- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 207896, 9 pages

http://dx.doi.org/10.1155/2012/207896

## New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

^{1}College of Science, Agricultural University of Hebei, Baoding 071001, China^{2}North China Electric Power University, Baoding 071003, China

Received 19 September 2012; Accepted 8 October 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Yaqin Zheng and Jinwei Shi. 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 methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

#### 1. Introduction

Let be a nonempty closed convex subset of a real Banach space . A mapping is said to be nonexpansive, if for all .

Now we know that the involved operators in the many practical applications can be reduced to the nonexpansive mappings, that is, there are a large number of applied areas which are closely related to the nonexpansive mappings, for example, inverse problem, partial differential equations, image recovery, and signal processing. Based on these facts, recently, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations. For related works, please see [1–26] and the references therein.

In the present paper, we focus on a class of strictly pseudocontractive mappings which strictly includes the class of nonexpansive mappings. Recall that a mapping is said to be strictly pseudocontractive if there exists a constant and such that for all . We use to denote the set of fixed points of .

We know that the strict pseudocontractions have more powerful applications than nonexpansive mappings in solving inverse problems. There are some related references in the literature for strictly pseudocontractive mappings; see, for example, [27–30]. Motivated and inspired by the works in the literature, in the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

#### 2. Preliminaries

Let be the dual space of a Banach space . Let () be the generalized duality mapping from into given by where denotes the generalized duality pairing. In particular, is called the normalized duality mapping and it is usually denoted by . It is well known that is a uniformly smooth Banach space if and only if is single valued and uniformly continuous on any bounded subset of .

Let be a nonempty closed convex subset of a Banach space , and let be a nonempty subset of . Recall that a mapping is called a retraction from onto provided for all . A retraction is sunny provided for all and whenever . A sunny nonexpansive retraction is a sunny retraction which is also nonexpansive. Sunny non-expansive retractions are characterized as follows.

Lemma 2.1. *If is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality
**
for all and . *

Lemma 2.2 (see [31]). *Let be a real -uniformly smooth Banach space, and let . Then, one has
**
for all . *

Lemma 2.3 (see [16]). *Let and be two bounded sequences in Banach spaces, and let be a sequence in with . Suppose that for all and . Then . *

Lemma 2.4 (see [14]). * Let be a nonempty closed convex subset of a real -uniformly smooth and uniformly convex Banach space . Let be a strictly pseudocontractive mapping. Then is demiclosed. *

Lemma 2.5 (see [32]). *Assume is a sequence of nonnegative real numbers such that for where is a sequence in and is a sequence in such that *(i)*; *(ii)* or . ** Then . *

#### 3. Main Results

In this section, we will give our main results. In the sequel, we assume the following: (C1) is a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping from to ; (C2) is a nonempty closed convex subset of ; (C3) is a sunny nonexpansive retraction from onto ; (C4) is a -strict pseudocontraction; (C5) is a -contraction; (C6) is strongly positive (i.e., for some ) and linear bounded operator with for all and ; (C7).

First, we consider the following *VI*: finding such that
The set of solutions of (3.1) is denoted by . In the sequel, we assume that . Note that (3.1) has the unique solution.

Next, we propose our algorithm.

*Algorithm 3.1. *For the initial point , we generate a sequence via the following manner:
where is a sequence in and .

Theorem 3.2. *If the sequence satisfies and , then the sequence generated by (3.2) converges strongly to the unique solution of VI (3.1). *

*Proof. *First, by using Lemma 2.2, we know that is nonexpansive. is sunny nonexpansive. Thus, is nonexpansive. Let . From (3.2), we have
Thus,
This indicates that is bounded.

We write for all . So,
Hence,
It follows that
where is a constant satisfying . This implies that
By Lemma 2.3, we deduce
Therefore,
Note that . As a matter of fact, if , that is , then . Since is a self-mapping, it is clear that . So, . Therefore, . Conversely, if , that is , we also have . Thus, . Set . We observe that . Next, we estimate .

Since
we have

Next, we show
where .

First, we have
Since the sequence is bounded, hence is bounded. Thus, we can take a subsequence of such that weakly. Without loss of generality, we may assume that weakly. Note that is nonexpansive and . By using the demiclosed principle of nonexpansive mappings (see Lemma 2.4), we get . At the same time, is weakly sequentially continuous. Therefore,
Finally we show that . From (3.2), we have
It follows that
It can be checked easily that and . From Lemma 2.5, we deduce . This completes the proof.

*Algorithm 3.3. *For the initial point , we generate a sequence via the following manner:
where is a sequence in and .

Corollary 3.4. *If the sequence satisfies and , then the sequence generated by (3.18) converges strongly to the unique solution of VI: finding such that
*

*Algorithm 3.5. *For the initial point and , we generate a sequence via the following manner:
where is a sequence in and .

Corollary 3.6. *If the sequence satisfies and , then the sequence generated by (3.20) converges strongly to the unique solution of VI: finding such that
*

*Algorithm 3.7. *For the initial point , we generate a sequence via the following manner:
where is a sequence in and .

Corollary 3.8. *If the sequence satisfies and , then the sequence generated by (3.22) converges strongly to . *

#### Acknowledgments

J. Shi was supported in part by the scientific research fund of the Educational Commission of Hebei Province of China (no. 936101101) and the National Natural Science Foundation of China (no. 51077053).

#### References

- F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”
*Journal of Mathematical Analysis and Applications*, vol. 20, pp. 197–228, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Halpern, “Fixed points of nonexpanding maps,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 957–961, 1967. 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 - P.-L. Lions, “Approximation de points fixes de contractions,”
*Comptes Rendus de l'Académie des Sciences*, vol. 284, no. 21, pp. A1357–A1359, 1977. View at Zentralblatt MATH - R. Wittmann, “Approximation of fixed points of nonexpansive mappings,”
*Archiv der Mathematik*, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. E. Rhoades, “Fixed point iterations using infinite matrices,”
*Transactions of the American Mathematical Society*, vol. 196, pp. 161–176, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. E. Rhoades, “Comments on two fixed point iteration methods,”
*Journal of Mathematical Analysis and Applications*, vol. 56, no. 3, pp. 741–750, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,”
*Nonlinear Analysis A*, vol. 67, no. 5, pp. 1464–1473, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces,”
*Computers & Mathematics with Applications*, vol. 58, no. 3, pp. 605–617, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-P. Chancelier, “Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 353, no. 1, pp. 141–153, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 2, pp. 1402–1416, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. E. Chidume and C. O. Chidume, “Iterative approximation of fixed points of nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 288–295, 2006. 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. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,”
*Taiwanese Journal of Mathematics*, vol. 5, no. 2, pp. 387–404, 2001. View at Zentralblatt MATH - T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,”
*Fixed Point Theory and Applications*, no. 1, pp. 103–123, 2005. View at Zentralblatt MATH - W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”
*Journal of Optimization Theory and Applications*, vol. 118, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Moudafi, “Viscosity approximation methods for fixed-points problems,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 46–55, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 469–479, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Lopez, V. Martin, and H.-K. Xu, “Perturbation techniques for nonexpansive mappings with applications,”
*Nonlinear Analysis*, vol. 10, no. 4, pp. 2369–2383, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,”
*Nonlinear Analysis A*, vol. 68, no. 6, pp. 1687–1693, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm for approximating convex minimization problem,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 648–656, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, Y.-C. Liou, and J.-C. Yao, “An extragradient method for fixed point problems and variational inequality problems,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 38752, 12 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, M. Aslam Noor, K. Inayat Noor, Y.-C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,”
*Acta Applicandae Mathematicae*, vol. 110, no. 3, pp. 1211–1224, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yao, M. A. Noor, and Y.-C. Liou, “Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
*Nonlinear Analysis A*, vol. 67, no. 7, pp. 2258–2271, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 1, pp. 336–346, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 294, no. 1, pp. 73–81, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-C. Zeng, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 4, pp. 837–849, 2006. View at Zentralblatt MATH - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis A*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH