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

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 457024, 11 pages

http://dx.doi.org/10.1155/2012/457024

## Strong Convergence of the Iterative Methods for Hierarchical Fixed Point Problems of an Infinite Family of Strictly Nonself Pseudocontractions

^{1}Tongji Zhejiang College, Zhejiang 314000, China^{2}Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China

Received 20 August 2012; Accepted 11 September 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Wei Xu and Yuanheng Wang. 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

This paper deals with a new iterative algorithm for solving hierarchical fixed point problems of an infinite family of pseudocontractions in Hilbert spaces by , , and , where is a nonself -strictly pseudocontraction. Under certain approximate conditions, the sequence converges strongly to , which solves some variational inequality. The results here improve and extend some recent results.

#### 1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of . A mapping is called a contraction with coefficient if there exits a constant such that

A mapping is called nonexpansive if

A mapping is called -strictly pseudocontraction if there exits a constant such that

Write as the set of fixed points of , that is, . In 2000, Moudafi [1] introduced an iterative scheme for nonexpansive mappings where be a contraction on and the sequence started with arbitrary initial . In 2004, Xu [2] proved that the sequence generated by (1.4) converges strongly to a fixed point of under certain conditions on the parameters, which also solves the variational inequality

Recently, some authors studied the problems of fixed points of nonexpansive mappings with strongly positive operators, Lipschitizian, strongly monotone operators, and extragradient methods, and many convergence results were obtained (such as, see [3–9]).

In 2008, Yao et al. [10] introduced the following iterative scheme: where is a contraction on and is nonexpansive mapping. In 2012, Song et al. [11] analyzed the following iterative algorithm: where is a -strictly pseudocontraction, is a lipschitzian and strongly monotone operator, is a contraction, and is the metric projection from onto . Under certain conditions on the parameters, the sequence generated by (1.7) converges strongly to a fixed point of a countable family of -strictly pseudocontraction, which is the solution of some variational inequality.

On the other hand, in 2010, Yao et al. [12] introduced the iterative algorithm for solving hierarchical fixed point of nonexpansive mappings and gave the following theorem.

* Theorem YCL*

Let be a nonempty closed convex subset of a real Hilbert space . Let be a contraction with coefficient . Suppose the following conditions are satisfied:(i) and ;(ii) ;(iii) and .Then the sequence generated by
converges strongly to a point of , which is the unique solution of the variational inequality

Motivated and inspired by the iterative schemes (1.7) and (1.8), we introduce and study the hybrid iterative algorithm for solving some hierarchical fixed point problem of infinite family of strictly nonself pseudocontractions: where , , and are the same in (1.8), is a nonself -strictly pseudocontraction. Under certain conditions on the parameters, the sequence generated by (1.10) converges strongly to a common fixed point of infinite family of -strictly pseudocontractions, which solves the variational inequality So, our results extend and improve some results of other authors (such as [10–12]) from self-mappings to nonself-mappings, from nonexpansive mappings to -strictly pseudocontraction, and from one mapping to a infinite family mappings.

#### 2. Preliminaries

In this section, we recall some basic facts that will be needed in the proof of the main results.

Lemma 2.1 (see [13] demiclosedness principle). *Let be a nonempty closed convex subset of a real Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then ; in particular if , then .*

Lemma 2.2 (see [9]). *Let and be any points. The following results hold:*(1) *that if and only if there holds the relation:
*(2) *that if and only if there holds the relation:
*(3) *there holds the ration:
*

Lemma 2.3 (see [14]). *For all , the following inequality holds:
*

Lemma 2.4 (see [3]). *Let be a contraction with coefficient and let be a nonexpansive mapping. Then for all :*(1) *the mapping is strongly monotone with coefficient , that is,
*(2) *the mapping is monotone:
*

Lemma 2.5 (see [15]). *Let be a Hilbert space and let be a nonempty convex subset of . Let be a -strictly pseudocontractive mapping with . Then .*

Lemma 2.6 (see [16]). *Let be a Hilbert space and let be a nonempty convex subset of . Let be a -strictly pseudocontractive mapping. Define a mapping for all . Then as , is a nonexpansive mapping such that .*

Lemma 2.7 (see [11]). *Let be a Hilbert space and let be a nonempty convex subset of . Assume that is a countable family of -strictly pseudocontraction for some and such that . Assume that is a positive sequence such that . Then is a -strictly pseudocontraction with coefficient and .*

Lemma 2.8 (see [17]). *Let be a sequence of nonnegative real numbers satisfying the following relation: , where , or , then .*

#### 3. Main Results

In this section, we prove some strong convergence results on the iterative algorithm for solving hierarchical fixed point problem.

