Open Access

On total asymptotically nonexpansive mappings in CAT ( κ ) spaces

Journal of Inequalities and Applications20142014:336

https://doi.org/10.1186/1029-242X-2014-336

Received: 17 April 2014

Accepted: 9 August 2014

Published: 2 September 2014

Abstract

In this article, we obtain the demiclosed principle, fixed point theorems and convergence theorems for the class of total asymptotically nonexpansive mappings on CAT ( κ ) spaces with κ > 0 . Our results generalize the results of Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Tang et al. (Abstr. Appl. Anal. 2012:965751, 2012), Karapınar et al. (J. Appl. Math. 2014:738150, 2014) and many others.

Keywords

fixed point total asymptotically nonexpansive mapping demiclosed principle Δ-convergence CAT ( κ ) space

1 Introduction

For a real number κ, a CAT ( κ ) space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.

Fixed point theory in CAT ( κ ) spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on CAT ( 0 ) spaces (see, e.g., [311]). Since any CAT ( κ ) space is a CAT ( κ ) space for κ κ , all results for CAT ( 0 ) spaces immediately apply to any CAT ( κ ) space with κ 0 . However, there are only a few articles that contain fixed point results in the setting of CAT ( κ ) spaces with κ > 0 .

The concept of total asymptotically nonexpansive mappings was first introduced in Banach spaces by Alber et al. [12]. It generalizes the concept of asymptotically nonexpansive mappings introduced by Goebel and Kirk [13] as well as the concept of nearly asymptotically nonexpansive mappings introduced by Sahu [14]. In 2012, Chang et al. [15] studied the demiclosed principle and Δ-convergence theorems for total asymptotically nonexpansive mappings in the setting of CAT ( 0 ) spaces. Since then the convergence of several iteration procedures for this type of mappings has been rapidly developed and many of articles have appeared (see, e.g., [1624]). Among other things, under some suitable assumptions, Karapınar et al. [24] obtained the demiclosed principle, fixed point theorems, and convergence theorems for the following iteration.

Let K be a nonempty closed convex subset of a CAT ( 0 ) space X and T : K K be a total asymptotically nonexpansive mapping. Given x 1 K , and let { x n } K be defined by
x n + 1 = ( 1 α n ) x n α n T n ( ( 1 β n ) x n β n T n ( x n ) ) , n N ,

where { α n } and { β n } are sequences in [ 0 , 1 ] .

In this article, we extend Karapınar et al.’s results to the general setting of CAT ( κ ) space with κ > 0 .

2 Preliminaries

Let ( X , ρ ) be a metric space. A geodesic path joining x X to y X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [ 0 , l ] R to X such that c ( 0 ) = x , c ( l ) = y , and ρ ( c ( t ) , c ( t ) ) = | t t | for all t , t [ 0 , l ] . In particular, c is an isometry and ρ ( x , y ) = l . The image c ( [ 0 , l ] ) of c is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by [ x , y ] . This means that z [ x , y ] if and only if there exists α [ 0 , 1 ] such that
ρ ( x , z ) = ( 1 α ) ρ ( x , y ) and ρ ( y , z ) = α ρ ( x , y ) .
In this case, we write z = α x ( 1 α ) y . The space ( X , ρ ) is said to be a geodesic space (D-geodesic space) if every two points of X (every two points of distance smaller than D) are joined by a geodesic, and X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic joining x and y for each x , y X (for x , y X with ρ ( x , y ) < D ). A subset K of X is said to be convex if K includes every geodesic segment joining any two of its points. The set K is said to be bounded if
diam ( K ) : = sup { ρ ( x , y ) : x , y K } < .
Now we introduce the model spaces M κ n , for more details on these spaces the reader is referred to [25]. Let n N . We denote by E n the metric space R n endowed with the usual Euclidean distance. We denote by ( | ) the Euclidean scalar product in R n , that is,
( x | y ) = x 1 y 1 + + x n y n where  x = ( x 1 , , x n ) , y = ( y 1 , , y n ) .
Let S n denote the n-dimensional sphere defined by
S n = { x = ( x 1 , , x n + 1 ) R n + 1 : ( x | x ) = 1 } ,

with metric d S n ( x , y ) = arccos ( x | y ) , x , y S n .

