Journal of Applied Mathematics

Volume 2013 (2013), Article ID 274931, 5 pages

http://dx.doi.org/10.1155/2013/274931

Research Article

Reconsiderations on the Equivalence of Convergence between Mann and Ishikawa Iterations for Asymptotically Pseudocontractive Mappings

Haizhen Sun and Zhiqun Xue

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

Received 28 December 2012; Accepted 11 June 2013

Academic Editor: D. R. Sahu

Copyright © 2013 Haizhen Sun 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.

Our aim in this paper is to illustrate that the proof of main theorem of Rhoades and Şoltuz (2003) concerning the equivalence between the convergences of Ishikawa and Mann iterations for uniformly -Lipschitzian asymptotically pseudocontractive maps is incorrect and to provide its correct version.

1. Introduction and Preliminary

In 2003, Rhoades and Şoltuz [1] proved the equivalence between convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map. This result provided significant improvements of recent some important results. Their result is as follows.

Theorem R-S (see [1, Theorem 8]). Let be a closed convex subset of an arbitrary Banach space and and defined by (3) and (4) with and satisfying (5). Let be an asymptotically pseudocontractive and Lipschitzian map with selfmap of . Let be the fixed point of . If , the following two assertions are equivalent:

(i)Mann type iteration (3) converges to ,(ii)Ishikawa iteration (4) converges to .

However, after careful reading of the paper of Rhoades and Şoltuz [1], we find that there exists a serious gap in the proof of Theorem 8 of [1], which happens to be main theorem of the paper. Note: in the proof of Theorem 8 of [1] the following mistakes occurred. “Using (6) with ” in line 19 of page 684 cannot obtain

The reason is that the following conditions are not equivalent:

(a1) is asymptotically pseudocontractive map,(a2), where and are from (1).

The aim of this paper is for us to provide its correct version. For this, we need the following definitions and lemmas.

Throughout this paper, suppose that is an arbitrary real Banach space and is a nonempty closed convex subset of . Let denote the normalized duality mapping from to defined by

where , , and denote the dual space of , the generalized duality pairing, and the single-valued normalized duality mapping, respectively.

Definition 1 (see [1]). Let be a mapping.

is called uniformly -Lipschitz if there is a constant such that, for all ,

is called asymptotically nonexpansive with a sequence and if for each such that

is called asymptotically pseudocontractive map with a sequence and if, for each , there exists such that

Obviously, an asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly -Lipschitz. Conversely, it is not true in general.

Definition 2 (see [2]). For arbitrary given , the sequences defined by

are called modified Mann and Ishikawa iterations, respectively, where , are two real sequences of and satisfy some conditions.

Lemma 3 (see [2]). Let be a real Banach space and be a normalized duality mapping. Then

for all and .

Lemma 4 (see [3]). Let be a strictly increasing and continuous function with , and let , , and be three nonnegative real sequences satisfying the following inequality:

where with , . Then as .

2. Main Results

Now we prove the following theorem which is the main result of this paper.

Theorem 5. Let be a real Banach space, be a nonempty closed convex subset of , and be a uniformly -Lipschitz asymptotically pseudocontractive mapping with a sequence such that . Let be two real numbers sequences in and satisfy the conditions (i)   as ; (ii)  . For some , let and be modified Mann and Ishikawa iterative sequences defined by (6) and (7), respectively. If , , and there exists a strictly increasing continuous function with such that

where , then the following two assertions are equivalent:

(1-1) the modified Mann iteration (6) converges strongly to the fixed point of  ;(1-2) the modified Ishikawa iteration (7) converges strongly to the fixed point of  .

Proof. We only need to prove (1-1) (1-2), that is, as as . Without loss of generality, . Since is a uniformly -Lipschitz, then .

Step 1. For any , is bounded.

Set , for all , , then :

And there exists and such that . Indeed, if as , then, ; if with , then, for , there exists a sequence such that as with . Hence there exists a natural number such that for , and then we redefine and .

Set , and then, from , we obtain that

Denote , . 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, and then . Now denote

Because as , without loss of generality, let for any . So we have

so

Using Lemma 3 and the above formula, we obtain

Since as , without loss of generality, let . Then (17) implies that

and this is a contradiction. Hence ; that is, is a bounded sequence.

Step 2. We show that as .

By Step 1, we obtain that is a bounded sequence, and denote . Applying (6), (7), and Lemma 3, we have

Observe that

where as .

Substituting (20) and (21) into (19), we obtain

Since as , without loss of generality, we may assume that

for any . Then, (22) implies that

Let , , . Then (24) leads to

By Lemma 4, we obtain . That is, as . From the inequality , we get as . This completes the proof.

Remark 6. The error in the proof of Theorem 8 of [1] has been pointed out and corrected, but it is not easy what the author really wants to obtain the proof of Theorem 8 in [1] at present.

Remark 7. The proof method of Theorem 5 is quite different from that of [1] and others.

Acknowledgments

This work was supported by Hebei Provincial Natural Science Foundation (Grant no. A2011210033). And authors thank the reviewers for good suggestions and valuable comments of the paper.

References

B. E. Rhoades and S. M. Şoltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map,” Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 681–688, 2003.

S. S. Zhang, “Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces,” Acta Mathematicae Applicatae Sinica, vol. 24, no. 2, pp. 236–241, 2001.

C. Moore and B. V. C. Nnoli, “Iterative solution of nonlinear equations involving set-valued uniformly accretive operators,” Computers & Mathematics with Applications, vol. 42, no. 1-2, pp. 131–140, 2001.