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Strong convergence of an Ishikawa-type algorithm in CAT ( 0 ) spaces

Fixed Point Theory and Applications20132013:207

https://doi.org/10.1186/1687-1812-2013-207

  • Received: 7 May 2013
  • Accepted: 8 July 2013
  • Published:

Abstract

We study strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive type maps to their common fixed point on a CAT ( 0 ) space. Our work provides an affirmative answer to the question of Tan and Xu (Proc. Am. Math. Soc. 122:733-739, 1994); in particular, strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive maps without the rate of convergence condition is obtained on a nonlinear domain.

MSC:47H09, 47H10, 49M05.

Keywords

  • asymptotically nonexpansive type map
  • common fixed point
  • Ishikawa-type algorithm
  • uniform equicontinuity
  • strong convergence

1 Introduction

A CAT ( 0 ) space is simply a geodesic metric space whose each geodesic triangle is at least as thin as its comparison triangle in the Euclidean plane. In 2004, Kirk [1] proved a fixed point theorem for a nonexpansive map defined on a subset of a CAT ( 0 ) space. Since then, approximation of fixed points of nonlinear maps on a CAT ( 0 ) space has rapidly developed (see, e.g., [25]).

We describe briefly the needed details for a CAT ( 0 ) space. A metric space ( X , d ) is said to be a length space if any two points of X are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points of X is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, d is said to be a length metric (otherwise known as an inner metric or intrinsic metric). In case no rectifiable path joins two points of the space, the distance between them is taken to be ∞.

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 d ( c ( t ) , c ( t ) ) = | t t | for all t , t [ 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. We say that X is: (i) a geodesic space if any two points of X are joined by a geodesic, and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each x , y X , which we will denote by [ x , y ] , called the segment joining x to y.

A geodesic triangle Δ ( x 1 , x 2 , x 3 ) in a geodesic metric space ( X , d ) consists of three points in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle Δ ( x 1 , x 2 , x 3 ) in ( X , d ) is a triangle Δ ¯ ( x 1 , x 2 , x 3 ) : = Δ ( x ¯ 1 , x ¯ 2 , x ¯ 3 ) in R 2 such that d R 2 ( x ¯ i , x ¯ j ) = d ( x i , x j ) for i , j { 1 , 2 , 3 } . Such a triangle always exists (see [6]).

A geodesic metric space is said to be a CAT ( 0 ) space if all geodesic triangles of appropriate size satisfy the following CAT ( 0 ) comparison axiom.

Let Δ be a geodesic triangle in X and let Δ ¯ R 2 be a comparison triangle for Δ. Then Δ is said to satisfy the CAT ( 0 ) inequality if for all x , y Δ and all comparison points x ¯ , y ¯ Δ ¯ ,
d ( x , y ) d ( x ¯ , y ¯ ) .
If x, y 1 , y 2 are points of a CAT ( 0 ) space and y 0 is the midpoint of the segment [ y 1 , y 2 ] , which we will denote by y 1 y 2 2 , then the CAT ( 0 ) inequality implies
d ( x , y 1 y 2 2 ) 2 1 2 d ( x , y 1 ) 2 + 1 2 d ( x , y 2 ) 2 1 4 d ( y 1 , y 2 ) 2 .
The above inequality is the (CN) inequality of Bruhat and Titz [7] and it was extended in [8] as follows:
d ( z , α x ( 1 α ) y ) 2 α d ( z , x ) 2 + ( 1 α ) d ( z , y ) 2 α ( 1 α ) d ( x , y ) 2

for any α [ 0 , 1 ] and x , y , z X .

Let us recall that a geodesic metric space is a CAT ( 0 ) space if and only if it satisfies the (CN) inequality (see [6], p. 163). Moreover, if X is a CAT ( 0 ) metric space and x , y X , then for any α [ 0 , 1 ] , there exists a unique point α x ( 1 α ) y [ x , y ] such that
d ( z , α x ( 1 α ) y ) α d ( z , x ) + ( 1 α ) d ( z , y )

for any z X and [ x , y ] = { α x ( 1 α ) y : α [ 0 , 1 ] } .

