Open Access

The Mann algorithm in a complete geodesic space with curvature bounded above

Fixed Point Theory and Applications20132013:336

https://doi.org/10.1186/1687-1812-2013-336

Received: 28 June 2013

Accepted: 15 November 2013

Published: 13 December 2013

Abstract

The purpose of this paper is to prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded above by a positive number. The first one for nonexpansive mappings improves the recent result of He et al. (Nonlinear Anal. 75:445-452, 2012). The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

MSC:49J53.

Keywords

CAT ( κ ) spacefixed pointMann algorithmnonexpansive mappingquasi-nonexpansive mappingΔ-convergence

1 Introduction

For a real number κ, a CAT ( κ ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding comparison triangle in a model space with curvature κ. The concept of these spaces has been studied by a large number of researchers. We know that any CAT ( κ ) space is a CAT ( κ ) space for κ > κ (see [1]), thus all results for CAT ( 0 ) spaces can immediately be applied to any CAT ( κ ) with κ 0 . Moreover, CAT ( κ ) spaces with positive κ can be treated as CAT ( 1 ) spaces by changing the scale of the space. So we are interested in CAT ( 1 ) spaces.

One of the most important analytical problems is the existence of fixed points for nonlinear mappings. In the case for nonexpansive mappings in a CAT ( κ ) space was proved by Kirk [2, 3] for κ 0 , and by Espánola and Fernández-León [4] for κ > 0 . In the cases when at least one fixed point exists, it is natural to wonder whether such a fixed point can be approximated by iterations. There are many methods for approximating fixed points of a nonexpansive mapping T. One of the most successful methods is the Mann algorithm [5] which is defined in a geodesic space X by x 1 X and
x n + 1 = t n x n ( 1 t n ) T x n for all  n N ,
(1.1)

where { t n } is a sequence in [ 0 , 1 ] . By using this algorithm, He et al. [6] proved the following result.

Theorem 1.1 Let X be a complete CAT ( 1 ) space and T : X X be a mapping. Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0′) T is nonexpansive with F ( T ) ;

  • (C1′) d ( x 1 , F ( T ) ) < π / 4 ;

  • (C2) n = 1 t n ( 1 t n ) = .

Then the sequence { x n } Δ-converges to a fixed point of T.

Motivated by these results, we prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in complete CAT ( 1 ) spaces. The first one for nonexpansive mappings improves Theorem 1.1. The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

2 Preliminaries

Let X be a metric space with a metric d and let x , y X with d ( x , y ) = l . A geodesic path from x to y is an isometry c : [ 0 , l ] X such that c ( 0 ) = x and c ( l ) = y . The image of a geodesic path from x to y is called a geodesic segment joining x and y. Let r ( 0 , ] . If for every x , y X with d ( x , y ) < r , a geodesic from x to y exists, then we say that X is r-geodesic. Moreover, if such a geodesic is unique for each pair of points, then X is said to be r-uniquely geodesic.

A geodesic segment joining x and y is not necessarily unique in general. When it is unique, this geodesic segment is denoted by [ x , y ] . We write z [ x , y ] if and only if there exists t [ 0 , 1 ] such that d ( z , x ) = ( 1 t ) d ( x , y ) and d ( z , y ) = t d ( x , y ) . In this case, we will write z = t x ( 1 t ) y for simplicity. A geodesic triangle ( x , y , z ) consists of three points x , y , z X and geodesic segments [ y , z ] , [ z , x ] and [ x , z ] joining two of them. We write w ( x , y , z ) if w [ y , z ] [ z , x ] [ x , y ] .

To define a CAT ( κ ) space, we use the following notation called model space. For κ = 0 , the two-dimensional model space M κ 2 = M 0 2 is the Euclidean space R 2 with the metric induced from the Euclidean norm. For κ > 0 , M κ 2 is the two-dimensional sphere ( 1 / κ ) S 2 whose metric is a length of a minimal great arc joining each two points. For κ < 0 , M κ 2 is the two-dimensional hyperbolic space ( 1 / κ ) H 2 with the metric defined by a usual hyperbolic distance.

The diameter of M κ 2 is denoted by D κ , that is, D κ = π / κ if κ > 0 and D κ = if κ 0 . We know that M κ 2 is a D κ -uniquely geodesic space for each κ R .