Let E n , 1 denote the vector space R n + 1 endowed with the symmetric bilinear form which associates to vectors u = ( u 1 , , u n + 1 ) and v = ( v 1 , , v n + 1 ) the real number u | v defined by
u | v = u n + 1 v n + 1 + i = 1 n u i v i .
Let H n denote the hyperbolic n-space defined by
H n = { u = ( u 1 , , u n + 1 ) E n , 1 : u | u = 1 , u n + 1 > 0 } ,
with metric d H n such that
cosh d H n ( x , y ) = x | y , x , y H n .
Definition 2.1 Given κ R , we denote by M κ n the following metric spaces:
  1. (i)

    if κ = 0 , then M 0 n is the Euclidean space E n ;

     
  2. (ii)

    if κ > 0 , then M κ n is obtained from the spherical space S n by multiplying the distance function by the constant 1 / κ ;

     
  3. (iii)

    if κ < 0 , then M κ n is obtained from the hyperbolic space H n by multiplying the distance function by the constant 1 / κ .

     
A geodesic triangle ( x , y , z ) in a geodesic space ( X , ρ ) consists of three points x, y, z in X (the vertices of ) and three geodesic segments between each pair of vertices (the edges of ). A comparison triangle for a geodesic triangle ( x , y , z ) in ( X , ρ ) is a triangle ¯ ( x ¯ , y ¯ , z ¯ ) in M κ 2 such that
ρ ( x , y ) = d M κ 2 ( x ¯ , y ¯ ) , ρ ( y , z ) = d M κ 2 ( y ¯ , z ¯ ) , and ρ ( z , x ) = d M κ 2 ( z ¯ , x ¯ ) .

If κ 0 , then such a comparison triangle always exists in M κ 2 . If κ > 0 , then such a triangle exists whenever ρ ( x , y ) + ρ ( y , z ) + ρ ( z , x ) < 2 D κ , where D κ = π / κ . A point p ¯ [ x ¯ , y ¯ ] is called a comparison point for p [ x , y ] if ρ ( x , p ) = d M κ 2 ( x ¯ , p ¯ ) .

A geodesic triangle ( x , y , z ) in X is said to satisfy the CAT ( κ ) inequality if for any p , q ( x , y , z ) and for their comparison points p ¯ , q ¯ ¯ ( x ¯ , y ¯ , z ¯ ) , one has
ρ ( p , q ) d M κ 2 ( p ¯ , q ¯ ) .

Definition 2.2 If κ 0 , then X is called a CAT ( κ ) space if and only if X is a geodesic space such that all of its geodesic triangles satisfy the CAT ( κ ) inequality.

If κ > 0 , then X is called a CAT ( κ ) space if and only if X is D κ -geodesic and any geodesic triangle ( x , y , z ) in X with ρ ( x , y ) + ρ ( y , z ) + ρ ( z , x ) < 2 D κ satisfies the CAT ( κ ) inequality.

Notice that in a CAT ( 0 ) space ( X , ρ ) if x , y , z X , then the CAT ( 0 ) inequality implies
( CN ) ρ 2 ( x , 1 2 y 1 2 z ) 1 2 ρ 2 ( x , y ) + 1 2 ρ 2 ( x , z ) 1 4 ρ 2 ( y , z ) .
This is the (CN) inequality of Bruhat and Tits [26]. This inequality is extended by Dhompongsa and Panyanak [27] as
( CN ) ρ 2 ( x , ( 1 α ) y α z ) ( 1 α ) ρ 2 ( x , y ) + α ρ 2 ( x , z ) α ( 1 α ) ρ 2 ( y , z )
for all α [ 0 , 1 ] and x , y , z X . In fact, if X is a geodesic space, then the following statements are equivalent:
  1. (i)

    X is a CAT ( 0 ) space;

     
  2. (ii)

    X satisfies (CN);

     
  3. (iii)

    X satisfies (CN).

