ISRN Mathematical Analysis

Volume 2011, Article ID 687184, 9 pages

http://dx.doi.org/10.5402/2011/687184

## Robustness of Krasnoselski-Mann's Algorithm for Asymptotically Nonexpansive Mappings

^{1}Department of Mathematics, Nanchang University, Nanchang 330031, China^{2}Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China

Received 22 February 2011; Accepted 11 April 2011

Academic Editors: V. Kravchenko and A. Peris

Copyright © 2011 Yu-Chao Tang and Li-Wei Liu. 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

Iterative approximation of fixed points of nonexpansive mapping is a very active theme in many aspects of mathematical and engineering areas, in particular, in image recovery and signal processing. Because the errors usually occur in few places, it is necessary to show that whether the iterative algorithm is robust or not. In the present work, we prove that Krasnoselski-Mann's algorithm is robust for asymptotically nonexpansive mapping in a Banach space setting. Our results generalize the corresponding results existing in the literature.

#### 1. Introduction

Many practical problems can be formulated as the fixed point problem of , where is a nonexpansive mapping. Iterative methods as a powerful tool are often used to approximate the fixed points of such mapping. It has been show that the methods used to find fixed points of nonexpansive mapping covered a widely applied mathematics problems, such as the convex feasibility problem [1–3] and the split feasibility problem [4–6]. It is recommended for interested reader to [7] for an extensive study on the theory about iterative fixed point theory.

Let be a real Banach space. is called a nonexpansive mapping if for any , . Krasnoselski-Mann's iteration method for finding fixed points of is defined by where is a sequence in .

In 2001, Combettes [8] considered a parallel projection method algorithm in signal synthesis problems in a real Hilbert space as follows: where , are positive weights such that , is the projection of a signal onto a closed convex subset of , and stands for the error made in computing the projection onto at each iteration . He firstly proved that the sequence generated by (1.2) converges weakly to a point in , where .

Kim and Xu [9] generalized the results of Combettes [8] from Hilbert spaces to uniformly convex Banach spaces and obtained its equivalent form as follows: where , , and is nonexpansive. They proved that the weak convergence of the (1.3) in a uniformly convex Banach space. More precisely, they proved that the following main theorems.

Theorem 1.1 (see [9]). *Assume that is a uniformly convex Banach space. Assume, in addition, that either has the Kadec-Klee property or satisfies Opial's property. Let be a nonexpansive mapping such that denotes the set of fixed points of , that is, ). Given an initial guess . Let be generated by (1.3) and satisfy the following properties: *(i)*,*(ii)*. **Then the sequence converges weakly to a fixed point of .*

Theorem 1.2 (see [9]). *Let be a nonempty closed convex subset of a Hilbert space and a nonexpansive mapping with . Given an initial guess . Let be generated by either
**
or
**
where the sequences and are such that *(i)*,*(ii)*.**Then converges weakly to a fixed point of .*

Very recently, Ceng et al. [10] extended the algorithm (1.3) of Kim and Xu [9] to Krasnoselski-Mann's algorithm with perturbed mapping defined by the following: where , and is a strongly accretive and strictly pseudocontractive mapping.

An important generalization of the class of nonexpansive mapping is asymptotically nonexpansive mapping (i.e., for , if there exists a sequence , such that for all and ), which was introduced by Goebel and Kirk [11]; they proved that if is a nonempty closed, convex, and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point. The class of asymptotically nonexpansive mapping has been studied by many authors and some recent results can be found in [12–17] and references cited therein.

Inspired and motivated by the above works, the purpose of this paper is to extend the results of Kim and Xu [9] from nonexpansive mapping to asymptotically nonexpansive mapping. We prove that the Krasnoselski-Mann iterative sequence converges weakly to the fixed point of asymptotically nonexpansive mapping.

#### 2. Preliminaries

In this section, we collect some useful results which will be used in the following section.

We use the following notations:(i) for weak convergence and for strong convergence,(ii) denotes the weak -limit set of .

It is well known that a Hilbert space satisfies Opial's condition [18]; that is, for each sequence in which converges weakly to a point , one has for all , .

Recall that given a closed convex subset of of a real Hilbert space , the nearest point projection from onto assigns to each its nearest point denoted by in from to ; that is, is the unique point in with the property

A Banach space is said to have the Kadec-Klee property [19] if for any sequence in , and imply that .

A mapping is said to be demiclosed at zero if whenever is a sequence in such that converges weakly to and converges strongly to zero, then .

Lemma 2.1 (see [20]). *Let be a real uniformly convex Banach space, let be a nonempty closed convex subset of , and let be an asymptotically nonexpansive mapping with a sequence and ; then is demiclosed at zero.*

Lemma 2.2 (see [21]). *Given a number , a real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , , such that
**
for all and such that and .*

Lemma 2.3 (see [22]). *Let be a real uniformly convex Banach space such that its dual has Kadec-Klee property. Let be a bounded sequence in and . Suppose that that
**
exists for all . Then .*

Lemma 2.4 (see [23]). *Let , , and be sequences of nonnegative real numbers satisfying the inequality
**
If , then (i) exists. (ii) In particular, if , one has .*

#### 3. Main Results

We state our first theorem as follows.

Theorem 3.1. *Suppose that is a uniformly convex Banach space, and has the Kadec-Klee property or satisfies Opial's property. Let be an asymptotically nonexpansive mapping with . For any , the sequence is generated by the following Krasnoselski-Mann's algorithm:
**
where and satisfy the following conditions: *(i)*, for some and for all ;*(ii)*.**If , then the sequence converges weakly to a fixed point of .*

For the sake of convenience, we need the following lemmas.

