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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 327434, 17 pages
http://dx.doi.org/10.1155/2012/327434
Research Article

Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces

1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 26 October 2011; Accepted 8 December 2011

Academic Editor: Rudong Chen

Copyright © 2012 P. Pasom and B. Panyanak. 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.

Abstract

Let be a nonempty bounded closed convex subset of a complete CAT(0) space . We prove that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive mappings on is nonempty closed and convex. We also show that, under some suitable conditions, the sequence defined by , converges to a common fixed point of where they are asymptotic pointwise nonexpansive mappings on , are sequences in for all and is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.

1. Introduction

A mapping on a subset of a Banach space is said to be asymptotic pointwise nonexpansive if there exists a sequence of mappings such that where , for all . This class of mappings was introduced by Kirk and Xu [1], where it was shown that if is a bounded closed convex subset of a uniformly convex Banach space , then every asymptotic pointwise nonexpansive mapping always has a fixed point. In 2009, Hussain and Khamsi [2] extended Kirk-Xu's result to the case of metric spaces, specifically to the so-called CAT(0) spaces. Recently, Kozlowski [3] defined an iterative sequence for an asymptotic pointwise nonexpansive mapping by and where and are sequences in and is an increasing sequence of natural numbers. He proved, under some suitable assumptions, that the sequence defined by (1.2) converges weakly to a fixed point of where is a uniformly convex Banach space which satisfies the Opial condition and converges strongly to a fixed point of provided is a compact mapping for some . On the other hand, Khan et al. [4] studied the iterative process defined by where are asymptotically quasi-nonexpansive mappings on and are sequences in for all .

In this paper, motivated by the results mentioned above, we ensure the existence of common fixed points for a family of asymptotic pointwise nonexpansive mappings in a CAT(0) space. Furthermore, we obtain and strong convergence theorems of a sequence defined by where are asymptotic pointwise nonexpansive mappings on a subset of a complete CAT(0) space and are sequences in for all , and is an increasing sequence of natural numbers. We also note that our method can be used to prove the analogous results for uniformly convex Banach spaces.

2. Preliminaries

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. It is well-known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [5]), -trees (see [6]), Euclidean buildings (see [7]), and the complex Hilbert ball with a hyperbolic metric (see [8]). For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [5].

Fixed point theory in CAT(0) spaces was first studied by Kirk (see [9, 10]). 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 papers have appeared (see, e.g., [2, 1122] and the references therein). It is worth mentioning that fixed point theorems in CAT(0) spaces (specially in -trees) can be applied to graph theory, biology, and computer science (see, e.g., [6, 2326]).

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.

CAT(0): Let be a geodesic triangle in , and let be a comparison triangle for . Then, is said to satisfy the CAT(0) inequality if for all and all comparison points ,

Let , by Lemma  2.1(iv) of [14] for each , there exists a unique point such that We will use the notation for the unique point satisfying (2.2). We now collect some elementary facts about CAT(0) spaces.

Lemma 2.1. Let be a complete CAT(0) space.[5, Proposition  2.4] If is a nonempty closed convex subset of , then, for every , there exists a unique point such that . Moreover, the map is a nonexpansive retract from onto .[14, Lemma  2.4] For and , we have [14, Lemma  2.5] For and , we have

We now give the concept of -convergence and collect some of its basic properties. Let be a bounded sequence in a CAT(0) space . For , we set The asymptotic radius of is given by and the asymptotic center of is the set

It is known from Proposition  7 of [27] that, in a CAT(0) space, consists of exactly one point.

Definition 2.2 (see [28, 29]). 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, we write and call the -limit of .

Lemma 2.3. Let be a complete CAT(0) space.(i)[28, page 3690] Every bounded sequence in has a -convergent subsequence.(ii)[30, Proposition  2.1] If is a closed convex subset of a complete CAT(0) space and if is a bounded sequence in , then the asymptotic center of is in .(iii)[14, 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 .

Recall that a mapping is said to be nonexpansive [31] if where is called asymptotically nonexpansive [32] if there is a sequence of positive numbers with the property and such thatwhere is called an asymptotic pointwise nonexpansive mapping [1] if there exists a sequence of functions such that where . The following implications hold. A point is called a fixed point of if . We shall denote by the set of fixed points of . The existence of fixed points for asymptotic pointwise nonexpansive mappings in CAT(0) spaces was proved by Hussain and Khamsi [2] as the following result.