Let κ R . For ( x , y , z ) in a geodesic space X satisfying that d ( x , y ) + d ( y , z ) + d ( x , z ) < 2 D κ , there exist points x ¯ , y ¯ , z ¯ M κ 2 such that d ( x , y ) = d M κ 2 ( x ¯ , y ¯ ) , d ( y , z ) = d M κ 2 ( y ¯ , z ¯ ) and d ( x , z ) = d M κ 2 ( x ¯ , z ¯ ) . We call the triangle having vertices x ¯ , y ¯ and z ¯ in M κ 2 a comparison triangle of ( x , y , z ) . Notice that it is unique up to an isometry of M κ 2 . For a specific choice of comparison triangles, we denote it by ( x ¯ , y ¯ , z ¯ ) . A point p ¯ [ x ¯ , y ¯ ] is called a comparison point for p [ x , y ] if d ( x , p ) = d M κ 2 ( x ¯ , p ¯ ) .

Let κ R and X be a D κ -geodesic space. If for any x , y , z X with d ( x , y ) + d ( y , z ) + d ( x , z ) < 2 D κ , for any p , q ( x , y , z ) , and for their comparison points p ¯ , q ¯ ( x ¯ , y ¯ , z ¯ ) , the inequality
d ( p , q ) d M κ 2 ( p ¯ , q ¯ )

holds, then we call X a CAT ( κ ) space. It is easy to see that all CAT ( κ ) spaces are D κ -uniquely geodesic; consider the triangle such that two of its vertices are identical.

Let { x n } be a bounded sequence in a metric space X. For x X , let r ( x , { x n } ) : = lim sup n d ( x , x n ) and define the asymptotic radius r ( { x n } ) of { x n } by
r ( { x n } ) : = inf x X r ( x , { x n } ) .

An element z of X is said to be an asymptotic center of { x n } if r ( z , { x n } ) = r ( { x n } ) . We say that { x n } is Δ-convergent to x X if x is the unique asymptotic center of any subsequence of { x n } . The concept of Δ-convergence introduced by Lim in 1976 was shown by Kirk and Panyanak [7] in CAT ( 0 ) spaces to be very similar to the weak convergence in Banach space setting.

Remark 2.1
  1. (1)

    If { x n } is a sequence in a complete CAT ( κ ) space such that r ( { x n } ) < D κ / 2 , then its asymptotic center consists of exactly one point.

     
  2. (2)

    Every sequence { x n } whose asymptotic radius is less than D κ / 2 has a Δ-convergent subsequence (see [4, 7, 8]), that is, ω Δ ( { x n } ) : = { x X : there exists  { x n k } { x n } such that  { x n k } Δ -converges to  x } .

     

We note that Remark 2.1(1) was proved by Dhompongsa et al. [9] for the case κ 0 , and by Espánola and Fernández-León [4] for the case κ > 0 .

Let X be a metric space with a metric d. A mapping T : X X is called nonexpansive if
d ( T x , T y ) d ( x , y ) for all  x , y X .
A point x X is called a fixed point of T if x = T x . We denote by F ( T ) the set of fixed points of T. The mapping T is called quasi-nonexpansive if F ( T ) and
d ( T x , p ) d ( x , p ) for all  x X  and  p F ( T ) .

The following lemmas are essentially needed for our main results.

Lemma 2.2 ([[10], Lemma 1])

Let { a n } and { b n } be two sequences of nonnegative real numbers such that
a n + 1 a n + b n for all  n N .

If n = 1 b n < , then lim n a n exists.

Lemma 2.3 ([[11], Lemma 5.4])

Let ( x , y , z ) be a geodesic triangle in a CAT ( 1 ) space such that d ( x , y ) + d ( x , z ) + d ( y , z ) < 2 π . Let u = t z ( 1 t ) x and v = t z ( 1 t ) y for some t [ 0 , 1 ] . If d ( x , z ) M , d ( y , z ) M , and sin ( ( 1 t ) M ) sin M for some M ( 0 , π ) , then
d ( u , v ) sin ( 1 t ) M sin M d ( x , y ) .