Lemma 3.2. *Let be a real normed linear space and let be an asymptotically nonexpansive mapping with . Let be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that ; then the limit exists for .*

*Proof. *By (3.1), one has
Since and , we obtain from Lemma 2.4 that the limit exists. Furthermore, the sequence is bounded.

Lemma 3.3. *Let be a real uniformly convex Banach space and let be an asymptotically nonexpansive mapping with . Let be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that ; then exists for all and .*

*Proof. *Let ; then exists. It follows from Lemma 3.2 that exists. Next, we show that exists for any .

Let , for all . For any , one has
Set , . The rest of the proof is the same as Lemma 3.3 of [14, 16]. This completes the proof of Lemma 3.3.

Now, we give the proof of Theorem 3.1.

*Proof. *Let . With the help of Lemma 2.2 and the inequality , one has
which follows that
This implies that
Therefore . Since is strictly increasing and continuous function with , then . Also, one has the following inequalities:
On the other hand,
By (3.7)–(3.9), we obtain
that is, .

From Lemma 3.2, we know that is bounded. Since is a uniformly convex Banach space, has a convergent subsequence . By the demiclosedness principle of , we obtain . The rest of proof is followed by the standard argument in Theorem 3.3 of Kim and Xu [9]. This completes the proof.

Theorem 3.4. *Let be a nonempty closed convex subset of a Hilbert space and let be an asymptotically nonexpansive mapping with . For any , the sequence is generated by either
**
or
**
where the sequences and are such that *(i)*, for some and for all ;*(ii)*. **If , then converges weakly to a fixed point of .*

*Proof. *Let . By (3.11), one has
Notice the condition (ii) and ; by Lemma 2.4, exists. Hence, is bounded.

By the well-known inequality , for all and , we obtain
That is,
This implies that
Therefore . We also have
It follows from (3.9) and (3.10) that . Since a Hilbert space must be a uniformly convex Banach space and satisfy Opial's property, then the rest of proof is the same as Theorem 3.1. So it is omitted.

#### Acknowledgment

This work was supported by The National Natural Science Foundations of China (60970149) and The Natural Science Foundations of Jiangxi Province (2009GZS0021, 2007GQS2063).

#### References

- Heinz H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,”
*SIAM Review*, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Butnariu, Y. Censor, P. Gurfil, and E. Hadar, “On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces,”
*SIAM Journal on Optimization*, vol. 19, no. 2, pp. 786–807, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Maruster and C. Popirlan, “On the Mann-type iteration and the convex feasibility problem,”
*Journal of Computational and Applied Mathematics*, vol. 212, no. 2, pp. 390–396, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,”
*Inverse Problems*, vol. 20, no. 1, pp. 103–120, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,”
*Inverse Problems*, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. K. Xu, “A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem,”
*Inverse Problems*, vol. 22, no. 6, pp. 2021–2034, 2006. View at Publisher · View at Google Scholar · View at Scopus - C. E. Chidume,
*Geometric Properties of Banach Spaces and Nonlinear Iterations*, vol. 1965 of*Lecture Notes in Mathematics*, Springer, London, UK, 2009. - P. L. Combettes, “On the numerical robustness of the parallel projection method in signal synthesis,”
*IEEE Signal Processing Letters*, vol. 8, no. 2, pp. 45–47, 2001. View at Publisher · View at Google Scholar · View at Scopus - T. H. Kim and H. K. Xu, “Robustness of Mann's algorithm for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 327, no. 2, pp. 1105–1115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. C. Ceng, Y. C. Liou, and J. C. Yao, “Robustness of Mann type algorithm with perturbed mapping for nonexpansive mappings in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 734181, 21 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
*Proceedings of the American Mathematical Society*, vol. 35, pp. 171–174, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Abbas, S. H. Khan, and B. E. Rhoades, “Simpler is also better in approximating fixed points,”
*Applied Mathematics and Computation*, vol. 205, no. 1, pp. 428–431, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Abbas and B. E. Rhoades, “A fixed point result for asymptotically nonexpansive mappings on an unbounded set,”
*Carpathian Journal of Mathematics*, vol. 25, no. 2, pp. 141–146, 2009. View at Google Scholar - C. E. Chidume and B. Ali, “Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 330, no. 1, pp. 377–387, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Hao, S. Y. Cho, and X. L. Qin, “Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 218573, 11 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M. O. Osilike, A. Udomene, D. I. Igbokwe, and B. G. Akuchu, “Demiclosedness principle and convergence theorems for $k$-strictly asymptotically pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1334–1345, 2007. View at Publisher · View at Google Scholar - H. K. Pathak, Y. J. Cho, and S. M. Kang, “Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 5, pp. 1929–1938, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 591–597, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Goebel and W. A. Kirk,
*Topics in Metric Fixed Point Theory*, vol. 28 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar - K. K. Tan and H. K. Xu, “Fixed point iteration processes for asymptotically nonexpansive mappings,”
*Proceedings of the American Mathematical Society*, vol. 122, no. 3, pp. 733–739, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Google Scholar · View at Scopus - J. G. Falset, W. Kaczor, T. Kuczumow, and S. Reich, “Weak convergence theorems for asymptotically nonexpansive mappings and semigroups,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 43, no. 3, pp. 377–401, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,”
*Mathematical and Computer Modelling*, vol. 32, no. 10, pp. 1181–1191, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH