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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 386983, 9 pages
On the Convergence of Multistep Iteration for Uniformly Continuous -Hemicontractive Mappings
1Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2Hajvery University, 43-52 Industrial Area, Gulberg III, Lahore 54660, Pakistan
3Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
Received 18 July 2012; Revised 25 August 2012; Accepted 28 August 2012
Academic Editor: Xiaolong Qin
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.
It is shown that the convergence of the multistep iterative process with errors is obtained for uniformly continuous -hemicontractive mappings in real Banach spaces. We also revise the problems of C. E. Chidume and C. O. Chidume (2005).
Let be a real Banach space with norm and let be its dual space. The normalized duality mapping is defined by where denotes the generalized duality pairing. The single-valued-normalized duality mapping is denoted by .
A mapping with domain and range in is said to be strongly pseudocontractive if there is a constant , and for all , such that The mapping is called -pseudocontractive if there exists a strictly increasing continuous function with such that holds for all . It is well known that the strongly pseudocontractive mapping must be the -pseudocontractive mapping in the special case in which , but the converse is not true in general. That is, the class of strongly pseudocontractive mappings is a proper subclass of the class of -pseudocontractive mappings. Let . If the inequalities (1.2) and (1.3) hold for any and , then the corresponding mapping is called strongly hemicontractive and -hemicontractive, respectively.
Let . An operator is called strongly quasiaccretive, -quasiaccretive if and only if is strongly hemicontractive, -hemicontractive, respectively, where denotes the identity mapping on . That is, if is -quasi-accretive, then and there exists a strictly increasing continuous function with such that holds for all and . Many authors have studied extensively the strongly convergence problems of the iterative algorithms for the class of operators.
In 2004, Rhoades and Soltuz  introduced the multistep iteration as follows.
Let be a nonempty closed convex subset of real Banach space and let be a mapping. The multistep iteration is defined by where in satisfy certain conditions. Obviously, the iteration defined above is generalization of Mann, Ishikawa, and Noor iterations.
Inspired and motivated by the work of Xu  and the iteration above, we discuss the following multistep iteration with errors: where in with , are the bounded sequences of .
In 2005, C. E. Chidume and C. O. Chidume  proved the convergence theorems for fixed points of uniformly continuous generalized -hemicontractive mappings and published in . However, there exists a gap in the proof course of their theorems.
The aim of this paper is to show the convergence of the multistep iteration with errors for fixed points of uniformly continuous -hemicontractive mappings and revise the results of C. E. Chidume and C. O. Chidume . For this, we need the following Lemmas.
Lemma 1.1 (see ). Let be a real Banach space and let be a normalized duality mapping. Then for all and for all .
Lemma 1.2 (see ). Let , and be three nonnegative real sequences and let be a strictly increasing and continuous function with satisfying the following inequality: where with , . Then as .
2. Main Results
Theorem 2.1. Let be an arbitrary real Banach space, a nonempty closed convex subset of , and a uniformly continuous -hemicontractive mapping with . Let be real sequences in and satisfy the conditions:(i);(ii) as ;(iii), .
Proof. Since is -hemicontractive mapping, then there exists a strictly increasing continuous function with such that
for , that is
Choose some and such that and denote that , is the range of . Indeed, if as , then ; if with (here, we only give a example. If , then ), then for , there exists a sequence in such that as with . Furthermore, we obtain that as . So is the bounded sequence. Hence, there exists natural number such that for , then we redefine and . This is to ensure that is defined well.
Step 1. We show that is a bounded sequence.
Set , then from above formula , we obtain that . Denote Since is the uniformly continuous, so is a bounded mapping. We let Next, we want to prove that . If , then . Now, assume that it holds for some , that is, . We prove that . Suppose that it is not the case, then . Since is uniformly continuous, then for , there exists such that when . Denote Since as for . Without loss of generality, we assume that for any . Since , let . Now, estimate for . By using (1.6), we have then . Similarly, we have then . We have then . Therefore, we get And we have Further, by using uniform continuity of , we have In view of Lemma 1.1 and the above formulas, we obtain which is a contradiction. Hence, , that is, is a bounded sequence; it leads to that are all bounded sequences as well.
Step 2. We want to prove as .
Since as and are bounded. From (2.9), we obtain By (2.11), we have where . By Lemma 1.2, we obtain that .
Theorem 2.2. Let be an arbitrary real Banach space and let be a uniformly continuous -quasi-accretive operator with . Let be real sequences in and satisfy the conditions:(i);(ii) as ;(iii), . For some , let be any bounded sequences of , and let be the multistep iterative sequence with errors defined by where is defined by for all . Then (2.14) converges strongly to the fixed point of .
Corollary 2.4. Let be an arbitrary real Banach space, a nonempty closed convex subset of , and a uniformly continuous -hemicontractive mapping with . Let be real sequences in and satisfy the conditions (i) ; (ii) as ; (iii) and . For some , let be any bounded sequence of , and let be Mann iterative sequence with errors defined by . Then converges strongly to the fixed point of .
Corollary 2.5. Let be an arbitrary real Banach space and let be a uniformly continuous -quasi-accretive operator with . Let be real sequences in and satisfy the conditions (i); (ii) as ; (iii) and . For some , let be any bounded sequence of , and let be Mann iterative sequence with errors defined by . where is defined by for all . Then converges strongly to the fixed point of .
Remark 2.6. It is mentioned to notice that there exists a serious shortcoming in the proof process of Theorem 2.3 of . That is, does not hold in line 15 of Claim 2 of page 552. The reason is that the conditions and , as can not obtain .
Counterexample, let the iteration parameters be in the following: Then, , , but .
Application 1. Let be a real number space with the usual norm and . Define by for all . Then is uniformly continuous with . Define by then is a strictly increasing function with . For all , we obtain that Therefore, is a -hemicontractive mapping. Set for all .
This work is supported by Hebei Natural Science Foundation under Grant no. A2011210033.
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