Theorem 2.4. Let be a nonempty bounded closed convex subset of a complete CAT(0) space . Suppose that is an asymptotic pointwise nonexpansive mapping. Then, is nonempty closed and convex.

3. Existence Theorems

Let be a metric space and a family of subsets of . Then, we say that defines a convexity structure on if it contains the closed balls and is stable by intersection.

Definition 3.1 (see [2]). Let be a convexity structure on . We will say that is compact if any family of elements of has a nonempty intersection provided for any finite subset .

Let be a complete CAT(0) space. We denote by the family of all closed convex subsets of . Then, is a compact convexity structure on (see, e.g., [2]).

The following theorem is an extension of Theorem  4.3 in [33]. For an analog of this result in uniformly convex Banach spaces, see [34].

Theorem 3.2. Let be a nonempty bounded closed and convex subset of a complete CAT(0) space . Then, for any commuting family of asymptotic pointwise nonexpansive mappings on , the set of common fixed points of is nonempty closed and convex.

Proof. Let be the family of all finite intersections of the fixed point sets of mappings in the commutative family . We first show that has the finite intersection property. Let . By Theorem 2.4, is a nonempty closed and convex subset of . We assume that is nonempty closed and convex for some with . For and with , we have Thus, is a fixed point of , which implies that ; therefore, is invariant under . Again, by Theorem 2.4, has a fixed point in , that is, By induction, . Hence, has the finite intersection property. Since is compact, Obviously, the set is closed and convex.

As a consequence of Lemma 2.1(i) and Theorem 3.2, we obtain an analog of Bruck's theorem [35].

Corollary 3.3. Let be a nonempty bounded closed and convex subset of a complete CAT(0) space . Then, for any commuting family of nonexpansive mappings on , the set of common fixed points of is a nonempty nonexpansive retract of .

4. Convergence Theorems

Throughout this section, stands for a complete CAT(0) space. Let be a closed convex subset of . We shall denote by the class of all asymptotic pointwise nonexpansive mappings from into . Let , without loss of generality, we can assume that there exists a sequence of mappings such that for all ,  , and , we have Let . Again, without loss of generality, we can assume that for all , and . We define , then, for each , we have .

The following definition is a mild modification of [3, Definition  2.3].

Definition 4.1. Define as a class of all such that .
Let , and let be bounded away from 0 and 1 for all , and an increasing sequence of natural numbers. Let , and define a sequence in as

We say that the sequence in (4.4) is well defined if . As in [3], we observe that for every . Hence, we can always choose a subsequence which makes well defined.

Lemma 4.2 (see [36, Lemma  2.2]). Let and be sequences of nonnegative real numbers satisfying Then, (i) exists, (ii) if , then .

Lemma 4.3 (see [37, 38]). Suppose is a sequence in for some and , are sequences in such that ,  , and for some . Then,

Lemma 4.4. Let be a nonempty closed convex subset of and . Let and be such that in (4.4) is well defined. Assume that . Then,(a)there exists a sequence in such that and ,   for all and all ,(b)there exists a constant such that ,   for all and .

Proof. (a) Let and for all . Since , we have . Now, Suppose that holds for some . Then, By induction, we have This implies This completes the proof of (a).
(b) We observe that holds for all and . Thus, by (a), for , we have The proof is complete by setting .

Theorem 4.5. Let be a nonempty closed convex subset of and . Let and be such that in (4.4) is well defined. Assume that . Then, converges to some point in if and only if , where .

Proof. The necessity is obvious. Now, we prove the sufficiency. From Lemma 4.4(a), we have This implies Since , then . By Lemma 4.2(ii), we get . Next, we show that is a Cauchy sequence. From Lemma 4.4(b), there exists such that Since , for each , there exists such that Hence, there exists such that By (4.14) and (4.16), for , we have This shows that is a Cauchy sequence and so converges to some . We next show that . Let . Then, for each , there exists such that Since , there exists such that Thus, there exists such that By (4.18) and (4.20), for each , we have Since is arbitrary, we have for all . Hence, .

As an immediate consequence of Theorem 4.5, we obtain the following.