     
Let R ( 0 , 2 ] . Recall that a geodesic space ( X , ρ ) is said to be R-convex for R (see [28]) if for any three points x , y , z X , we have
ρ 2 ( x , ( 1 α ) y α z ) ( 1 α ) ρ 2 ( x , y ) + α ρ 2 ( x , z ) R 2 α ( 1 α ) ρ 2 ( y , z ) .
(1)

It follows from (CN) that a geodesic space ( X , ρ ) is a CAT ( 0 ) space if and only if ( X , ρ ) is R-convex for R = 2 . The following lemma is a consequence of Proposition 3.1 in [28].

Lemma 2.3 Let κ > 0 and ( X , ρ ) be a CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Then ( X , ρ ) is R-convex for R = ( π 2 ε ) tan ( ε ) .

The following lemma is also needed.

Lemma 2.4 ([[25], p.176])

Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Then
ρ ( ( 1 α ) x α y , z ) ( 1 α ) ρ ( x , z ) + α ρ ( y , z )

for all x , y , z X and α [ 0 , 1 ] .

We now collect some elementary facts about CAT ( κ ) spaces. Most of them are proved in the setting of CAT ( 1 ) spaces. For completeness, we state the results in CAT ( κ ) with κ > 0 .

Let { x n } be a bounded sequence in a CAT ( κ ) space ( X , ρ ) . For x X , we set
r ( x , { x n } ) = lim sup n ρ ( x , x n ) .
The asymptotic radius r ( { x n } ) of { x n } is given by
r ( { x n } ) = inf { r ( x , { x n } ) : x X } ,
and the asymptotic center A ( { x n } ) of { x n } is the set
A ( { x n } ) = { x X : r ( x , { x n } ) = r ( { x n } ) } .

It is known from Proposition 4.1 of [8] that in a CAT ( κ ) space X with diam ( X ) < π 2 κ , A ( { x n } ) consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.

Definition 2.5 ([6, 29])

A sequence { x n } in X is said to Δ-converge to x X if x is the unique asymptotic center of { u n } for every subsequence { u n } of { x n } . In this case we write Δ - lim n x n = x and call x the Δ-limit of { x n } .

Lemma 2.6 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Then the following statements hold:
  1. (i)

    [[8], Corollary  4.4] Every sequence in X has a Δ-convergent subsequence;

     
  2. (ii)

    [[8], Proposition  4.5] If { x n } X and Δ - lim n x n = x , then x k = 1 conv ¯ { x k , x k + 1 , } , where conv ¯ ( A ) = { B : B A  and  B  is closed and convex } .

     

By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[27], Lemma 2.8]).

Lemma 2.7 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . If { x n } is a sequence in X with A ( { x n } ) = { x } and { u n } is a subsequence of { x n } with A ( { u n } ) = { u } and the sequence { ρ ( x n , u ) } converges, then x = u .

Definition 2.8 Let K be a nonempty subset of a CAT ( κ ) space ( X , ρ ) . A mapping T : K K is called total asymptotically nonexpansive if there exist nonnegative real sequences { ν n } , { μ n } with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : [ 0 , ) [ 0 , ) with ψ ( 0 ) = 0 such that
ρ ( T n ( x ) , T n ( y ) ) ρ ( x , y ) + ν n ψ ( ρ ( x , y ) ) + μ n for all  n N , x , y K .
A point x K is called a fixed point of T if x = T ( x ) . We denote with F ( T ) the set of fixed points of T. A sequence { x n } in K is called approximate fixed point sequence for T (AFPS in short) if
lim n ρ ( x n , T ( x n ) ) = 0 .
Algorithm 1 The sequence { x n } defined by x 1 K and
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

is called an Ishikawa iterative sequence (see [30]).

If β n = 0 for all n N , then Algorithm 1 reduces to the following.

Algorithm 2 The sequence { x n } defined by x 1 K and
x n + 1 = ( 1 α n ) x n α n T n ( x n ) , n N ,

is called a Mann iterative sequence (see [31]).

The following lemma is also needed.

Lemma 2.9 ([[32], Lemma 1])

Let { s n } and { t n } be sequences of nonnegative real numbers satisfying
s n + 1 s n + t n for all  n N .