Lemma 2.4 ([[12], Corollary 2.2])

Let ( x , y , z ) be a geodesic triangle in a CAT ( 1 ) space such that d ( x , y ) + d ( x , z ) + d ( y , z ) < 2 π . Let u = t x ( 1 t ) y for some t [ 0 , 1 ] . Then
cos d ( u , z ) sin d ( x , y ) cos d ( x , z ) sin ( t d ( x , y ) ) + cos d ( y , z ) sin ( ( 1 t ) d ( x , y ) ) .

Lemma 2.5 ([[11], Lemma 3.1])

Let ( x , y , z ) be a geodesic triangle in a CAT ( 1 ) space such that d ( x , y ) + d ( x , z ) + d ( y , z ) < 2 π , and let t [ 0 , 1 ] . Then
cos d ( t x ( 1 t ) y , z ) t cos d ( x , z ) + ( 1 t ) cos d ( y , z ) .

3 Main results

We start with some propositions which are common tools for proving the main results in the next two subsections.

Proposition 3.1 Let { x n } be a sequence of a complete CAT ( 1 ) space X such that r ( { x n } ) < π / 2 . Suppose that lim n d ( x n , z ) exists for all z ω Δ ( { x n } ) . Then { x n } Δ-converges to an element of ω Δ ( { x n } ) . Moreover, ω Δ ( { x n } ) consists of exactly one point.

Proof Let x be the asymptotic center of { x n } and let { x n k } be any subsequence of { x n } with the asymptotic center y. We show that y = x and hence { x n } Δ-converges to x as desired. Since r ( { x n k } ) r ( { x n } ) < π / 2 , there exists a subsequence { x n k l } of { x n k } such that { x n k l } Δ-converges to z for some z X . Clearly, z ω Δ ( { x n } ) and it follows from the assumption that lim n d ( x n , z ) exists. Let u X . Then
lim n d ( x n , z ) = lim k d ( x n k , z ) = lim l d ( x n k l , z ) lim sup l d ( x n k l , u ) lim sup k d ( x n k , u ) lim sup n d ( x n , u ) .
Since
lim n d ( x n , z ) lim sup n d ( x n , u ) for all  u X ,
we have z = x . Since
lim k d ( x n k , z ) lim sup k d ( x n k , u ) for all  u X ,

we have z = y . This implies that y = x . □

Proposition 3.2 Let X be a complete CAT ( 1 ) space and { T n } : X X be a countable family of quasi-nonexpansive mappings with F : = n = 1 F ( T n ) . Let { x n } be a sequence in X such that d ( x 1 , F ) < π / 2 and
x n + 1 = t n x n ( 1 t n ) T n x n for all  n N ,
where { t n } is a sequence in [ 0 , 1 ] . Then
  1. (i)

    { x n } is well defined and r ( { x n } ) < π / 2 . In particular, ω Δ ( { x n } ) and d ( x n + 1 , p ) d ( x n , p ) for all n N whenever d ( x 1 , p ) < π / 2 and p F .

     
  2. (ii)

    If ω Δ ( { x n } ) F , then { x n } Δ-converges to an element of F.

     
Proof (i) Let p F be such that d ( x 1 , p ) < π / 2 . Since d ( T 1 x 1 , p ) d ( x 1 , p ) < π / 2 , we have that d ( x 1 , T 1 x 1 ) < π . This implies that x 2 is well defined. It follows from Lemma 2.5 that
cos d ( x 2 , p ) = cos d ( t 1 x 1 ( 1 t 1 ) T 1 x 1 , p ) t 1 cos d ( x 1 , p ) + ( 1 t 1 ) cos d ( T 1 x 1 , p ) t 1 cos d ( x 1 , p ) + ( 1 t 1 ) cos d ( x 1 , p ) = cos d ( x 1 , p ) ,
and we have
d ( x 2 , p ) min { d ( x 1 , x 2 ) + d ( x 1 , p ) , d ( T x 1 , x 2 ) + d ( T x 1 , p ) } = min { ( 1 t 1 ) d ( x 1 , T x 1 ) + d ( x 1 , p ) , t 1 d ( x 1 , T x 1 ) + d ( T x 1 , p ) } min { 1 t 1 , t 1 } d ( x 1 , T x 1 ) + d ( x 1 , p ) 1 2 d ( x 1 , T x 1 ) + d ( x 1 , p ) < π 2 + π 2 = π .
Therefore, d ( x 2 , p ) d ( x 1 , p ) < π / 2 . Using mathematical induction, we can conclude that the sequence { x n } is well defined and
d ( x n + 1 , p ) d ( x n , p ) d ( x 1 , p ) < π / 2 for all  n N .

