- W. Laowang
^{1}and - B. Panyanak
^{1}Email author

**2009**:730132

https://doi.org/10.1155/2009/730132

© W. Laowang and B. Panyanak. 2009

**Received: **12 December 2008

**Accepted: **3 April 2009

**Published: **5 May 2009

## Abstract

We show strong and convergence for Mann iteration of a multivalued nonexpansive mapping whose domain is a nonempty closed convex subset of a CAT(0) space. The results we obtain are analogs of Banach space results by Song and Wang [2009, 2008]. Strong convergence of Ishikawa iteration are also included.

## 1. Introduction

where is the distance from the point to the set

A point
is called a *fixed point* of
if
We denote by
the set of all fixed points of

In 2005, Sastry and Babu [1] introduced the Mann and Ishikawa iterations for multivalued mappings as follows: let be a real Hilbert space and be a multivalued mapping for which . Fix and define

- (A)
- (B)

They proved the following results.

Theorem 1.1.

Let be a nonempty compact convex subset of a Hilbert space Suppose is nonexpansive and has a fixed point Assume that (i) and (ii) Then the sequence of Mann iterates defined by (A) converges to a fixed point of

Theorem 1.2.

Let be a nonempty compact convex subset of a Hilbert space Suppose that a nonexpansive map has a fixed point Assume that (i) (ii) and (iii) Then the sequence of Ishikawa iterates defined by (B) converges to a fixed point of

In 2007, Panyanak [2] extended Sastry-Babu's results to uniformly convex Banach spaces as the following results.

Theorem 1.3.

Let be a nonempty compact convex subset of a uniformly convex Banach spaces Suppose that a nonexpansive map has a fixed point Let be the sequence of Mann iterates defined by (A). Assume that (i) and (ii) Then the sequence converges to a fixed point of

Theorem 1.4.

Let be a nonempty compact convex subset of a uniformly convex Banach spaces Suppose that a nonexpansive map has a fixed point . Let be the sequence of Ishikawa iterates defined by (B). Assume that (i) (ii) and (iii) Then the sequence converges to a fixed point of

Recently, Song and Wang [3, 4] pointed out that the proof of Theorem 1.4 contains a gap. Namely, the iterative sequence defined by (B) depends on the fixed point Clearly, if and then the sequence defined by is different from the one defined by Thus, for defined by , we cannot obtain that is a decreasing sequence from the monotony of . Hence, the conclusion of Theorem 1.4 (also Theorem 1.3) is very dubious.

Motivated by solving the above gap, they defined the modified Mann and Ishikawa iterations as follows.

*The sequence of Mann iterates*is defined as follows: let and such that Choose and Let

They obtained the following results.

Theorem 1.5 (see [3, Theorem 2.3]).

Then the sequence strongly converges to a fixed point of

Theorem 1.6 (see [3, Theorem 2.4]).

Then the sequence strongly converges to a fixed point of

Theorem 1.7 (see [3, Theorem 2.5]).

Then the sequence weakly converges to a fixed point of

Theorem 1.8 (see [4, Theorem 1]).

Let be a nonempty compact convex subset of a uniformly convex Banach space Suppose that is a multivalued nonexpansive mapping and satisfying for any fixed point Let be the sequence of Ishikawa iterates defined by (1.13). Assume that (i) (ii) and (iii) Then the sequence strongly converges to a fixed point of

Theorem 1.9 (see [4, Theorem 2]).

Let be a nonempty closed convex subset of a uniformly convex Banach space Suppose that is a multivalued nonexpansive mapping that satisfy Condition I. Let be the sequence of Ishikawa iterates defined by (1.13). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

In this paper, we study the iteration processes defined by (1.8) and (1.13) in a CAT(0) space and give analogs of Theorems 1.5–1.9 in this setting.

## 2. Spaces

A metric space is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in is at least as "thin" as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT space. Other examples include Pre-Hilbert spaces, -trees (see [8]), Euclidean buildings (see [9]), the complex Hilbert ball with a hyperbolic metric (see [10]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry (see Bridson and Haefliger [8]). Burago, et al. [11] contains a somewhat more elementary treatment, and Gromov [12] a deeper study.

Fixed point theory in a CAT(0) space was first studied by Kirk (see [13] and [14]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see, e.g., [15–24]). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT( ) space with since any CAT( ) space is a CAT( ) space for every (see [8], page 165).

Let
be a metric space. A *geodesic path* joining
to
(or, more briefly, a *geodesic* from
to
) is a map
from a closed interval
to
such that
and
for all
In particular,
is an isometry and
The image
of
is called a *geodesic* (or *metric*) *segment* joining
and
. When it is unique this geodesic is denoted by
. The space
is said to be a *geodesic space* if every two points of
are joined by a geodesic, and
is said to be *uniquely geodesic* if there is exactly one geodesic joining
and
for each
A subset
is said to be *convex* if
includes every geodesic segment joining any two of its points.

A *geodesic triangle*
in a geodesic space
consists of three points
in
(the *vertices* of
) and a geodesic segment between each pair of vertices (the *edges* of
). A *comparison triangle* for geodesic triangle
in
is a triangle
in the Euclidean plane
such that
for

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

*inequality*if for all and all comparison points

From now on we will use the notation for the unique point satisfying (2.2). By using this notation Dhompongsa and Panyanak [24] obtained the following lemma which will be used frequently in the proof of our main theorems.

Lemma 2.1.

This is the (CN) inequality of Bruhat and Tits [25]. In fact (cf. [8, page 163]), a geodesic metric space is a CAT(0) space if and only if it satisfies (CN).

The following lemma is a generalization of the (CN) inequality which can be found in [24].

Lemma 2.2.

The preceding facts yield the following result.

Proposition 2.3.

Let be a geodesic space. Then the following are equivalent:

The existence of fixed points for multivalued nonexpansive mappings in a CAT(0) space was proved by S. Dhompongsa et al. [17], as follows.

Theorem 2.4.

## 3. The Setting

The notion of asymptotic centers in a Banach space can be extended to a CAT(0) space as well, simply replacing with It is known (see, e.g., [18, Proposition 7]) that in a CAT(0) space, consists of exactly one point.

Next we provide the definition and collect some basic properties of -convergence.

Definition 3.1 (see [23]).

A sequence in a CAT(0) space is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case one must write and call the -limit of

Remark 3.2.

*Opial's condition*([26]) if given whenever converges weakly to

Proposition 3.3.

Let be a reflexive Banach space satisfying Opial's condition and let be a bounded sequence in and let Then converges weakly to if and only if for all subsequence of

Proof.

( ) Let be a subsequence of . Then converges weakly to By Opial's condition ( ) Suppose for all subsequence of and assume that does not converge weakly to Then there exists a subsequence of such that for each is outside a weak neighborhood of Since is bounded, without loss of generality we may assume that converges weakly to By Opial's condition a contradiction.

Lemma 3.4.

(i) Every bounded sequence in has a convergent subsequence (see [23, page 3690]). (ii) If is a closed convex subset of and if is a bounded sequence in then the asymptotic center of is in (see [17, Proposition 2.1]).

Now, we define the sequences of Mann and Ishikawa iterates in a CAT(0) space which are analogs of the two defined in Banach spaces by Song and Wang [3, 4].

Definition 3.5.

*The sequence of Mann iterates*is defined as follows: let and such that Choose and Let

Definition 3.6.

*The sequence of Ishikawa iterates*is defined as follows: let , and such that Choose and Let

Lemma 3.7.

for some sequence in Then has a fixed point. Moreover, if converges for each , then strongly converges to a fixed point of

Proof.

This implies that is a fixed point of Since the limit of exists and we have This show that the sequence strongly converges to

Before proving our main results we state a lemma which is an analog of Lemma 2.2 of [27]. The proof is metric in nature and carries over to the present setting without change.

Lemma 3.8.

## 4. Strong and Convergence of Mann Iteration

Theorem 4.1.

- (i)
- (ii)

Then the sequence strongly converges to a fixed point of

Proof

Case 1.

By Lemma 3.7, converges to a point Since the limit of exists, it must be the case that and hence the conclusion follows.

Case 2.

so the conclusion follows from Lemma 3.7.

Theorem 4.2.