A subset C of a CAT ( 0 ) space X is convex if for any x , y C , we have [ x , y ] C .

Complete CAT ( 0 ) spaces are known as Hadamard spaces (see [9]). The reader interested in a more general nonlinear domain, namely 2-uniformly convex hyperbolic space containing a CAT ( 0 ) space as a special case, is referred to Dehaish [10] and Dehaish et al. [11].

Let C be a nonempty subset of a metric space ( X , d ) . Then a selfmap T on C is:
  1. (i)

    uniformly L-Lipschitzian if for some L > 0 , d ( T n x , T n y ) L d ( x , y ) for x , y C , n 1 ;

     
  2. (ii)

    uniformly Hölder continuous if for some positive constants L and α, d ( T n x , T n y ) L d ( x , y ) α for x , y C , n 1 ;

     
  3. (iii)

    uniformly equicontinuous if for any ε > 0 , there exists δ > 0 such that d ( T n x , T n y ) ε whenever d ( x , y ) δ for x , y C , n 1 or, equivalently, T is uniformly equicontinuous if and only if d ( T n x n , T n y n ) 0 whenever d ( x n , y n ) 0 as n ;

     
  4. (iv)

    asymptotically nonexpansive if there is a sequence { k n } [ 1 , ) with lim n k n = 1 such that d ( T n x , T n y ) k n d ( x , y ) for x , y C , n 1 ;

     
  5. (v)

    asymptotically nonexpansive in the intermediate sense provided T is uniformly continuous and lim sup n sup x , y C { d ( T n x , T n y ) d ( x , y ) } 0 for n 1 , and

     
  6. (vi)

    of asymptotically nonexpansive type in the sense of Xu [12] if lim sup n sup x C { d ( T n x , T n y ) d ( x , y ) } 0 for each y C , n 1 ;

     
  7. (vii)

    of asymptotically nonexpansive type in the sense of Chang et al. [13] if lim sup n sup x C { d ( T n x , T n y ) 2 d ( x , y ) 2 } 0 for each y C , n 1 .

     

The map T is semi-compact if for any bounded sequence { x n } in C with d ( x n , T x n ) 0 as n , there is a subsequence { x n i } of { x n } such that x n i x C as n i .

It is not difficult to see that nonexpansive map, asymptotically nonexpansive map, asymptotically nonexpansive map in the intermediate sense and asymptotically nonexpansive type map in the sense of Xu [12] all are special cases of asymptotically nonexpansive type map in the sense of Chang et al. [13]. Moreover, a uniformly L-Lipschitzian map is uniformly Hölder continuous, and a uniformly Hölder continuous map is uniformly equicontinuous. However, the converse statements are not true as indicated below.

Example 1.1 Take X = R and C = [ 0 , 1 ] . Define T : C C by T x = ( 1 x 3 2 ) 2 3 for all x C . Then T is uniformly equicontinuous, but it is neither uniformly L-Lipschitzian nor uniformly Hölder continuous.

In uniformly convex Banach spaces, the convergence of an Ishikawa-type algorithm and a Mann-type algorithm of nonexpansive maps, asymptotically nonexpansive maps and asymptotically nonexpansive maps in the intermediate sense to their fixed points have been studied by a number of researchers [12, 1424]. For the iterative construction of fixed points of some other classes of nonlinear maps, see [2527].

The sequence { k n } in definition (iv) satisfies the rate of convergence condition if n = 1 ( k n 1 ) < . This condition has been extensively used in iterative construction of fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces and CAT ( 0 ) spaces (see, e.g., [4, 5, 21, 28, 29]).

Chang et al. [13] established strong convergence of an Ishikawa-type algorithm as well as a Mann-type algorithm to a fixed point of an asymptotically nonexpansive type map.