Then lim n d ( x n , p ) exists which is less than π / 2 , and so r ( { x n } ) < π / 2 . This implies that ω Δ ( { x n } ) .

(ii) Let u ω Δ ( { x n } ) F . Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. Notice that r ( u , { x n k } ) = r ( { x n k } ) r ( { x n } ) < π / 2 . Thus there is N N such that d ( x N , u ) < π / 2 . Similar to the first step, we have that d ( x n + 1 , u ) d ( x n , u ) for all n N . This implies that lim n d ( x n , u ) exists. It follows immediately from Proposition 3.1 that { x n } Δ-converges to an element of F and the proof is finished. □

3.1 Countable nonexpansive mappings

The following concept is introduced by Aoyama et al. [13]. Let X be a complete metric space and { T n } be a countable family of mappings from X into itself with F : = n = 1 F ( T n ) . We say that ( { T n } , T ) satisfies AKTT-condition if

  • n = 1 sup { d ( T n + 1 y , T n y ) : y Y } < for each bounded subset Y of X;

  • T x : = lim n T n x for all x X and F ( T ) = F .

Remark 3.3 Assume that ( { T n } , T ) satisfies AKTT-condition.
  1. (1)

    For each x X , we have { T n x } is a Cauchy sequence and hence the mapping T above is well defined.

     
  2. (2)

    If { x n } is bounded, then n = 1 d ( T n + 1 x n , T n x n ) < .

     
Theorem 3.4 Let X be a complete CAT ( 1 ) space and { T n } be a countable family of mappings from X into itself. Let { x n } be a sequence in X defined by x 1 X and
x n + 1 = t n x n ( 1 t n ) T n x n for all  n N ,

where { t n } is a sequence in [ 0 , 1 ] . Suppose that

  • ( C0 n ) T n is nonexpansive for all n N and F : = n = 1 F ( T n ) ;

  • (C1) d ( x 1 , F ) < π / 2 ;

  • (C2) n = 1 t n ( 1 t n ) = ;

  • (C3′) ( { T n } , T ) satisfies AKTT-condition.

Then the sequence { x n } Δ-converges to a common fixed point of { T n } .

Proof We first show that lim n d ( x n , T n x n ) exists. Using the nonexpansiveness of T n and the definition of { x n } , we obtain that
d ( x n + 1 , T n + 1 x n + 1 ) d ( x n + 1 , T n x n ) + d ( T n x n , T n x n + 1 ) + d ( T n x n + 1 , T n + 1 x n + 1 ) d ( x n + 1 , T n x n ) + d ( x n , x n + 1 ) + d ( T n x n + 1 , T n + 1 x n + 1 ) = d ( x n , T n x n ) + d ( T n x n + 1 , T n + 1 x n + 1 )
for all n N . It follows from Lemma 2.2 and n = 1 d ( T n x n + 1 , T n + 1 x n + 1 ) < that
lim n d ( x n , T n x n )

exists.

Next, we show that lim n d ( x n , T n x n ) = 0 . Assume that lim n d ( x n , T n x n ) > 0 . Thus, without loss of generality, there is a positive real number A such that
A d ( x n , T n x n ) < π for all  n N .
To get the right inequality of the preceding expression, let p F be such that d ( x 1 , p ) < π / 2 . By Proposition 3.2(i), we have
d ( x n , T n x n ) d ( x n , p ) + d ( T n x n , p ) 2 d ( x n , p ) < 2 d ( x 1 , p ) < π .
Put A n : = d ( x n , T n x n ) for all n N . By elementary trigonometry and Lemma 2.4, we get that
cos d ( x n + 1 , p ) sin A n = cos d ( t n x n ( 1 t n ) T n x n , p ) sin A n cos d ( x n , p ) sin ( t n A n ) + cos d ( T n x n , p ) sin ( ( 1 t n ) A n ) cos d ( x n , p ) ( sin ( t n A n ) + sin ( ( 1 t n ) A n ) ) ,
and it follows that
cos d ( x n + 1 , p ) cos d ( x n , p ) cos d ( x n , p ) ( sin ( t n A n ) + sin ( ( 1 t n ) A n ) sin A n 1 ) = 2 cos d ( x n , p ) sin ( t n A n / 2 ) sin ( ( 1 t n ) A n / 2 ) cos ( A n / 2 ) 2 cos d ( x 1 , p ) sin ( t n A / 2 ) sin ( ( 1 t n ) A / 2 ) cos ( A / 2 ) 2 t n ( 1 t n ) cos d ( x 1 , p ) sin 2 ( A / 2 ) cos ( A / 2 )
for all n N . Notice that cos d ( x 1 , p ) , cos ( A / 2 ) , and sin ( A / 2 ) are positive. Consequently,
n = 1 t n ( 1 t n ) cos ( A / 2 ) 2 cos d ( x 1 , p ) sin 2 ( A / 2 ) n = 1 ( cos d ( x n + 1 , p ) cos d ( x n , p ) ) < ,
which is a contradiction. Then we get that lim n d ( x n , T n x n ) = 0 and hence
d ( x n , T x n ) d ( x n , T n x n ) + d ( T n x n , T x n ) d ( x n , T n x n ) + sup { d ( T n y , T y ) : y Y } 0 ,

