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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 909187, 15 pages
http://dx.doi.org/10.1155/2012/909187
Research Article

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

1Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 15 August 2012; Accepted 27 August 2012

Academic Editor: Yongfu Su

Copyright © 2012 Zhiqun Xue et al. 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 an arbitrary uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping with . Let be four real sequences in and satisfy the conditions: (i) , ; (ii) as and ; (iii) . For some , let be any bounded sequences in , and let be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of is equivalent to that of .

1. Introduction and Preliminary

Let be a real Banach space and let be its dual space. The normalized duality mapping is defined by where denotes the generalized duality pairing. It is well known that(i)if is a smooth Banach space, then the mapping is single-valued;(ii) for all and ;(iii)if is a uniformly smooth Banach space, then the mapping is uniformly continuous on any bounded subset of . Throughout this paper, we denote that is the single-valued normalized duality mapping, is a nonempty closed convex subset of , is a mapping, and is the unit mapping .In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings as follows.

Definition 1.1. A mapping is said to be asymptotically nonexpansive if for each where with .
Schu [2], in 1991, gave the definition of asymptotically pseudocontractive mappings and proved the correlation results.

Definition 1.2. The mapping is called asymptotically pseudocontractive with the sequence if and only if , and for all and all , there exists such that
It is easy to find that every asymptotically nonexpansive mapping is asymptotically pseudocontractive. However, the converse is not true in general. See example of [3].
Recently, Colao [4] combined the proof ideas of the papers of Chang [5] and C. E. Chidume and C. O. Chidume [6] and then showed the equivalent theorem results of the convergence between Mann and Ishikawa iterations with errors for generalized strongly asymptotically -pseudocontractive mapping with bounded range. In fact, he proved the following theorem.

Theorem 1.3. Let be a uniformly smooth Banach space, and let be generalized strongly asymptotically -pseudocontractive mapping with fixed point and bounded range. Let and be the sequences defined by (1.4) and (1.5), respectively, where satisfy(H1)  and ,(H2) ,and the sequences are bounded in , then for any initial point , the following two assertions are equivalent.(1)The modified Ishikawa iteration sequence with errors (1.4) converges to ;(2)The modified Mann iteration sequence with errors (1.5) converges to .

The aim of this paper is to prove the equivalence of convergent results of above Ishikawa and Mann iterations with errors for generalized asymptotically -strongly pseudocontractive mappings without bounded range assumptions in uniformly smooth real Banach spaces. For this, we need the following concepts and lemmas.

Definition 1.4 (see [4]). The mapping is called generalized asymptotically -strongly pseudocontractive if where , is converging to one and is strictly increasing continuous function with .

Definition 1.5 (see [4]). For arbitrary given , modified Ishikawa iterative process with errors defined by where are any bounded sequences in ; are four real sequences in and satisfy , for all . If , we define modified Mann iterative process with errors by where is any bounded sequence in .

Lemma 1.6 (see [7]). Let be a uniformly smooth real Banach space and let be a normalized duality mapping. Then for all .

Lemma 1.7 (see [8]). Let be a nonnegative sequence which satisfies the following inequality: where with , . Then as .

2. Main Results

First of all, we give a new concept.

Definition 2.1. A mapping is called uniformly generalized Lipschitz if there exists a constant such that It is mentioned to notice that if has bounded range, then it is uniformly generalized Lipschitz. In fact, since , then , thus , where . On the contrary, it is not true in general (See [6]).

In the following, we prove the main theorems of this paper.

Theorem 2.2. Let be an arbitrary uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping with . Let be four real sequences in and satisfy the following conditions:(i), ;(ii) as and ;(iii). For some , let be any bounded sequences in , and let and be Ishikawa and Mann iterative sequences with errors defined by (1.7) and (1.8), respectively. Then the following conclusions are equivalent:(1) converges strongly to the unique fixed point of ;(2) converges strongly to the unique fixed point of .