If n = 1 t n < , then lim n s n exists.

3 Main results

3.1 Existence theorems

Theorem 3.1 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let K be a nonempty closed convex subset of X, and let T : K K be a continuous total asymptotically nonexpansive mapping. Then T has a fixed point in K.

Proof Fix x K . We can consider the sequence { T n ( x ) } n = 1 as a bounded sequence in K. Let ϕ : K [ 0 , ) be a function defined by
ϕ ( u ) : = lim sup n ρ ( T n ( x ) , u ) for all  u K .
Then there exists w K such that ϕ ( w ) = inf { ϕ ( u ) : u K } . Since T is total asymptotically nonexpansive, for each n , m N , we have
ρ ( T n + m ( x ) , T m ( w ) ) ρ ( T n ( x ) , w ) + ν m ψ ( ρ ( T n ( x ) , w ) ) + μ m .
(2)
Let M = diam ( K ) . Taking n in (2), we get that
ϕ ( T m ( w ) ) ϕ ( w ) + ν m ψ ( M ) + μ m .
This implies that
lim m ϕ ( T m ( w ) ) ϕ ( w ) .
(3)
In view of (1), we have
ρ ( T n ( x ) , 1 2 T m ( w ) 1 2 T h ( w ) ) 2 1 2 ρ ( T n ( x ) , T m ( w ) ) 2 + 1 2 ρ ( T n ( x ) , T h ( w ) ) 2 R 8 ρ ( T m ( w ) , T h ( w ) ) 2 .
Taking n , we get that
ϕ ( w ) 2 ϕ ( 1 2 T m ( w ) 1 2 T h ( w ) ) 2 1 2 ϕ ( T m ( w ) ) 2 + 1 2 ϕ ( T h ( w ) ) 2 R 8 ρ ( T m ( w ) , T h ( w ) ) 2 ,
yielding
R 8 ρ ( T m ( w ) , T h ( w ) ) 2 1 2 ϕ ( T m ( w ) ) 2 + 1 2 ϕ ( T h ( w ) ) 2 ϕ ( w ) 2 .
(4)
By (3) and (4), we have lim m , h ρ ( T m ( w ) , T h ( w ) ) 2 0 . Therefore, { T n ( w ) } n = 1 is a Cauchy sequence in K and hence converges to some point v K . Since T is continuous,
T ( v ) = T ( lim n T n ( w ) ) = lim n T n + 1 ( w ) = v .

 □

From Theorem 3.1 we shall now derive a result for CAT ( 0 ) spaces which can also be found in [24].

Corollary 3.2 Let ( X , ρ ) be a complete CAT ( 0 ) space and K be a nonempty bounded closed convex subset of X. If T : K K is a continuous total asymptotically nonexpansive mapping, then T has a fixed point.

Proof It is well known that every convex subset of a CAT ( 0 ) space, equipped with the induced metric, is a CAT ( 0 ) space (cf. [25]). Then ( K , ρ ) is a CAT ( 0 ) space and hence it is a CAT ( κ ) space for all κ > 0 . Notice also that K is R-convex for R = 2 . Since K is bounded, we can choose ε ( 0 , π / 2 ) and κ > 0 so that diam ( K ) π / 2 ε κ . The conclusion follows from Theorem 3.1. □

3.2 Demiclosed principle

Theorem 3.3 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let K be a nonempty closed convex subset of X, and let T : K K be a uniformly continuous total asymptotically nonexpansive mapping. If { x n } is an AFPS for T such that Δ - lim n x n = w , then w K and w = T ( w ) .