where Y : = { x n } .

Finally, we show that { x n } Δ-converges to an element of F ( T ) . To apply Proposition 3.2(ii), we show that ω Δ ( { x n } ) F ( T ) . Let u ω Δ ( { x n } ) . Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. Clearly, u is the unique asymptotic center of { x n k } . Using the nonexpansiveness of T and d ( x n , T x n ) 0 , we get that
lim sup k d ( x n k , T u ) lim sup k d ( x n k , T x n k ) + lim sup k d ( T x n k , T u ) lim sup k d ( x n k , u ) .

This implies that T u = u , that is, ω Δ ( { x n } ) F ( T ) . This completes the proof. □

As an immediate consequence of Theorem 3.4, we obtain the following result.

Corollary 3.5 Let X be a complete CAT ( 1 ) space and T : X X be a mapping. Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0′) T is nonexpansive with F ( T ) ;

  • (C1) d ( x 1 , F ( T ) ) < π / 2 ;

  • (C2) n = 1 t n ( 1 t n ) = .

Then the sequence { x n } Δ-converges to a fixed point of T.

Remark 3.6 Our Corollary 3.5 improves Theorem 3.1 of He et al. [6] (see Theorem 1.1) because (C1′) of Theorem 1.1 implies (C1) of Corollary 3.5. Moreover, (C1) is sharp in the sense that if d ( x 1 , F ( T ) ) = π / 2 , then we may construct the Mann algorithm for a nonexpansive mapping which is not Δ-convergent.

Example 3.7 Let S 2 be the unit sphere of the Euclidean space R 3 with the geodesic metric. Let T : S 2 S 2 be defined by
T ( x , y , z ) = ( y , x , z ) for all  ( x , y , z ) S 2 .
Then T is nonexpansive and F ( T ) = { ( 0 , 0 , 1 ) , ( 0 , 0 , 1 ) } . Let { x n } be a sequence in S 2 defined by x 1 = ( 1 , 0 , 0 ) and
x n + 1 = 1 2 x n 1 2 T x n for all  n N .
Then d ( x 1 , F ( T ) ) = π / 2 and
x n + 1 = ( cos n π 4 , sin n π 4 , 0 ) for all  n N .

It is easy to see that { x 8 n + 1 } has the unique asymptotic center which is { ( 1 , 0 , 0 ) } and { x 8 n + 3 } has the unique asymptotic center which is { ( 0 , 1 , 0 ) } . Hence, { x n } is not Δ-convergent.

3.2 Countable quasi-nonexpansive mappings

In this subsection, we give a supplement result to Theorem 3.4. Obviously, every nonexpansive mapping with a fixed point is quasi-nonexpansive. Moreover, if T is nonexpansive, then T is Δ-demiclosed [11], that is, if for any Δ-convergent sequence { x n } in X, its Δ-limit belongs to F ( T ) whenever lim n d ( x n , T x n ) = 0 .

In the following theorem, we deal with quasi-nonexpansive mappings satisfying Δ-demiclosedness. This interesting class of mappings includes the metric projections [11]. However, there are many metric projections such that they are not nonexpansive.

