Journal of Applied Mathematics

Volume 2012 (2012), Article ID 395098, 11 pages

doi:10.1155/2012/395098

Research Article

The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ-Pseudocontractive Mappings in Normed Linear Spaces

Guiwen Lv and Zhiqun Xue

Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received 16 February 2012; Accepted 31 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Guiwen Lv and Zhiqun Xue. 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.

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

1. Introduction

Let be a real normed linear space, its dual space, and the normalized duality mapping defined by

where denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by .

Definition 1.1. A mapping is said to be

(1)strongly accretive if, for all , there exist a constant and such that

(2)strongly accretive if there exist and a strictly increasing function with such that

(3)generalized accretive if, for all , there exist and a strictly increasing function with such that

Definition 1.2. Let . The mapping is called strongly quasi-accretive if, for all , there exist a constant and such that ; is called strongly quasi-accretive if, for all , there exists a function such that , where is as in Definition 1.1. Finally, is called generalized quasi-accretive if, for each , there exist and a strictly increasing function with such that .

Closely related to the class of accretive-type mappings are those of pseudocontractive types.

Definition 1.3. A mapping with domain and range is said to be

(1)strongly pseudocontractive if there exist a constant and such that, for each ,

(2)strongly pseudocontractive if there exist and a strictly increasing function with such that

(3)generalized pseudocontractive if, for all , there exist and a strictly increasing function with such that

Definition 1.4. Let . The mapping is called generalized -hemi-pseudocontractive if, for all , there exist and a strictly increasing function with such that

Obviously, a mapping is strongly pseudocontractive, strongly pseudocontractive, generalized -pseudocontractive, and generalized -hemicontractive if and only if is strongly accretive, strongly accretive, generalized accretive, and generalized quasi-accretive, respectively.

It is shown in [1] that the class of strongly pseudocontractive mappings is a proper subclass of -strongly pseudocontractive mappings. Furthermore, an example in [2] shows that the class of -strongly pseudocontractive mappings with the nonempty fixed point set is a proper subclass of -hemicontractive mappings. Hence, the class of generalized -hemicontractive mappings is the most general among those defined above.

Definition 1.5. The mapping is called Lipschitz if exists a constant such that

It is clear that if is Lipschitz then it must be uniformly continuous. Otherwise, it is not true.

The following iteration schemes were introduced by Xu [3] in 1998. Let be a nonempty convex subset of . For any given , the sequence is defined by

is called the Ishikawa iteration sequence with errors, where are arbitrary bounded sequences in and , and are six real sequences in such that for all and satisfy certain conditions.

If , for all , then, from (1.10), we get the Mann iteration sequence with errors defined by

where is an arbitrary bounded sequence in .

Numerous convergence results have been proved through iterative methods of approximating fixed points of Lipschitz pseudocontractive- (accretive-) type nonlinear mappings [3–10]. Most of these results have been extended to uniformly continuous mappings by some authors. Recently, C. E. Chidume and C. O. Chidume in [11] gave the most general result for uniformly continuous generalized -hemicontractive mappings in normed linear. Their results are as follows.

Theorem 1.6 (see [11, Theorem 2.3]). Let be a real normed linear space, nonempty subset of , and a uniformly continuous generalized hemicontractive mapping, that is, there exist and a strictly increasing function , such that, for all , there exists such that

(a)If is a fixed point of , then and so has at most one fixed point in .(b)Suppose there exists such that the sequence defined by

is contained in , where , and are real sequences satisfying the following conditions:

(i), (ii), (iii), (iv).

Then, converges strongly to . In particular, if is a fixed point of in , then converges strongly to .

Unfortunately, the control conditions (iii) and (iv) cannot assure that the result in [11] holds. In the proof course of p552, let be any given, we can choose (also in view of conditions (iii) and (iv)) an integer such that for all the following inequality holds, where . The inequality above implies that . But conditions (iii) and (iv) of [11] can not assure that . On the one hand, let ; then , but . On the other hand, set ; then , but .

The purpose of this paper is that we obtain the convergence result of the Mann iteration with errors, and we also prove the equivalence of convergence between the Ishikawa iteration with errors defined by (1.10) and the Mann iteration with errors defined by (1.11). We also show that the Ishikawa iteration with errors defined by (1.10) converges to the unique fixed point of . Our results extend and improve the corresponding results of [3–12]. For this, in the sequel, we will need the following lemmas.

Lemma 1.7 (see [13]). Let be a real normed space. Then, for all , the following inequality holds:

Lemma 1.8 (see [14]). Let be a strictly increasing continuous function with and and nonnegative three sequences that satisfy the following inequality:

where , and . Then as .

2. Main Results

Theorem 2.1. Let be a nonempty closed convex subset of a real normed linear space . Suppose that is a uniformly continuous generalized hemicontractive mapping with . Let be a sequence in defined iteratively from some by (1.11), where is an arbitrary bounded sequence in and and are three sequences in satisfying the following conditions:

(i), (ii), (iii), (iv).