We shall follow the idea of a geodesic path, namely, there exists a unique point α x ( 1 α ) y for any x , y C and α [ 0 , 1 ] , to construct an Ishikawa-type algorithm of two asymptotically nonexpansive type maps on a nonempty subset C of a CAT ( 0 ) space.
x 1 C , x n + 1 = ( 1 α n ) x n α n S n y n , y n = ( 1 β n ) x n β n T n x n , n 1 ,
(1.1)

where 0 α n , β n 1 .

When T = I (the identity map) in (1.1), it reduces to the following Mann-type algorithm:
x 1 C , x n + 1 = ( 1 α n ) x n α n T n y n , n 1 ,
(1.2)

where 0 α n 1 .

The purpose of this paper is to approximate a common fixed point of asymptotically nonexpansive type maps in a special kind of a metric space, namely a CAT ( 0 ) space. Our work is a significant generalization of the corresponding results in [5], and it provides analogues of the related results of Chang et al. [13] in uniformly convex Banach spaces. One of our results (Theorem 2.4) gives an affirmative answer to a famous question of Tan and Xu [30] on a nonlinear domain for common fixed points.

2 Fixed point approximation

We begin with the following asymptotic regularity result.

Lemma 2.1 Let C be a nonempty bounded closed convex subset of a CAT ( 0 ) space X. Let S , T : C C be uniformly equicontinuous. Then for the sequence { x n } in (1.1) satisfying
lim n d ( x n , S n x n ) = 0 = lim n d ( x n , T n x n ) ,
we have that
lim n d ( x n , S x n ) = 0 = lim n d ( x n , T x n ) .
Proof Since S is uniformly equicontinuous and
d ( x n , y n ) = d ( x n , ( 1 β n ) x n β n T n x n ) ( 1 β n ) d ( x n , x n ) + β n d ( x n , T n x n ) = β n d ( x n , T n x n ) 0 ,
therefore,
d ( S n x n , S n y n ) 0 .
Now
d ( x n , x n + 1 ) = d ( x n , ( 1 α n ) x n α n S n y n ) α n d ( x n , S n y n ) d ( x n , S n x n ) + d ( S n x n , S n y n )
gives that
lim n d ( x n , x n + 1 ) = 0 .
(2.1)
Clearly,
d ( x n , S x n ) d ( x n , x n + 1 ) + d ( x n + 1 , S n + 1 x n + 1 ) + d ( S n + 1 x n + 1 , S n + 1 x n ) + d ( S n + 1 x n , S x n ) ,
(2.2)
applying limsup to both sides of (2.2), using the uniformly equicontinuous property of S and (2.1), we get that
lim sup n d ( x n , S x n ) 0
and hence
lim n d ( x n , S x n ) = 0 .
Similarly,
lim n d ( x n , T x n ) = 0 .
That is,
lim n d ( x n , S x n ) = 0 = lim n d ( x n , T x n ) .

 □

Our main result is as follows.

Theorem 2.2 Let C be a nonempty, bounded, closed and convex subset of a CAT ( 0 ) space X. Let S , T : C C be uniformly equicontinuous and asymptotically nonexpansive type maps such that F ( S ) F ( T ) . Suppose that 0 < δ α n , β n 1 δ for some δ ( 0 , 1 ) , where { α n } and { β n } are the control parameters of the iteration scheme { x n } in (1.1). If S or T is semi-compact, then { x n } converges strongly to a common fixed point of S and T.