Theorem 3.8 Let X be a complete CAT ( 1 ) space and { T n } be a countable family of mappings from X into itself. Let { x n } be a sequence in X defined by x 1 X and
x n + 1 = t n x n ( 1 t n ) T n x n for all  n N ,

where { t n } is a sequence in ( 0 , 1 ) . Suppose that

  • (C0n) T n is quasi-nonexpansive for all n N and F = n = 1 F ( T n ) ;

  • (C1) d ( x 1 , F ) < π / 2 ;

  • (C2′) lim inf n t n ( 1 t n ) > 0 ;

  • (C3) there exists a mapping T : X X such that
    • { T n } converges uniformly to T on each bounded subset of X;

    • F ( T ) = F ;

  • (C4) T is Δ-demiclosed.

Then the sequence { x n } Δ-converges to a common fixed point of { T n } .

Remark 3.9 Let us compare Theorems 3.4 and 3.8:
  1. (1)

    ( C0 n ) (C0n);

     
  2. (2)

    (C3′) (C3);

     
  3. (3)

    (C0′) and (C3′) (C4);

     
  4. (4)

    (C2′) (C2).

     
Proof of Theorem 3.8 We first show that lim n d ( x n , T n x n ) = 0 . Let p F ( T ) be such that d ( x 1 , p ) < π / 2 . We have that A : = lim n d ( x n , p ) exists which is less than π / 2 and r ( { x n } ) < π / 2 by Proposition 3.2(i). Put B : = lim sup n d ( x n , T n x n ) . Notice that B < π . Since lim inf n t n ( 1 t n ) > 0 , we may assume that there exists a subsequence { n k } of such that lim k d ( x n k , T n k x n k ) = B and t n k t ( 0 , 1 ) . Using the quasi-nonexpansiveness of T n and Lemma 2.4, we get that
cos d ( x n k + 1 , p ) sin d ( x n k , T n k x n k ) = cos d ( t n k x n k ( 1 t n k ) T n k x n k , p ) sin d ( x n k , T n k x n k ) cos d ( x n k , p ) sin ( t n k d ( x n k , T n k x n k ) ) + cos d ( T n k x n k , p ) sin ( ( 1 t n k ) d ( x n k , T n k x n k ) ) cos d ( x n k , p ) ( sin ( t n k d ( x n k , T n k x n k ) ) + sin ( ( 1 t n k ) d ( x n k , T n k x n k ) ) ) .
Letting k yields
cos A sin B cos A ( sin t B + sin ( 1 t ) B ) .
Using elementary trigonometry, we get that B = 0 . Hence it follows that
lim n d ( x n , T n x n ) = 0 .
By condition (C3), we get that
d ( x n , T x n ) d ( x n , T n x n ) + d ( T n x n , T x n ) d ( x n , T n x n ) + sup { d ( T n y , T y ) : y Y } 0 ,

where Y : = { x n } .

Finally, we show that ω Δ ( { x n } ) F ( T ) . Let u ω Δ ( { x n } ) . Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. It follows from the Δ-demiclosedness of T and d ( x n , T x n ) 0 that T u = u , that is, ω Δ ( { x n } ) F ( T ) . Hence the result follows from Proposition 3.2(ii). The proof is now finished. □

As an immediate consequence of Theorem 3.8, we obtain the following result.

Corollary 3.10 Let X be a complete CAT ( 1 ) space and T : X X be a mapping with F ( T ) . Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0) T is quasi-nonexpansive and Δ-demiclosed;

  • (C1) d ( x 1 , F ( T ) ) < π / 2 ;

  • (C2′) lim inf n t n ( 1 t n ) > 0 .

Then the sequence { x n } Δ-converges to a fixed point of T.

Question 3.11 We do not know whether the conclusion of Corollary 3.10 holds if (C2′) is replaced by the more general condition (C2).

Let { T n } : X X be a countable family of quasi-nonexpansive mappings with F : = n = 1 F ( T n ) . We next show how to generate a family { W n } and a mapping W satisfying (C3′) and (C4), and hence Theorem 3.8 is applicable.

Theorem 3.12 Let X be a complete CAT ( 1 ) space such that d ( u , v ) < π / 2 for all u , v X , and let { T n } : X X be a countable family of quasi-nonexpansive mappings with F : = n = 1 F ( T n ) . Then there exist a family of quasi-nonexpansive mappings { W n } : X X and a quasi-nonexpansive mapping W : X X such that
  1. (i)

    ( { W n } , W ) satisfies AKTT-condition and F ( W ) = n = 1 F ( W n ) = F ;

     
  2. (ii)

    W is Δ-demiclosed whenever T n is Δ-demiclosed for all n N .