Proof By Lemma 2.6, w K . As in Theorem 3.1, we define ϕ ( u ) : = lim sup n ρ ( x n , u ) for each u K . Since lim n ρ ( x n , T ( x n ) ) = 0 , by induction we can show that lim n ρ ( x n , T m ( x n ) ) = 0 for all m N (cf. [16]). This implies that
ϕ ( u ) = lim sup n ρ ( T m ( x n ) , u ) for each  u K  and  m N .
(5)
In (5), taking u = T m ( w ) , we have
ϕ ( T m ( w ) ) = lim sup n ρ ( T m ( x n ) , T m ( w ) ) lim sup n ( ρ ( x n , w ) + ν m ψ ( ρ ( x n , w ) ) + μ m ) .
Hence
lim sup m ϕ ( T m ( w ) ) ϕ ( w ) .
(6)
In view of (1), we have
ρ ( x n , 1 2 w 1 2 T m ( w ) ) 2 1 2 ρ ( x n , w ) 2 + 1 2 ρ ( x n , T m ( w ) ) 2 R 8 ρ ( w , T m ( w ) ) 2 ,
where R = ( π 2 ε ) tan ( ε ) . Since Δ - lim n x n = w , letting n , we get that
ϕ ( w ) 2 ϕ ( 1 2 w 1 2 T m ( w ) ) 2 1 2 ϕ ( w ) 2 + 1 2 ϕ ( T m ( w ) ) 2 R 8 ρ ( w , T m ( w ) ) 2 ,
yielding
ρ ( w , T m ( w ) ) 2 4 R [ ϕ ( T m ( w ) ) 2 ϕ ( w ) 2 ] .
(7)
By (6) and (7), we have lim m ρ ( w , T m ( w ) ) = 0 . Since T is continuous,
T ( w ) = T ( lim m T m ( w ) ) = lim m T m + 1 ( w ) = w .

 □

As we have observed in Corollary 3.2, we can derive the following result from Theorem 3.3.

Corollary 3.4 ([[24], Theorem 12])

Let ( X , ρ ) be a complete CAT ( 0 ) space, K be a nonempty bounded closed convex subset of X, and T : K K be a uniformly continuous total asymptotically nonexpansive mapping. If { x n } is an AFPS for T such that Δ - lim n x n = w , then w K and w = T ( w ) .

3.3 Convergence theorems

We begin this section by proving a crucial lemma.

Lemma 3.5 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let K be a nonempty closed convex subset of X, and T : K K be a uniformly continuous total asymptotically nonexpansive mapping with n = 1 ν n < and n = 1 μ n < . Let x 1 K and { x n } be a sequence in K defined by
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

where { α n } and { β n } are sequences in ( 0 , 1 ) such that lim inf n α n β n ( 1 β n ) > 0 . Then { x n } is an AFPS for T and lim n ρ ( x n , p ) exists for all p F ( T ) .