Proof For any p F ( S ) F ( T ) , by the (CN)-inequality, we have
d ( x n + 1 , p ) 2 = d ( α n x n α n S n y n , p ) 2 ( 1 α n ) d ( x n , p ) 2 + α n d ( S n y n , p ) 2 α n ( 1 α n ) d ( x n , S n y n ) 2 = d ( x n , p ) 2 + α n { d ( S n y n , p ) 2 d ( y n , p ) 2 } + α n { d ( y n , p ) 2 d ( x n , p ) 2 } α n ( 1 α n ) d ( x n , S n y n ) 2 .
That is,
d ( x n + 1 , p ) 2 d ( x n , p ) 2 + α n { d ( S n y n , p ) 2 d ( y n , p ) 2 } + α n { d ( y n , p ) 2 d ( x n , p ) 2 } α n ( 1 α n ) d ( x n , S n y n ) 2 .
(2.3)
Next we consider the third term on the right side of (2.3):
d ( y n , p ) 2 d ( x n , p ) 2 = d ( ( 1 β n ) x n β n T n x n , p ) 2 d ( x n , p ) 2 ( 1 β n ) d ( x n , p ) 2 + β n d ( T n x n , p ) 2 d ( x n , p ) 2 β n ( 1 β n ) d ( x n , T n x n ) 2 = β n { d ( T n x n , p ) 2 d ( x n , p ) 2 } β n ( 1 β n ) d ( x n , T n x n ) 2 .
That is,
α n { d ( y n , p ) 2 d ( x n , p ) 2 } α n β n { d ( T n x n , p ) 2 d ( x n , p ) 2 } α n β n ( 1 β n ) d ( x n , T n x n ) 2 .
(2.4)
Substituting (2.4) into (2.3) and using 0 < δ α n , β n 1 δ , we have
d ( x n + 1 , p ) 2 d ( x n , p ) 2 α n ( 1 α n ) 2 d ( S n y n , p ) 2 α n β n ( 1 β n ) 2 d ( x n , T n x n ) 2 + α n { d ( S n y n , p ) 2 d ( y n , p ) 2 ( 1 α n ) 2 d ( S n y n , p ) 2 } + α n β n { d ( T n x n , p ) 2 d ( x n , p ) 2 ( 1 β n ) 2 d ( x n , T n x n ) 2 } d ( x n , p ) 2 δ 2 2 d ( S n y n , p ) 2 δ 3 2 d ( x n , T n x n ) 2 + ( 1 δ ) { d ( S n y n , p ) 2 d ( y n , p ) 2 δ 2 d ( S n y n , p ) 2 } + ( 1 δ ) 2 { d ( T n x n , p ) 2 d ( x n , p ) 2 δ 2 d ( x n , T n x n ) 2 } .
(2.5)
Next we prove that
lim n d ( x n , S n y n ) = 0 = lim n d ( x n , T n x n ) .

Assume that lim sup n d ( x n , S n y n ) > 0 and lim sup n d ( x n , T n x n ) > 0 .

Then there exist subsequences (we use the same notation for a subsequence as well) of { x n } , { y n } and μ 1 > 0 , μ 2 > 0 such that d ( x n , S n y n ) μ 1 > 0 and d ( x n , T n x n ) μ 2 > 0 .

Now from (2.5) it follows that
d ( x n + 1 , p ) 2 d ( x n , p ) 2 δ 2 μ 1 2 2 δ 3 μ 2 2 2 + ( 1 δ ) { d ( S n y n , p ) 2 d ( y n , p ) 2 δ μ 1 2 2 } + ( 1 δ ) 2 { d ( T n x n , p ) 2 d ( x n , p ) 2 δ μ 2 2 2 } .
(2.6)
For an asymptotically nonexpansive type map T, we have that
lim sup n sup x C { d ( T n x , p ) 2 d ( x , p ) 2 } 0 .
That is,
lim n sup m n { sup x C ( d ( T m x , p ) 2 d ( x , p ) 2 ) } 0 .
Hence, for given δ μ i 2 2 > 0 ( i = 1 , 2 ), there exists a positive integer n 0 such that
sup n n 0 { sup x C ( d ( T n x , p ) 2 d ( x , p ) 2 ) } < δ μ i 2 2 .
Since { x n } and { y n } are sequences in C, therefore, for n n 0 , it follows that
d ( S n y n , p ) 2 d ( y n , p ) 2 < δ μ 1 2 2
and
d ( T n x n , p ) 2 d ( x n , p ) 2 < δ μ 2 2 2 .
In the light of the two inequalities above, (2.6) reduces to
δ 2 μ 1 2 2 + δ 3 μ 2 2 2 d ( x n , p ) 2 d ( x n + 1 , p ) 2 for all  n n 0 .
(2.7)
Let m n 0 be any positive integer. Obtain m n 0 inequalities from (2.7) and then, summing up these inequalities, we get
( δ 2 μ 1 2 2 + δ 3 μ 2 2 2 ) ( m n 0 ) d ( x n 0 , p ) 2 d ( x m + 1 , p ) 2 d ( x n 0 , p ) 2 < .
If m , then
= d ( x n 0 , p ) 2 < ,

