# Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces

- Zeqing Liu
^{1}, - Jeong Sheok Ume
^{2}Email author and - Shin Min Kang
^{3}

**2009**:520976

https://doi.org/10.1155/2009/520976

© Zeqing Liu et al. 2009

**Received: **9 May 2009

**Accepted: **14 December 2009

**Published: **14 December 2009

## Abstract

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 [1] and Kirk [2] 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 [9] pointed out that the Picard iteration schemes for nonexpansive mappings need not converge. Senter and Dotson [10] 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 [7] established that the Mann iteration methods can be used to approximate fixed points of nonexpansive mappings in Banach spaces. Deng [3] 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 [3], Ishikawa [7], and Senter and Dotson [10].

Assume that is a nonempty subset of a normed linear space and denotes the family of all nonempty convex compact subsets of , and is the Hausdorff metric induced by the norm . For , , , , , and , let

It is easy to see that and for all and . Hence is convex. Hu and Huang [12] proved that if is a Banach space, then is a complete metric space. Now we introduce the following concepts in hyperspaces.

Definition 1.1.

Let be a nonempty subset of and let be a mapping. Assume that , , , and are arbitrary real sequences in satisfying and for and and are any bounded sequences of the elements in .

is called the Ishikawa iteration sequence with errors provided that .

is called the Ishikawa iteration sequence provided that .

is called the Mann iteration sequence with errors provided that .

is called the Mann iteration sequence provided that

Definition 1.2.

Let be a nonempty subset of . A mapping is said to be

(ii)quasi-nonexpansive if and for all and .

Definition 1.3.

Let be a nonempty subset of . A mapping with is said to be satisfy the following.

(i)Condition A if there is a continuous function with and for , such that for all .

(ii)Condition B if there is a nondecreasing function with and for , such that for all

Remark 1.4.

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 [8]. 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 [12]).

Let be a Banach space and a compact subset of . Then is compact, where stands for the closure of .

Lemma 1.6 (see [4]).

Suppose that , , and are three sequences of nonnegative numbers such that for all . If and converge, then exists.

Lemma 1.7 (see [14]).

Let be a metric space. Let and be compact subsets of . Then for any , there exists such that , where is the Hausdorff metric induced by .

Lemma 1.8.

Proof.

Thus (1.6) follows from (1.10) and (1.11). This completes the proof.

Lemma 1.9.

Let be a normed linear space and a nonempty closed subset of . If is quasi-nonexpansive, then is closed.

Proof.

## 2. Main Results

Our results are as follows.

Theorem 2.1.

If the Ishikawa iteration sequence with errors is bounded, then

Proof.

That is, (2.14) holds for . Hence (2.14) holds for all .

as . Thus (2.3) and (2.20) yield that , which is absurd. Hence . This completes the proof.

Theorem 2.2.

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 .

Proof.

for . It follows from Lemma 1.6, (2.2), (2.23), and (2.24) that exists. Using (2.21) we get that . This completes the proof.

Theorem 2.3.

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 .

Proof.

for . That is, converges to . This completes the proof.

A proof similar to that of Theorem 2.3 gives the following result and is thus omitted.

Theorem 2.4.

Then the Ishikawa iteration sequence converges to a fixed point of in .

Let be a nonempty subset of . It is easy to see that is isometric to . Thus Theorems 2.1–2.4 yield the following results.

Corollary 2.5.

Let be a nonempty subset of a normed linear space . Assume that is nonexpansive and . Suppose that (2.1) and (2.2) hold. If the Ishikawa iteration sequence with errors is bounded, then .

Remark 2.6.

Corollary 2.5 extends Theorem 1 in [3] and Lemma 2 in [7] from the Ishikawa iteration scheme and Mann iteration scheme into the Ishikawa iteration scheme with errors, respectively.

Corollary 2.7.

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 .

Remark 2.8.

Theorem 3 in [3] and Theorem 1 in [7] and [8] are special cases of Corollary 2.7.

Corollary 2.9.

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 .

Corollary 2.10.

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 .

Remark 2.11.

Corollary 2.10 extends, improves, and unifies Theorem 4 in [3], Theorem 2 in [7] and [8] in the following ways:

(i)the Mann iteration method in [7, 8], and Ishikawa iteration method in [3] are replaced by the more general Ishikawa iteration method with errors;

(ii)the nonexpansive mappings in [3, 7, 8] are replaced by the more general quasi-nonexpansive mappings.

## 3. Examples and Problems

Now we construct a few nontrivial examples to illustrate the results in Section 2. The following example reveals that Corollary 2.10 extends properly Theorem 4 in [3], Theorem 2 in [7] and [8].

Example 3.1.

The examples below show that Theorems 2.1–2.4 extend substantially Corollaries 2.5–2.10, respectively.

Example 3.2.

for . That is, the conditions of Theorems 2.1 and 2.2 are fulfilled. Hence we can invoke our Theorems 2.1 and 2.2 show that the Ishikawa iteration sequence with errors converges to and .

Example 3.3.

for . Therefore the conditions of Theorem 2.3 are fulfilled.

Example 3.4.

We conclude with the following problems.

Problem 3.5.

Can Condition A in Theorem 2.3 be replaced by Condition B?

Problem 3.6.

Can the boundedness of in Theorem 2.3 be removed?

Problem 3.7.

Can Theorem 2.4 be extended to the Ishikawa iteration method with errors?

## Declarations

### Acknowledgment

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00042).

## Authors’ Affiliations

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