Then, the iteration sequence converges strongly to the unique fixed point of .

Proof. Let . The uniqueness of the fixed point of comes from Definition 1.4.

First, we prove that there exists with such that . In fact, if , then we are done. Otherwise, there exists the smallest positive integer such that . We denote , and then we obtain that . Indeed, if , then . If with , then, for , there exists a sequence such that as with , and we also obtain that the sequence is bounded. So there exists such that for , and then we redefine . Let .

Next for we will prove by induction. Clearly, holds. Suppose that , for some ; then we want to prove . If it is not the case, then . Since is a uniformly continuous mapping, setting , there exists such that whenever and is a bounded operator. Set . Since , without loss of generality, let

From (1.10), we have

Applying Lemma 1.7, the recursion (1.11), and the inequalities above, we obtain

which is a contraction with the assumption . Then, , that is, the sequence is bounded. It leads to

Again using Lemma 1.7, we have

where

Therefore, (2.5) becomes

From Lemma 1.8, we obtain .

Theorem 2.2. Let be a nonempty closed convex subset of a real normed linear space and a uniformly continuous generalized pseudocontractive mapping with . Let be two sequences in defined iteratively from some by (1.11) and (1.10), where are three arbitrary bounded sequences in and , and are six sequences in satisfying the following conditions:

(i), (ii), (iii), (iv).

Then, the following two assertions are equivalent:

(1)Iteration (1.11) converges strongly to the unique fixed point of ,(2)Iteration (1.10) converges strongly to the unique fixed point of .

Proof. Let . The uniqueness of comes from Definition 1.4. If the Ishikawa iteration with errors converges to , then setting , for all in (1.10), we can get the convergence of the Mann iteration with errors. Conversely, we only prove that (1)(2), that is, if the Mann iteration with errors converges to , we want to prove the convergence of the Ishikawa iteration with errors.

First, we will prove that there exists with such that .

In fact, if , then we are done. Otherwise, there exists the smallest positive integer such that . We denote , and then we obtain that . Indeed if , then . If with , then for there exists a sequence such that as with , and we also obtain that the sequence is bounded. So there exists such that , and then we redefine . Let .

Second, we will prove that the sequence is a bounded sequence.

Set

Since is uniformly continuous, for , there exists such that whenever . Now let . By the control conditions (iii) and (iv), without loss of generality, set , for all.

Observe that if , we obtain . Indeed

Next, by induction, we prove . Clearly, from (2.1), we obtain , that is, . Set , for some ; then we will prove that . If it is not the case, we assume that . From (1.10), we obtain the following inequalities:

Since is a uniformly continuous mapping, then

Since , we let

From (2.9)–(2.12), we have

which is a contradiction with the assumption . Hence, , that is, the sequence is bounded. From (2.9), the sequence is also bounded. So we obtain

Since is a uniformly continuous mapping, then

Using the recursion formula (1.10), we compute as follows:

where

Since the sequence is bounded, from (2.15), (2.16), and the control condition (iv), we get . Then, in (2.18), set , and . From Lemma 1.8, we obtain . Since , using inequality , then , and we complete the proof of Theorem 2.1.

From Theorems 2.1 and 2.2, we obtain the following corollary.

Corollary 2.3. Let be a nonempty convex subset of a real normed linear space . is a uniformly continuous generalized hemicontractive mapping with . Let be a sequence in defined iteratively from some by (1.10), where are arbitrary bounded sequences in and , and are sequences in satisfying the following conditions:

(i), (ii), (iii), (iv).

Then, the iteration sequence defined by (1.10) converges strongly to the unique fixed point of .

Corollary 2.4. Let be a uniformly continuous generalized accretive mapping with . For some , the iteration sequences in are defined as follows:

where for all . , and are arbitrary bounded sequences in and , and are sequences in satisfying the following conditions:

(i), (ii), (iii), (iv).

Then, the following two assertions are equivalent:

(1) converges strongly to the unique fixed point of ,(2) converges strongly to the unique fixed point of .

Proof. Let , and observe that is a uniformly continuous generalized accretive mapping if and only if is a uniformly continuous generalized pseudocontractive mapping. The result follows from Theorem 2.2.

Remark 2.5. Our results improve and extend the corresponding results of [11] in the following sense.

(i)We point out the gaps of C. E. Chidume and C. O. Chidume [11] in their proof and provide a counterexample.(ii)In the proof method, Theorem 2.1 differs from Theorem  2.3 in [11].(iii)The control conditions and in Theorem  2.3 in [11] are replaced by the condition . Under the new condition, we obtain the convergence theorem of the Mann iteration sequence with errors.(iv)We also obtain the equivalence of convergence results between the Ishikawa and Mann iterations with errors.(v)The Mann iteration sequence with errors is extended to the Ishikawa iteration sequence with errors.

Acknowledgments

This paper is supported by Hebei Province Natural Science Foundation (A2011210033) and Shijiazhuang Tiedao University Foundation (Q64).

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