a contradiction.

This proves that lim sup n d ( x n , S n y n ) = 0 = lim sup n d ( x n , T n x n ) .

That is,
lim n d ( x n , S n y n ) = 0 = lim n d ( x n , T n x n ) .
As
d ( x n , S n x n ) d ( x n , S n y n ) + d ( S n x n , S n y n ) ,
d ( x n , y n ) 0 and S is uniformly equicontinuous. So, by taking lim sup on both sides, we get
lim n d ( x n , S n x n ) = 0 .
Now, Lemma 2.1 implies that
lim n d ( x n , S x n ) = 0 = lim n d ( x n , T x n ) .
(2.8)
Since T is semi-compact, therefore there exists a subsequence { x n i } of { x n } and q C such that
x n i q .
(2.9)
Now, by the uniform equicontinuity of S and T and hence continuity, it follows from (2.8) that
d ( q , S q ) = 0 = d ( q , T q ) .

This gives that q is a common fixed point of S and T.

We now proceed to establish strong convergence of { x n } to q.

Since
d ( T n i x n i , q ) d ( T n i x n i , x n i ) + d ( x n i , q ) ,
therefore
T n i x n i q as  n i .
(2.10)
Clearly,
d ( y n i , q ) = d ( ( 1 β n i ) x n i β n i T n i x n i , q ) ( 1 β n i ) d ( x n i , q ) + β n i d ( T n i x n i , q ) .
Therefore, from (2.9) and (2.10), it follows that
y n i q as  n i .

Next we prove that S n i y n i q as n i .

Since S : C C is of asymptotically nonexpansive type and { y n i } is a sequence in C, therefore we have
lim sup n i { d ( S n i y n i , q ) 2 d ( y n i , q ) 2 } lim sup n i sup x C { d ( S n i x , q ) 2 d ( x , q ) 2 } lim sup n sup x C { d ( S n x , q ) 2 d ( x , q ) 2 } 0 .
(2.11)
As y n i q as n i , it follows from (2.11) that
lim sup n i d ( S n i y n i , q ) 2 0 .
That is,
S n i y n i q as  n i .
Replace p by q in (2.5) to get
d ( x n i + 1 , q ) 2 d ( x n i , q ) 2 δ 2 2 d ( S n i y n i , q ) 2 δ 3 2 d ( x n i , T n i x n i ) 2 + ( 1 δ ) { d ( S n i y n i , q ) 2 d ( y n i , q ) 2 δ 2 d ( S n i y n i , q ) 2 } + ( 1 δ ) 2 { d ( T n i x n i , q ) 2 d ( x n i , q ) 2 δ 2 d ( x n i , T n i x n i ) 2 } ,

which gives that x n i + 1 q as n i .

Continuing in this way, by induction, we can prove that for any m 0 ,
x n i + m q as  n i .

By induction, one can prove that m = 0 { x n i + m } converges to q as i ; in fact { x n } n = n 1 = m = 0 { x n i + m } i = 1 gives that x n q as n . □

We need the following lemma to approximate a common fixed point of two asymptotically nonexpansive maps.

Lemma 2.3 Every asymptotically nonexpansive selfmap T on a nonempty bounded subset C of a metric space X is uniformly equicontinuous and of asymptotically nonexpansive type.

