.

Lemma 4.8 Let C be a nonempty subset of a CAT ( 0 ) space X. Let T : C X be an α-nonexpansive mapping for some α < 1 such that F ( T ) . Then T is quasi-nonexpansive.

Lemma 4.9 Let C be a nonempty closed and convex subset of a CAT ( 0 ) space X. Let T : C X be an α-nonexpansive mapping for some α < 1 . Then the following assertions hold.
  1. (i)
    If 0 α < 1 , then
    d ( x , T y ) 2 1 + α 1 α d ( x , T x ) 2 + 2 1 α ( α d ( x , y ) + d ( T x , T y ) ) d ( x , T x ) + d ( x , y ) 2 , x , y C .
     
  2. (ii)
    If α < 0 , then
    d ( x , T y ) 2 d ( x , T x ) 2 + 2 1 α [ ( α ) d ( T x , y ) + d ( T x , T y ) ] d ( x , T x ) + d ( x , y ) 2 , x , y C .
     
Lemma 4.10 Let C be a nonempty closed and convex subset of a CAT ( 0 ) space X. Let T : C C be an α-nonexpansive mapping for some α < 1 . Let a sequence { x n } with x 1 in C be defined by (4.2) such that { β n } and { γ n } are arbitrary sequences in [ 0 , 1 ] . Let z F ( T ) . Then the following assertions hold:
  1. (1)

    max { d ( x n + 1 , z ) , d ( y n , z ) } d ( x n , z ) for n = 1 , 2 ,  .

     
  2. (2)

    lim n d ( x n , z ) exists.

     
  3. (3)

    lim n d ( x n , F ( T ) ) exists.

     

Lemma 4.11 [15]

Let C be a nonempty convex subset of a CAT ( 0 ) space X and let T : C C be a quasi-nonexpansive map whose fixed point set is nonempty. Then F ( T ) is closed, convex and hence contractible.

The following result is deduced from Lemmas 4.8 and 4.11.

Lemma 4.12 Let C be a nonempty convex subset of a CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping with a nonempty fixed point set F ( T ) for some α < 1 . Then F ( T ) is closed, convex, and hence contractible.

Lemma 4.13 Let C be a nonempty closed and convex subset of a complete CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping for some α < 1 . If { x n } is a sequence in C such that d ( T x n , x n ) 0 and Δ - lim n x n = z for some z in X, then z C and T z = z .

Proof It follows from Lemma 4.4 that z C .

Let 0 α < 1 . By Lemma 4.9(i), we deduce that
d ( x n , T z ) 2 1 + α 1 α d ( x n , T x n ) 2 + 2 1 α ( α d ( x n , z ) + d ( T x n , T z ) ) d ( x n , T x n ) + d ( x n , z ) 2
for all n in . Thus we have
lim sup n d ( x n , T z ) lim sup n d ( x n , z ) .
Let α < 0 . Then, by Lemma 4.9(ii), we have
d ( x n , T z ) 2 d ( x n , T x n ) 2 + 2 1 α [ ( α ) d ( T x n , z ) + d ( T x n , T z ) ] d ( x n , T x n ) + d ( x n , z ) 2
for all n in . This implies again that
lim sup n d ( x n , T z ) lim sup n d ( x n , z ) .

By the uniqueness of asymptotic centers, T z = z . □

5 Fixed point and convergence theorems in CAT ( 0 ) spaces

In this section, we extend our results in Section 3 to CAT ( 0 ) spaces.

Theorem 5.1 Let C be a nonempty closed and convex subset of a complete CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] such that 0 < lim inf k γ n k lim sup k γ n k < 1 for a subsequence { γ n k } of { γ n } . In the case α 0 , we assume also that lim sup k β n k < 1 . Let { x n } be a sequence with x 1 in C defined by (4.2). Then the fixed point set F ( T ) if and only if { x n } is bounded and lim k d ( T x n k , x n k ) = 0 .

Proof Suppose that F ( T ) and z in F ( T ) is arbitrarily chosen. By Lemma 4.10, lim n d ( x n , z ) exists and { x n } is bounded. Let
lim n d ( x n , z ) = l .
(5.1)
It follows from Lemmas 4.8 and 4.1(ii) that
d ( T y n , z ) d ( y n , z ) = d ( β n T x n ( 1 β n ) x n , z ) β n d ( T x n , z ) + ( 1 β n ) d ( x n , z ) β n d ( x n , z ) + ( 1 β n ) d ( x n , z ) = d ( x n , z ) .
Thus, we have
lim sup n d ( T y n , z ) lim sup n d ( y n , z ) lim sup n d ( x n , z ) = l .
(5.2)
On the other hand, it follows from (4.2) and (5.1) that
lim n d ( γ n T y n ( 1 γ n ) x n , z ) = lim n d ( x n + 1 , z ) = l .
(5.3)
In view of (5.1)-(5.3) and Lemma 4.5, we conclude that
lim k d ( T y n k , x n k ) = 0 .

