Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces
© Zeqing Liu et al. 2009
Received: 9 May 2009
Accepted: 14 December 2009
Published: 14 December 2009
This paper provides a few convergence results of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature.
1. Introduction and Preliminaries
Browder  and Kirk  established that a nonexpansive mapping which maps a closed bounded convex subset of a uniformly convex Banach space into itself has a fixed point in . Since then, many researchers have studied, under various conditions, the convergence of the Mann and Ishikawa iteration methods dealing with nonexpansive and quasi-nonexpansive mappings (see [3–11] and the references therein). Rhoades  pointed out that the Picard iteration schemes for nonexpansive mappings need not converge. Senter and Dotson  obtained conditions under which the Mann iteration schemes generated by nonexpansive and quasi-nonexpansiv mappings in uniformly convex Banach spaces, converge to fixed points of these mappings, respectively. Ishikawa  established that the Mann iteration methods can be used to approximate fixed points of nonexpansive mappings in Banach spaces. Deng  obtained similar results for Ishikawa iteration processes in normed linear spaces and Banach spaces.
Our aim is to prove several convergence theorems of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. Our results presented in this paper extend substantially the results due to Deng , Ishikawa , and Senter and Dotson .
It is easy to see that and for all and . Hence is convex. Hu and Huang  proved that if is a Banach space, then is a complete metric space. Now we introduce the following concepts in hyperspaces.
In case , where is a nonempty subset of , and is a mapping, then Definitions 1.1, 1.2, and 1.3(ii) reduce to the corresponding concepts in [1–11, 13]. It is well known that every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive, but the converse is not true; see . Examples 3.1 and 3.4 in this paper reveal that the class of nonexpansive mappings with nonempty fixed point set is a proper subclass of quasi-nonexpansive mappings with both Condition A and Condition B.
The following lemmas play important roles in this paper.
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Lemma 1.7 (see ).
Thus (1.6) follows from (1.10) and (1.11). This completes the proof.
2. Main Results
Our results are as follows.
Let be a Banach space and a nonempty closed subset of . Assume that is nonexpansive and there exists a compact subset of such that If (2.1) and (2.2) hold, then has a fixed point in . Moreover, given , the Ishikawa iteration sequence with errors converges to a fixed point of .
Let be a Banach space and a nonempty closed subset of . Suppose that is a qusi-nonexpansive mapping and satisfies Condition A. Assume that (2.1) and (2.2) hold and is in . If is bounded, then the Ishikawa iteration sequence with errors converges to a fixed point of in .
A proof similar to that of Theorem 2.3 gives the following result and is thus omitted.
Let be a nonempty closed subset of a Banach space . Assume that is nonexpansive and there exists a compact subset of with . Suppose that (2.1) and (2.2) hold. Then has a fixed point in . Moreover for any , the Ishikawa iteration sequence with errors converges to a fixed point of .
Let be a nonempty closed subset of a Banach space and let be quasi-nonexpansive. Assume that (2.1) and (2.2) hold and satisfies Condition A. If is bounded, then for any , the Ishikawa iteration sequence with errors converges to a fixed point of in .
Let be a nonempty closed subset of a Banach space and let be quasi-nonexpansive. Assume that (2.32) holds and is in . If satisfies Condition B, then the Ishikawa iteration sequence converges to a fixed point of in .
3. Examples and Problems
The examples below show that Theorems 2.1–2.4 extend substantially Corollaries 2.5–2.10, respectively.
We conclude with the following problems.
Can Condition A in Theorem 2.3 be replaced by Condition B?
Can Theorem 2.4 be extended to the Ishikawa iteration method with errors?
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00042).
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