     

To prove Theorem 3.12, we need the following lemmas.

Lemma 3.13 ([14])

Let X be a complete CAT ( 1 ) space such that d ( u , v ) < π / 2 for all u , v X , and let S , T : X X be quasi-nonexpansive mappings with F ( S ) F ( T ) . Then, for each 0 < t < 1 , F ( S ) F ( T ) = F ( t S ( 1 t ) T ) and the mapping t S ( 1 t ) T is quasi-nonexpansive.

The following lemma is essentially proved in [11]. For the sake of completeness, we show the proof.

Lemma 3.14 Let X be a complete CAT ( 1 ) space such that d ( u , v ) < π / 2 for all u , v X , and let S , T : X X be quasi-nonexpansive mappings with F ( S ) F ( T ) . Let { x n } be a sequence of X. If d ( x n , 1 2 S x n 1 2 T x n ) 0 , then d ( x n , S x n ) 0 and d ( x n , T x n ) 0 .

Proof Put W : = 1 2 S 1 2 T . By Lemma 3.13, we have that W is quasi-nonexpansive and F ( W ) = F ( S ) F ( T ) . Let p F ( W ) . By Lemma 2.4 and the quasi-nonexpansiveness of S and T, we get that
2 cos d ( W x n , p ) sin d ( S x n , T x n ) 2 cos d ( S x n , T x n ) 2 = cos d ( 1 2 S x n 1 2 T x n , p ) sin d ( S x n , T x n ) cos d ( S x n , p ) sin d ( S x n , T x n ) 2 + cos d ( T x n , p ) sin d ( S x n , T x n ) 2 2 cos d ( x n , p ) sin d ( S x n , T x n ) 2 .
This implies that
d ( S x n , T x n ) = 0 or cos d ( S x n , T x n ) 2 cos d ( x n , p ) cos d ( W x n , p ) .
It follows from d ( x n , W x n ) 0 that cos d ( x n , p ) cos d ( W x n , p ) 1 , that is, d ( S x n , T x n ) 0 . Thus
d ( x n , S x n ) d ( x n , 1 2 S x n 1 2 T x n ) + d ( 1 2 S x n 1 2 T x n , S x n ) = d ( x n , 1 2 S x n 1 2 T x n ) + d ( S x n , T x n ) 2 0 .

Hence d ( x n , S x n ) 0 . Similarly, d ( x n , T x n ) 0 and the proof is finished. □

We are now ready to prove Theorem 3.12.

Proof of Theorem 3.12 Put S n : = 1 2 I 1 2 T n for all n N . We define a family of mappings { W n } : X X by
W 1 x = S 1 x ; W 2 x = 1 2 S 1 x 1 2 S 2 x ; W 3 x = 1 2 S 1 x 1 2 ( 1 2 S 2 x 1 2 S 3 x ) ; W n x = 1 2 S 1 x 1 2 ( 1 2 S 2 x 1 2 ( 1 2 ( 1 2 S n 1 x 1 2 S n x ) ) ) ;

It follows from Lemma 3.13 that W n is quasi-nonexpansive for all n N and n = 1 F ( W n ) = n = 1 F ( T n ) .