By simply replacing with d ( , ) in the proof of Theorem 3.2, we have the desired result lim k d ( T x n k , x n k ) = 0 . The proof in the converse direction follows similarly. □

Theorem 5.2 Let C be a nonempty closed and convex subset of a complete CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] such that 0 < lim inf k γ n k lim sup k γ n k < 1 for a subsequence { γ n k } of { γ n } . In the case α 0 , we assume also that lim sup k β n k < 1 . Let { x n } be a sequence with x 1 in C defined by (4.2). If F ( T ) , then { x n k } Δ-converges to a fixed point of T.

Proof It follows from Theorem 5.1 that { x n } is bounded and lim k d ( T x n k , x n k ) = 0 . Denote by ω w ( x n k ) : = A ( C , { u n } ) , where the union is taken over all subsequences { u n } of { x n k } . We prove that ω w ( x n k ) F ( T ) . Let u ω w ( x n k ) . Then there exists a subsequence { u n } of { x n k } such that A ( C , { u n } ) = { u } . In view of Lemmas 4.3 and 4.4, there exists a subsequence { v n } of { u n } such that Δ - lim n v n = v for some v in C. Since lim n d ( T v n , v n ) = 0 , Lemma 4.13 implies that v F ( T ) . By Lemma 4.10, lim n d ( x n , v ) exists. We claim that u = v . For else, the uniqueness of asymptotic centers implies that
lim sup n d ( v n , v ) < lim sup n d ( v n , u ) lim sup n d ( u n , u ) < lim sup n d ( u n , v ) = lim sup n d ( x n , v ) = lim sup n d ( v n , v ) ,

which is a contradiction. Thus, we have u = v F ( T ) and hence ω w ( x n k ) F ( T ) .

Now, we prove that { x n k } Δ-converges to a fixed point of T. It suffices to show that ω w ( x n k ) consists of exactly one point. Let { u n } be a subsequence of { x n k } . In view of Lemmas 4.3 and 4.4, there exists a subsequence { v n } of { u n } such that Δ - lim n v n = v for some v in C. Let A ( C , { u n } ) = { u } and A ( C , { x n k } ) = { x } . By the argument mentioned above, we have u = v and v F ( T ) . We show that x = v . If it is not the case, then the uniqueness of asymptotic centers implies that
lim sup n d ( v n , v ) < lim sup n d ( v n , x ) lim sup n d ( x n , x ) < lim sup n d ( x n , v ) = lim sup n d ( v n , v ) ,

which is a contradiction. Thus we have the desired result. □

Theorem 5.3 Let C be a nonempty compact convex subset of a complete CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] such that 0 < lim inf k γ n k lim sup k γ n k < 1 for a subsequence { γ n k } of { γ n } . In the case α 0 , we assume also that lim sup k β n k < 1 . Let { x n } be a sequence with x 1 in C defined by (4.2). Then { x n } converges in metric to a fixed point of T.

Proof Using Lemmas 4.7 and 4.9 and replacing with d ( , ) in the proof of Theorem 3.4, we conclude the desired result. □

As in the proof of Theorem 3.5, we can verify the following result.

Theorem 5.4 Let C be a nonempty compact convex subset of a complete CAT ( 0 ) space X and let T : C C be an α-nonexpansive mapping for some α < 1 . Let { β n } and { γ n } be sequences in [ 0 , 1 ] such that 0 < lim inf k γ n k lim sup k γ n k < 1 for a subsequence { γ n k } of { γ n } . In the case α 0 , we assume also that lim sup k β n k < 1 . Let { x n } be a sequence with x 1 in C defined by (4.2). If T satisfies condition (I), then { x n } converges in metric to a fixed point of T.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the referees for their careful reading and valuable suggestions. This research was partially supported by the Grants NSC 99-2115-M-110-007-MY3 (for N.-C. Wong) and NSC 99-2221-E-037-007-MY3 (for J.-C. Yao).

Authors’ Affiliations

(1)
Department of Mathematics, Yasouj University
(2)
Department of Applied Mathematics, National Sun Yat-Sen University
(3)
Center of Fundamental Science, Kaohsiung Medical University
(4)
Department of Mathematics, King Abdulaziz University

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