Let be a nonempty closed convex subset of a complete CAT(0) space Suppose that is a multivalued nonexpansive mapping that satisfies Condition I. Let be the sequence of Mann iterates defined by (3.7). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

Proof.

The proof of remaining part closely follows the proof of of [2, Theorem 3.8], simply replacing with .

Next we show a convergence theorem of Mann iteration in a CAT(0) space setting which is an analog of Theorem 1.7. For this we need more lemmas.

Lemma 4.3 (see [24, Lemma 2.8]).

If is a bounded sequence in a complete CAT (0)space with and is a subsequence of with and the sequence converges, then

Lemma 4.4.

Proof.

Since and hence by (4.19). Therefore is a fixed point of

Lemma 4.5.

Let be a closed convex subset of a complete CAT (0)space and let be a nonexpansive mapping. Suppose is a bounded sequence in such that and converges for all then Here where the union is taken over all subsequences of Moreover, consists of exactly one point.

Proof.

Let , then there exists a subsequence of such that By Lemma 3.4(i) and (ii) there exists a subsequence of such that . By Lemma 4.4, . By Lemma 4.3, This shows that Next, we show that consists of exactly one point. Let be a subsequence of with and let . Since is convergent by the assumption. By Lemma 4.3, This completes the proof.

Theorem 4.6.

Then the sequence converges to a fixed point of

Proof.

Thus by (4.9). By Lemma 4.5, consists of exactly one point and is contained in . This shows that converges to an element of

## 5. Strong Convergence of Ishikawa Iteration

The following lemma can be found in [2].

Lemma 5.1.

Let be two real sequences such that

Let be a nonnegative real sequence such that is bounded. Then has a subsequence which converges to zero.

The following theorem is an analog of Theorem 1.8.

Theorem 5.2.

Let be a nonempty compact convex subset of a complete CAT (0) space Suppose that is a multivalued nonexpansive mapping and satisfying for any fixed point Let be the sequence of Ishikawa iterates defined by (3.12). Assume that

Then the sequence strongly converges to a fixed point of

Proof.

By Lemma 3.7, converges to a point Since the limit of exists, it must be the case that and hence the conclusion follows.

The following theorem is an analog of Theorem 1.9.

Theorem 5.3.

Let be a nonempty closed convex subset of a complete CAT(0) space Suppose that is a multivalued nonexpansive mapping that satisfies Condition I. Let be the sequence of Ishikawa iterates defined by (3.12). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

Proof.

The proof of remaining part closely follows the proof of [2, Theorem 3.8], simply replacing with .

## Declarations

### Acknowledgments

We are grateful to Professor Sompong Dhompongsa for his suggestion and advice during the preparation of the article. The research was supported by the Commission on Higher Education and Thailand Research Fund under Grant MRG5080188.