Proof It follows from Theorem 3.1 that F ( T ) . Let p F ( T ) and M = diam ( K ) . Since T is total asymptotically nonexpansive, by Lemma 2.4 we have
ρ ( y n , p ) = ρ ( ( 1 β n ) x n β n T n ( x n ) , p ) ( 1 β n ) ρ ( x n , p ) + β n ρ ( T n ( x n ) , T n ( p ) ) ρ ( x n , p ) + β n ν n ψ ( M ) + β n μ n .
This implies that
ρ ( x n + 1 , p ) = ρ ( ( 1 α n ) x n α n T n ( y n ) , p ) ( 1 α n ) ρ ( x n , p ) + α n ρ ( T n ( y n ) , T n ( p ) ) ( 1 α n ) ρ ( x n , p ) + α n [ ρ ( y n , p ) + ν n ψ ( M ) + μ n ] ρ ( x n , p ) + α n ( 1 + β n ) ( ν n ψ ( M ) + μ n ) .
Since n = 1 ν n < and n = 1 μ n < , by Lemma 2.9 lim n ρ ( x n , p ) exists. Next, we show that { x n } is an AFPS for T. In view of (1), we have
ρ ( x n + 1 , p ) 2 = ρ ( ( 1 α n ) x n α n T n ( y n ) , p ) 2 ( 1 α n ) ρ ( x n , p ) 2 + α n ρ ( T n ( y n ) , p ) 2 ( 1 α n ) ρ ( x n , p ) 2 + α n [ ρ ( y n , p ) + ν n ψ ( M ) + μ n ] 2 ( 1 α n ) ρ ( x n , p ) 2 + α n ρ ( y n , p ) 2 + α n [ 2 ρ ( y n , p ) ( ν n ψ ( M ) + μ n ) + ( ν n ψ ( M ) + μ n ) 2 ] .
This implies that
ρ ( x n + 1 , p ) 2 ( 1 α n ) ρ ( x n , p ) 2 + α n ρ ( y n , p ) 2 + A ν n + B μ n for some  A , B 0 .
(8)
Again by (1), we have
ρ ( y n , p ) 2 = ρ ( ( 1 β n ) x n β n T n ( x n ) , p ) 2 ( 1 β n ) ρ ( x n , p ) 2 + β n ρ ( T n ( x n ) , T n ( p ) ) 2 R 2 β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 ( 1 β n ) ρ ( x n , p ) 2 + β n [ ρ ( x n , p ) + ν n ψ ( M ) + μ n ] 2 R 2 β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 ρ ( x n , p ) 2 + β n [ 2 ρ ( x n , p ) ( ν n ψ ( M ) + μ n ) + ( ν n ψ ( M ) + μ n ) 2 ] R 2 β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 .
Substituting this into (8), we get that
ρ ( x n + 1 , p ) 2 ρ ( x n , p ) 2 + α n β n [ 2 ρ ( x n , p ) ( ν n ψ ( M ) + μ n ) + ( ν n ψ ( M ) + μ n ) 2 ] R 2 α n β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 + A ν n + B μ n ,
yielding
R 2 α n β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 ρ ( x n , p ) 2 ρ ( x n + 1 , p ) 2 + C ν n + D μ n for some  C , D 0 .
Since n = 1 ν n < and n = 1 μ n < , we have
n = 1 α n β n ( 1 β n ) ρ ( x n , T n ( x n ) ) 2 < .
This implies by lim inf n α n β n ( 1 β n ) > 0 that
lim n ρ ( x n , T n ( x n ) ) = 0 .
(9)
By the uniform continuity of T, we have
lim n ρ ( T ( x n ) , T n + 1 ( x n ) ) = 0 .
(10)
It follows from (9) and the definitions of x n + 1 and y n that
ρ ( x n , x n + 1 ) ρ ( x n , T n ( y n ) ) ρ ( x n , T n ( x n ) ) + ρ ( T n ( x n ) , T n ( y n ) ) ρ ( x n , T n ( x n ) ) + ρ ( x n , y n ) + ν n ψ ( M ) + μ n ( 1 + β n ) ρ ( x n , T n ( x n ) ) + ν n ψ ( M ) + μ n 0 as  n .
(11)
By (9), (10), and (11), we have
ρ ( x n , T ( x n ) ) ρ ( x n , x n + 1 ) + ρ ( x n + 1 , T n + 1 ( x n + 1 ) ) + ρ ( T n + 1 ( x n + 1 ) , T n + 1 ( x n ) ) + ρ ( T n + 1 ( x n ) , T ( x n ) ) ρ ( x n , x n + 1 ) + ρ ( x n + 1 , T n + 1 ( x n + 1 ) ) + ρ ( x n + 1 , x n ) + ν n + 1 ψ ( M ) + μ n + 1 + ρ ( T n + 1 ( x n ) , T ( x n ) ) 0 as  n .

 □

Now, we are ready to prove our Δ-convergence theorem.

Theorem 3.6 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let K be a nonempty closed convex subset of X, and let T : K K be a uniformly continuous total asymptotically nonexpansive mapping with n = 1 ν n < and n = 1 μ n < . Let x 1 K and { x n } be a sequence in K defined by
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

where { α n } and { β n } are sequences in ( 0 , 1 ) such that lim inf n α n β n ( 1 β n ) > 0 . Then { x n } Δ-converges to a fixed point of T.