Corollary 4.6. Let be a nonempty closed convex subset of and . Let and be such that in (4.4) is well defined. Assume that . Then, converges to a point if and only if there exists a subsequence of which converges to .

Definition 4.7. A strictly increasing sequence is called quasiperiodic [39] if the sequence is bounded or equivalently if there exists a number such that any block of consecutive natural numbers must contain a term of the sequence . The smallest of such numbers will be called a quasiperiod of .

Lemma 4.8. Let be a nonempty closed convex subset of and . Let for some and be such that in (4.4) is well defined. Then,(i) exists for all ,(ii),   for all ,(iii) if the set is quasiperiodic, then ,   for all .

Proof. (i) Follows from Lemmas 4.2(i) and 4.4(a).
(ii) Let , then, by (i), we have exists. Let By (4.9) and (4.22), we get that Note that Thus, so that From (4.23) and (4.26), we have That is for each .
We also obtain from (4.23) that By Lemma 4.3, we get that For the case , by (4.1), we have But since , then Moreover, Again, by Lemma 4.3, we get that Thus, (4.30) and (4.34) imply that For , from (ii), we have If , then we have By (ii) and , we get By (4.36) and (4.38), we have By the construction of the sequence , we have from (4.35) that Next, we show that It is enough to prove that as though . Indeed, let be a quasiperiod of , and let be given. Then, there exists such that By the quasiperiodicity of , for each , there exists such that . Without loss of generality, we can assume that (the proof for the other case is identical). Let . Then, . Since by (4.40), there exists such that This implies that By the definition of , we have
Let . Then, for , we have from (4.42), (4.44), and (4.45) that To prove that as though . Since is quasiperiodic, for each , we have From this, together with (4.39) and (4.40), we can obtain that as through .

The following lemmas can be found in [3] (see also [2]).

Lemma 4.9. Let be a nonempty closed convex subset of , and let . If , then for every .

Lemma 4.10. Let be a nonempty closed convex subset of , and let . Suppose is a bounded sequence in such that and -. Then, .

By using Lemmas 2.3 and 4.10, we can obtain the following result. We omit the proof because it is similar to the one given in [38].

Lemma 4.11. Let be a closed convex subset of , and let be an asymptotic pointwise nonexpansive mapping. Suppose is a bounded sequence in such that and converges for each , then . Here, where the union is taken over all subsequences of . Moreover, consists of exactly one point.

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

Theorem 4.12. Let be a nonempty closed convex subset of and . Let for some and be such that in (4.4) is well defined. Suppose that and the set is quasiperiodic. Then, -converges to a common fixed point of the family .

Proof. Let , by Lemma 4.8,   existsm and hence is bounded. Since for all , then by Lemma 4.11   for all , and hence . Since consists of exactly one point, then -converges to an element of .

Before proving our strong convergence theorem, we recall that a mapping is said to be semicompact if is closed and, for any bounded sequence in with , there exists a subsequence of and such that .

Theorem 4.13. Let be a nonempty closed convex subset of and such that is semicompact for some and . Let for some and be such that in (4.4) is well defined. Suppose that and the set is quasiperiodic. Then, converges to a common fixed point of the family .

Proof. By Lemma 4.8, we have Let be such that is semicompact. Thus, by Lemma 4.9, We can also find a subsequence of such that . Hence, from (4.48), we have Thus, , and, by Corollary 4.6, converges to . This completes the proof.

5. Concluding Remarks

One may observe that our method can be used to obtain the analogous results for uniformly convex Banach spaces. Let be a nonempty closed convex subset of a Banach space and fix . Define a sequence in as where ,   are sequences in for all , and is an increasing sequence of natural numbers.

Theorem 5.1. Let be a uniformly convex Banach space with the Opial property, and let be a nonempty closed convex subset of . Let ,   for some , and let be such that in (5.1) is well defined. Suppose that and the set is quasiperiodic. Then, converges weakly to a common fixed point of the family .

Theorem 5.2. Let be a nonempty closed convex subset of a uniformly convex Banach space and such that is semicompact for some and . Let for some , and let be such that in (5.1) is well defined. Suppose that and the set is quasiperiodic. Then, converges strongly to a common fixed point of the family .

Acknowledgments

This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author also thanks the Graduate School of Chiang Mai University, Thailand.

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