Proof Let T : C C be an asymptotically nonexpansive map with a sequence { k n } [ 1 , ) such that lim n k n = 1 . Let ε > 0 . Then, for each γ > 0 , there exists a positive integer n 0 such that k n 1 < γ for all n n 0 . Put s = max { 1 + γ , k 1 , k 2 , , k n 0 } . Then d ( T n x , T n y ) k n d ( x , y ) s d ( x , y ) for x , y C , n 1 . Choose δ = ε s . Then d ( T n x , T n y ) ε whenever d ( x , y ) δ for x , y C , n 1 , proving that T is uniformly equicontinuous.

The second part of the lemma follows from
lim sup n sup x C { d 2 ( T n x , T n y ) d 2 ( x , y ) } lim n ( k n 1 ) sup x C d 2 ( x , y ) = 0 . sup x C d 2 ( x , y ) = 0 .

 □

By Theorem 2.2 and Lemma 2.3, we have the following result which is new in the literature and sets an analogue of Theorem 2 in [21] without the rate of convergence condition.

Theorem 2.4 Let C be a nonempty, bounded, closed and convex subset of a CAT ( 0 ) space X. Let S , T : C C be asymptotically nonexpansive maps with sequences { s n } , { t n } [ 1 , ) , respectively and F ( S ) F ( T ) . Suppose that 0 < δ α n , β n 1 δ for some δ ( 0 , 1 ) , where { α n } and { β n } are the control parameters of the sequence { x n } in (1.1). If S or T is semi-compact, then { x n } converges strongly to a common fixed point of S and T.

As every uniformly equicontinuous map is uniformly L-Lipschitzian, so the following result is immediate and it unifies Theorem 2.1 and Theorem 2.2 of Chang et al. [13] in Hadamard spaces.

Theorem 2.5 Let C be a nonempty, bounded, closed and convex subset of a CAT ( 0 ) space X. Let S , T : C C be uniformly L-Lipschitzian and asymptotically nonexpansive type maps such that F ( S ) F ( T ) . Suppose that 0 < δ α n , β n 1 δ for some δ ( 0 , 1 ) , where { α n } and { β n } are the control parameters of the sequence { x n } in (1.1). If S or T is semi-compact, then { x n } converges strongly to a common fixed point of S and T.

For S = T , Theorem 2.5 sets an analogue of Theorem 2.1 in [13].

Theorem 2.6 Let C be a nonempty, bounded, closed and convex subset of a CAT ( 0 ) space X. Let T : C C be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that F ( T ) . Suppose that 0 < δ α n , β n 1 δ for some δ ( 0 , 1 ) , where { α n } and { β n } are the control parameters of the sequence { x n } in (1.1) with S = T . If T is semi-compact, then { x n } converges strongly to a fixed point of T.

On taking S = I (the identity map) in Theorem 2.5, we obtain an analogue of Theorem 2.2 in [13].

Theorem 2.7 Let C be a nonempty, bounded, closed and convex subset of a CAT ( 0 ) space X. Let T : C C be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that F ( T ) . Suppose that 0 < δ α n , β n 1 δ for some δ ( 0 , 1 ) , where { α n } and { β n } are the control parameters of the sequence { x n } in (1.2). If T is semi-compact, then { x n } converges strongly to a fixed point of T.

Remark 2.8 (1) Tan and Xu [30] obtained only weak convergence theorems for asymptotically nonexpansive maps satisfying the rate of convergence condition and remarked, ‘We do not know whether our weak convergence Theorem 3.1 remains valid if k n is allowed to approach 1 slowly enough so that n = 1 ( k n 1 ) diverges’. Our Theorem 2.4 gives an affirmative answer to their question in CAT ( 0 ) spaces.

(2) Our results are generalizations in CAT ( 0 ) spaces of the corresponding basic results in [16, 21, 28, 29].

(3) Theorem 2.2 improves and generalizes Theorems 4.2-4.3 in [5].

Declarations

Acknowledgements

The author is grateful to King Fahd University of Petroleum & Minerals for supporting research project IN 121023.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia
(2)
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

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© Fukhar-ud-din; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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