Proof Let ω w ( x n ) : = A ( { u n } ) where the union is taken over all subsequences { u n } of { x n } . We can complete the proof by showing that ω w ( x n ) is contained in F ( T ) and ω w ( x n ) consists of exactly one point. Let u ω w ( x n ) , then there exists a subsequence { u n } of { x n } such that A ( { u n } ) = { u } . By Lemma 2.6, there exists a subsequence { v n } of { u n } such that Δ - lim n v n = v K . Hence v F ( T ) by Lemma 3.5 and Theorem 3.3. Since lim n ρ ( x n , v ) exists, u = v by Lemma 2.7. This shows that ω w ( x n ) F ( T ) . Next, we show that ω w ( x n ) consists of exactly one point. Let { u n } be a subsequence of { x n } with A ( { u n } ) = { u } , and let A ( { x n } ) = { x } . Since u ω w ( x n ) F ( T ) , by Lemma 3.5 lim n ρ ( x n , u ) exists. Again, by Lemma 2.7, x = u . This completes the proof. □

As a consequence of Theorem 3.6, we obtain the following.

Corollary 3.7 ([[24], Theorem 17])

Let ( X , ρ ) be a complete CAT ( 0 ) space, K be a nonempty bounded closed convex subset of X, and T : K K be a uniformly continuous total asymptotically nonexpansive mapping with n = 1 ν n < and n = 1 μ n < . Let x 1 K and { x n } be a sequence in K defined by
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

where { α n } and { β n } are sequences in ( 0 , 1 ) such that lim inf n α n β n ( 1 β n ) > 0 . Then { x n } Δ-converges to a fixed point of T.

Recall that a mapping T : K K is said to be semi-compact if K is closed and each bounded AFPS for T in K has a convergent subsequence. Now, we prove a strong convergence theorem for uniformly continuous total asymptotically nonexpansive semi-compact mappings.

Theorem 3.8 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let K be a nonempty closed convex subset of X, and let T : K K be a uniformly continuous total asymptotically nonexpansive mapping with n = 1 ν n < and n = 1 μ n < . Let x 1 K and { x n } be a sequence in K defined by
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

where { α n } and { β n } are sequences in ( 0 , 1 ) such that lim inf n α n β n ( 1 β n ) > 0 . Suppose that T m is semi-compact for some m N . Then { x n } converges strongly to a fixed point of T.

Proof By Lemma 3.5, lim n ρ ( x n , T ( x n ) ) = 0 . Since T is uniformly continuous, we have
ρ ( x n , T m ( x n ) ) ρ ( x n , T ( x n ) ) + ρ ( T ( x n ) , T 2 ( x n ) ) + + ρ ( T m 1 ( x n ) , T m ( x n ) ) 0
as n . That is, { x n } is an AFPS for T m . By the semi-compactness of T m , there exist a subsequence { x n j } of { x n } and p K such that lim j x n j = p . Again, by the uniform continuity of T, we have
ρ ( T ( p ) , p ) ρ ( T ( p ) , T ( x n j ) ) + ρ ( T ( x n j ) , x n j ) + ρ ( x n j , p ) 0 as  j .

That is, p F ( T ) . By Lemma 3.5, lim n ρ ( x n , p ) exists, thus p is the strong limit of the sequence { x n } itself. □

Corollary 3.9 ([[24], Theorem 22])

Let ( X , ρ ) be a complete CAT ( 0 ) space, K be a nonempty bounded closed convex subset of X, and T : K K be a uniformly continuous total asymptotically nonexpansive mapping with n = 1 ν n < and n = 1 μ n < . Let x 1 K and { x n } be a sequence in K defined by
x n + 1 = ( 1 α n ) x n α n T n ( y n ) , y n = ( 1 β n ) x n β n T n ( x n ) , n N ,

where { α n } and { β n } are sequences in ( 0 , 1 ) such that lim inf n α n β n ( 1 β n ) > 0 . Suppose that T m is semi-compact for some m N . Then { x n } converges strongly to a fixed point of T.

Remark 3.10 The results in this article also hold for the class of weakly total asymptotically nonexpansive mappings in the following sense. A mapping T : K K is called weakly total asymptotically nonexpansive if there exist nonnegative real sequences { ν n } , { μ n } with ν n 0 , μ n 0 as n and a nondecreasing function ψ : [ 0 , ) [ 0 , ) such that
ρ ( T n ( x ) , T n ( y ) ) ρ ( x , y ) + ν n ψ ( ρ ( x , y ) ) + μ n for all  n N , x , y K .

Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

Declarations

Acknowledgements

The author thanks Chiang Mai University for financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University

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