Proof. (1)(2) is obvious, that is, let , (1.7) turns into (1.8). We only need to show that (2)(1). Since is a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping, then there exists a strictly increasing continuous function with such that that is, for any . For convenience, denote .
Step 1. There exists and such that (range of ).
Indeed, if as , then ; if with , then, for , there exists a sequence in such that as with . Furthermore, there exists a natural number such that for , then we redefine such that . Hence, it is to ensure that is well defined.
Step 2. For any , is a bounded sequence.
Set . From (2.3), we have that is, . Thus, we obtain that . Denote Next, we want to prove that for any by induction. If , then . Now we assume that it holds for some , that is, . We prove that . Suppose that it is not the case, then . Since is uniformly continuous on bounded subset of , then, for , there exists such that when . Now denote Since as , and , without loss of generality, we assume that for any . Then we obtain the following estimates: Hence, ; .
Using Lemma 1.6 and formulas above, we obtain Substitute (2.10) into (2.9) this is a contradiction. Thus , that is, is a bounded sequence. So are all bounded sequences. Since as , without loss of generality, we let . Therefore, is also bounded.
Step 3. We want to prove as .
Set .
Again using Lemma 1.6, we have where , , and as .
Taking place (2.13) into (2.12), we have where as .
Set , then . If it is not the case, we assume that . Let , then , that is, . Thus, from (2.14) that which implies that Let . Then we get that Applying Lemma 1.7, we get that as . This is a contradiction and so . Therefore, there exists an infinite subsequence such that as . Since , then as . In view of the strictly increasing and continuity of , we have as . From (1.7), we have as . Next we want to prove as . Let , there exists such that , , , for any . First, we want to prove . Suppose it is not this case, then . Using (1.7), we may get the following estimates: Since is strictly increasing, then (2.20) leads to . From (2.14), we have is a contradiction. Hence, . Suppose that holds. Repeating the above course, we can easily prove that holds. Therefore, for any and , we obtain that , which means as . This completes the proof.

In order to make the existence of Theorem 2.2 more meaningful, we give the following theorem.

Theorem 2.3. Let be an arbitrary uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping with . Let be two real sequences in and satisfy the conditions (i) ; (ii) as and ; (iii) . For some , let be any bounded sequence in and let be modified Mann iterative sequence with errors defined by (1.8). Then converges strongly to the unique fixed point of .

Proof. Since is a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping, then there exists a strictly increasing continuous function with such that for any .
Step 1. There exists and such that , where . In fact, if as , then ; if with , then, for , there exists a sequence in such that as with . Furthermore, there exists a natural number such that for , then we redefine such that .
Step 2. For any , is bounded.
Set , we have . Let . Next, we prove that for any by induction. First is obvious. Suppose that holds. We prove that . If it is not the case, then . By uniformly continuity of on bounded subset, we choose , there exists such that when . Now denote Since as , and , without loss of generality, let for any . Then we have the following estimates from (1.8): Therefore, .
Using Lemma 1.6 and formulas above, we obtain this is a contradiction. Thus , that is, is a bounded sequence, so is also bounded. Denote .
Step 3. We prove as .
Again using Lemma 1.6, we have where and as .
Set , then . If it is not the case, we assume that . Let , then , that is, . Thus, from (2.14) that which implies that Let . Then we get that Applying Lemma 1.7, we get that as . This is a contradiction and so . Therefore, there exists an infinite subsequence such that as . Since , then as . In view of the strictly increasing and continuity of , we have as . Let   be any given, there exists such that , , , for any . First, we want to prove . Suppose it is not this case, then . Using (1.8), we may get the following estimates: Since is strictly increasing, then (2.32) leads to . From (2.27), we have is a contradiction. Hence, . Suppose that holds. Repeating the above course, we can easily prove that holds. Therefore, for any and , we obtain that , which means as . This completes the proof.

Theorem 2.4. Let be an arbitrary uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping with . Let be four real sequences in and satisfy the conditions (i) , ; (ii) as and ; (iii) . For some , let be two arbitrary bounded sequences in , and let be Ishikawa iterative sequence with errors defined by (1.7). Then (1.7) converges strongly to the unique fixed point of .

Proof. By Theorems 2.3 and 2.2, we obtain directly the result of Theorem 2.4.

Remark 2.5. Our Theorem 2.2 extends and improves Theorem 3.1 of [4] from the bounded range of to uniformly generalized Lipschitz mapping, and the proof course of Theorem 2.2 is quite different from that of [4].

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