-Convergence Problems for Asymptotically Nonexpansive Mappings in CAT(0) Spaces

Abstract

New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping of a closed convex subset of a CAT(0) space . Consider the iteration process , where is arbitrary and or for , where . It is shown that under certain appropriate conditions on   △-converges to a fixed point of .

1. Introduction and Preliminaries

Let be a nonempty subset of a metric space . A mapping is a contraction if there exists such that for all , we have . It is said to be nonexpansive if for all , we have . is said to be asymptotically nonexpansive if there exists a sequence with such that for all integers and all . Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is asymptotically nonexpansive with sequence , for all . There are, however, asymptotically nonexpansive mappings which are not nonexpansive (see, e.g., [1]). As a generalization of the class of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972 and has been studied by several authors (see, e.g., [3–5]). Goebel and Kirk proved that if is a nonempty closed convex and bounded subset of a uniformly convex Banach space (more general than a Hilbert space, i.e., CAT(0) space), then every asymptotically nonexpansive self-mapping of has a fixed point. The weak and strong convergence problems to fixed points of nonexpansive and asymptotically nonexpansive mappings have been studied by many authors.

We will denote by the set of fixed points of . In 1967, Halpern [6] introduced an explicit iterative scheme for a nonexpansive mapping on a subset of a Hilbert space by taking any point and defined the iterative sequence by where . He pointed out that under certain appropriate conditions on converges strongly to a fixed point of . In 1994, Tan and Xu [7] introduced the following iterative scheme for asymptotically nonexpansive mapping on uniformly convex Banach space: where . They proved that under certain appropriate conditions on converges weakly to a fixed point of .

In 2012, we [8] studied the viscosity approximation methods for nonexpansive mappings on CAT(0) space. For a contraction on , consider the iteration process , where is arbitrary and for , where . We proved that under certain appropriate conditions on converges strongly to a fixed point of which solves some variational inequality.

The purpose of this paper is to study the iterative scheme defined as follows: consider an asymptotically nonexpansive self-mapping of a closed convex subset of a CAT(0) space with coefficient . consider the iteration process , where is arbitrary and or for , where . We show that   -converges to a fixed point of under certain appropriate conditions on , and .

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 1. Let be a CAT(0) space. Then, one has the following:(i)(see [9, Lemma 2.4])  for each and , one has (ii) (see [10]) for each and one has (iii) (see [5, Lemma 3]) for each and , one has (iv) (see [9]) for each and , one has

Let be a complete CAT(0) space and let be a bounded sequence in a complete and for set

The asymptotic radius of is given by and the asymptotic center of is the set

It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, consists of exactly one point.

A sequence in 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. Assume that is a CAT(0) space. Then, one has the following:(i)(see [12]) every bounded sequence in has a -convergent subsequence;(ii)(see [13]) if is a closed convex subset of and is an asymptotically nonexpansive mapping, then the conditions   -converge to and , imply and .

Lemma 3 (see [14, 15]). Let be three nonnegative real sequences satisfying the following condition: where is some nonnegative integer, , . Then the limit exists.

2. -Convergence of the Iteration Sequences

In this section, we will study the -convergence of the iteration sequence for asymptotically nonexpansive mappings in CAT(0) spaces.

Suppose that be a CAT(0) space, a closed convex subset of , and an asymptotically nonexpansive mapping with coefficient . Firstly, we consider the iteration process: where and satisfy the following.(i)There exist positive integers , , and , , where , such that (ii)Consider .

We will prove that   -converges to a fixed point of .

Lemma 4. Let be a CAT(0) space, a closed convex subset of , an asymptotically nonexpansive mapping with coefficient , and . If , . Let , be generated by , , . Then the limit exists for all .

Proof. Taking , we have By Lemma 3, we can get that exists.

Remark 5. The above lemma implies that is bounded and so is the sequence . Moreover, let , then we have It follows that the sequences , , are bounded.

Proposition 6. Let be a CAT(0) space, a closed convex subset of , and an asymptotically nonexpansive mapping with coefficient . If , . Let , be generated by , , . Then under the hypotheses (i) and (ii), one can get that .

Proof. By the assumption, is nonempty. Take , by Lemma 1(iv), we have which implies that
Therefore, we have
Since and are bounded and for all . we have
By the conditions (i) and (ii), we have which implies that

Theorem 7. Let be a CAT(0) space, a closed convex subset of , and an asymptotically nonexpansive mapping with coefficient . If , . Let , be generated by . Then under the hypotheses (i) and (ii), one can get that   -converges to a fix point of .

Proof. We first show that . Indeed it follows that By the conditions (i) and (ii) and Proposition 6, we get .
And then, By Proposition 6, we get that .
We claim that . Indeed we have Thus,
Since is bounded, we may assume that   -converges to a point . By Lemma 2, we have .

Next we will consider another iteration process: where , and satisfy the following(H1) There exist positive integers and , such that (H2).

We will prove that also -converges to a fixed point of .

Lemma 8. Let be a CAT(0) space, a closed convex subset of , an asymptotically nonexpansive mapping with coefficient , and . If , . Let , be generated by . Then the limit exists for all .

Proof. Taking , we have By Lemma 3, we can get that exists.