## Authors’ Affiliations

## References

- Sastry KPR, Babu GVR:
**Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point.***Czechoslovak Mathematical Journal*2005,**55**(4):817–826. 10.1007/s10587-005-0068-zMATHMathSciNetView ArticleGoogle Scholar - Panyanak B:
**Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces.***Computers & Mathematics with Applications*2007,**54**(6):872–877. 10.1016/j.camwa.2007.03.012MATHMathSciNetView ArticleGoogle Scholar - Song Y, Wang H:
**Convergence of iterative algorithms for multivalued mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(4):1547–1556. 10.1016/j.na.2008.02.034MATHMathSciNetView ArticleGoogle Scholar - Song Y, Wang H:
**Erratum to "Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces" [Comput. Math. Appl. 54 (2007) 872–877].***Computers & Mathematics with Applications*2008,**55**(12):2999–3002. 10.1016/j.camwa.2007.11.042MATHMathSciNetView ArticleGoogle Scholar - Nadler SB Jr.:
**Multi-valued contraction mappings.***Pacific Journal of Mathematics*1969,**30:**475–488.MATHMathSciNetView ArticleGoogle Scholar - Assad NA, Kirk WA:
**Fixed point theorems for set-valued mappings of contractive type.***Pacific Journal of Mathematics*1972,**43**(3):553–562.MathSciNetView ArticleMATHGoogle Scholar - Senter HF, Dotson WG Jr.:
**Approximating fixed points of nonexpansive mappings.***Proceedings of the American Mathematical Society*1974,**44**(2):375–380. 10.1090/S0002-9939-1974-0346608-8MATHMathSciNetView ArticleGoogle Scholar - Bridson MR, Haefliger A:
*Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences*.*Volume 319*. Springer, Berlin, Germany; 1999:xxii+643.View ArticleMATHGoogle Scholar - Brown KS:
*Buildings*. Springer, New York, NY, USA; 1989:viii+215.MATHView ArticleGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Burago D, Burago Y, Ivanov S:
*A Course in Metric Geometry, Graduate Studies in Mathematics*.*Volume 33*. American Mathematical Society, Providence, RI, USA; 2001:xiv+415.MATHGoogle Scholar - Gromov M:
*Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics*.*Volume 152*. Birkhäuser, Boston, Mass, USA; 1999:xx+585.MATHGoogle Scholar - Kirk WA:
**Geodesic geometry and fixed point theory.**In*Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colección Abierta*.*Volume 64*. Universidad de Sevilla, Secretariado de Publicaciones, Sevilla, Spain; 2003:195–225.Google Scholar - Kirk WA:
**Geodesic geometry and fixed point theory. II.**In*International Conference on Fixed Point Theory and Applications*. Yokohama Publishers, Yokohama, Japan; 2004:113–142.Google Scholar - Kirk WA:
**Fixed point theorems in CAT(0) spaces and**-**trees.***Fixed Point Theory and Applications*2004,**2004**(4):309–316.MATHMathSciNetView ArticleGoogle Scholar - Dhompongsa S, Kaewkhao A, Panyanak B:
**Lim's theorems for multivalued mappings in CAT(0) spaces.***Journal of Mathematical Analysis and Applications*2005,**312**(2):478–487. 10.1016/j.jmaa.2005.03.055MATHMathSciNetView ArticleGoogle Scholar - Dhompongsa S, Kirk WA, Panyanak B:
**Nonexpansive set-valued mappings in metric and Banach spaces.***Journal of Nonlinear and Convex Analysis*2007,**8**(1):35–45.MATHMathSciNetGoogle Scholar - Dhompongsa S, Kirk WA, Sims B:
**Fixed points of uniformly Lipschitzian mappings.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(4):762–772. 10.1016/j.na.2005.09.044MATHMathSciNetView ArticleGoogle Scholar - Chaoha P, Phon-on A:
**A note on fixed point sets in CAT(0) spaces.***Journal of Mathematical Analysis and Applications*2006,**320**(2):983–987. 10.1016/j.jmaa.2005.08.006MATHMathSciNetView ArticleGoogle Scholar - Fujiwara K, Nagano K, Shioya T:
**Fixed point sets of parabolic isometries of CAT(0)-spaces.***Commentarii Mathematici Helvetici*2006,**81**(2):305–335.MATHMathSciNetView ArticleGoogle Scholar - Leustean L:
**A quadratic rate of asymptotic regularity for CAT(0)-spaces.***Journal of Mathematical Analysis and Applications*2007,**325**(1):386–399. 10.1016/j.jmaa.2006.01.081MATHMathSciNetView ArticleGoogle Scholar - Shahzad N, Markin J:
**Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces.***Journal of Mathematical Analysis and Applications*2008,**337**(2):1457–1464. 10.1016/j.jmaa.2007.04.041MATHMathSciNetView ArticleGoogle Scholar - Kirk WA, Panyanak B:
**A concept of convergence in geodesic spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(12):3689–3696. 10.1016/j.na.2007.04.011MATHMathSciNetView ArticleGoogle Scholar - Dhompongsa S, Panyanak B:
**On**-**convergence theorems in CAT(0) spaces.***Computers & Mathematics with Applications*2008,**56**(10):2572–2579. 10.1016/j.camwa.2008.05.036MATHMathSciNetView ArticleGoogle Scholar - Bruhat F, Tits J:
**Groupes réductifs sur un corps local.***Publications Mathématiques de L'IHÉS*1972,**41**(1):5–251.MATHMathSciNetView ArticleGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MATHMathSciNetView ArticleGoogle Scholar - Suzuki T:
**Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(1):103–123.MATHView ArticleMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.