We first show that n = 1 sup { d ( W n + 1 x , W n x ) : x X } < . For k N , put V k ( k ) : = S k and
V n ( k ) : = 1 2 S k 1 2 ( 1 2 S k + 1 1 2 ( 1 2 ( 1 2 S n 1 1 2 S n ) ) )
for all n > k . By Lemma 2.3, we have that
d ( W n + 1 x , W n x ) = d ( 1 2 S 1 x 1 2 V n + 1 ( 2 ) x , 1 2 S 1 x 1 2 V n ( 2 ) x ) ( 2 2 ) d ( 1 2 S 2 x 1 2 V n + 1 ( 3 ) x , 1 2 S 2 x 1 2 V n ( 3 ) x ) ( 2 2 ) 2 d ( 1 2 S 3 x 1 2 V n + 1 ( 4 ) x , 1 2 S 3 x 1 2 V n ( 4 ) x ) ( 2 2 ) n 1 d ( 1 2 S n x 1 2 S n + 1 x , S n x ) ( 2 2 ) n 1 π 2
for all x X and n N . Then
sup { d ( W n + 1 x , W n x ) : x X } ( 2 2 ) n 1 π 2
and the result follows. In particular, { W n x } is a Cauchy sequence for each x X . We now define the mapping W : X X by
W x : = lim n W n x for all  x X .
Next, we show that F ( W ) = n = 1 F ( T n ) . It is easy to see that n = 1 F ( T n ) F ( W ) . On the other hand, let q n = 1 F ( T n ) = n = 1 F ( S n ) and p F ( W ) . We prove that p F ( T k ) for all k N . Let k N be given. For any n > k , it follows from Lemma 2.5 that
cos d ( q , W n p ) = cos d ( q , 1 2 S 1 p 1 2 ( 1 2 S 2 p 1 2 ( 1 2 ( 1 2 S n 1 p 1 2 S n p ) ) ) ) 1 2 cos d ( q , S 1 p ) + 1 2 cos d ( q , 1 2 S 2 p 1 2 ( 1 2 ( 1 2 S n 1 p 1 2 S n p ) ) ) 1 2 cos d ( q , S 1 p ) + + 1 2 k cos d ( q , S k p ) + + 1 2 n 1 cos d ( q , S n 1 p ) + 1 2 n 1 cos d ( q , S n p ) ( 1 2 + + 1 2 k 1 + 1 2 k + 1 + + 1 2 n 1 + 1 2 n 1 ) cos d ( q , p ) + 1 2 k cos d ( q , S k p ) = ( 1 1 2 k ) cos d ( q , p ) + 1 2 k cos d ( q , S k p ) .
Letting n yields W n p W p = p and
cos d ( q , p ) = cos d ( q , W p ) ( 1 1 2 k ) cos d ( q , p ) + 1 2 k cos d ( q , S k p ) .
This implies that cos d ( q , p ) = cos d ( q , S k p ) . It follows from Lemma 2.4 that
cos d ( q , p ) sin d ( p , T k p ) = cos d ( q , S k p ) sin d ( p , T k p ) = cos d ( q , 1 2 p 1 2 T k p ) sin d ( p , T k p ) cos d ( q , p ) sin d ( p , T k p ) 2 + cos d ( q , T k p ) sin d ( p , T k p ) 2 2 cos d ( q , p ) sin d ( p , T k p ) 2 .

Using elementary trigonometry, we get that d ( p , T k p ) = 0 . Since k is arbitrary, we have p n = 1 F ( T n ) . Hence (i) is proved.

Finally, we prove (ii). We assume that T n is Δ-demiclosed for all n N . We show that W is Δ-demiclosed. Let { x n } X be such that lim n d ( x n , W x n ) = 0 and { x n } Δ-converges to x X . It follows from the definitions of { W n } and { V n ( 2 ) } that
W n = 1 2 S 1 1 2 V n ( 2 ) for all  n 2 .
Similar to the proof of the first and the second steps, we can define the quasi-nonexpansive mapping V : X X by
V x : = lim n V n ( 2 ) x for all  x X
and F ( V ) = n = 2 F ( T n ) . This implies that
W = 1 2 S 1 1 2 V and F ( W ) = F ( S 1 ) F ( V ) .
Then, by Lemma 3.14, we obtain that d ( x n , S 1 x n ) 0 and d ( x n , V x n ) 0 . Thus
d ( x n , T 1 x n ) = 2 d ( x n , 1 2 x n 1 2 T 1 x n ) = 2 d ( x n , S 1 x n ) 0 .

Since T 1 is Δ-demiclosed, we have x F ( T 1 ) . Continuing this procedure gives x n = 1 F ( T n ) = F ( W ) . This completes the proof. □

Declarations

Acknowledgements

The authors thank the referee for their suggestions which enhanced the presentation of the paper. The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from Japan Society for the Promotion of Science. The second author is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand. The last author is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0188/2552) and Khon Kaen University under the RGJ-Ph.D. scholarship.

Authors’ Affiliations

(1)
Department of Information Science, Toho University
(2)
Department of Mathematics, Faculty of Science, Khon Kaen University
(3)
Centre of Excellence in Mathematics, CHE

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