Theorem 3.1. *Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a (possibly nonself) contraction with coefficient , and let be a nonexpansive mapping. Let be a countable family of -strictly (possibly nonself) pseudocontraction with such that . Let the sequence be generated by (1.10) with , in . Suppose for each , , for all and for all. Assume that the parameters satisfy the following conditions:*(i) * and ;*(ii) *;*(iii) *, , and ;*(iv) * and .**Then the sequence converges strongly to , which solves the variational inequality
*

* Proof. *The proof is divided into four steps.*Step* 1. We show that the sequences and are bounded.

For each , write and by Lemma 2.7, we have is a -strictly pseudocontraction on and , for all . Therefore, the iterative algorithm (1.10) can be written as

By condition (ii), without loss of generality, we may assume , for all . Take and we estimate . For fixed approximate , define a mapping and by Lemma 2.6, is a nonexpansive mapping and . So

Together with (3.2) and (3.3), we get

Therefore, we obtain
which gives the results that the sequence is bounded and so are , , , .*Step* 2. Now we show that as . Let
Next we estimate . From (3.2),we have
where is a constant such that

From (3.2), we also obtain
Together with (3.7) and (3.9), we have
By Lemma 2.8 and conditions (i)–(iii), we immediately get as .*Step* 3. Next we prove that as .

Let . By Lemma 2.7 and condition (iv), we get the results that is a -strictly pseudocontraction with and as , for any ,
Because , , and , so we obtain as .*Step* 4. Now we show that , where .

Since the sequence is bounded, we take subsequence of such that and . Notice that and by Lemmas 2.1 and 2.5, we have . Then

Now, by Lemma 2.2, we get . Therefore, we have

Hence it follows that

Now, by Lemma 2.8, conditions (i)–(iii), and
we have as and also solves the variational inequality
This completes the proof.

From Theorem 3.1, if we take or , for all , we get the following corollary.

Corollary 3.2. * Let be a real Hilbert space and be a nonempty closed convex subset of . Let be a (possibly nonself) contraction with coefficient and let be a countable family of -strictly (possibly nonself) pseudocontraction with and such that . Let the sequence be generated by
**
with in . Suppose for each , , for all and for all . Assume that the parameters satisfied the following conditions:*(i) * and ;*(ii) *, and ;*(iii) * and .** Then the the sequence converges strongly to , which solves the variational inequality
*

*Remark 3.3. *Theorem 3.1 extends and improves Theorem YCL in the following way. The nonexpasnsive self-mapping is extended to a infinite family of nonself -strictly pseudocontraction . If we take in Theorem 3.1, then reduces to a nonexpasnsive (possibly nonself) mapping, thus Theorem 3.1 reduces to Theorem YCL.

#### Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This paper was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).

#### References

- 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 - 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 - G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher · View at Google Scholar - L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general iterative method with strongly positive operators for general variational inequalities,”
*Computers & Mathematics with Applications. An International Journal*, vol. 59, no. 4, pp. 1441–1452, 2010. View at Publisher · View at Google Scholar - M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 73, no. 3, pp. 689–694, 2010. View at Publisher · View at Google Scholar - L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces,”
*Computers & Mathematics with Applications*, vol. 61, no. 9, pp. 2447–2455, 2011. View at Publisher · View at Google Scholar - Y. H. Yao and Y. C. Liou, “Composite algorithms for minimization over the solutions of equilibrium problems
and fixed point problems,”
*Abstract and Applied Analysis*, vol. 2010, Article ID 763506, 19 pages, 2010. View at Publisher · View at Google Scholar - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient iterative methods for variational inequalities,”
*Applied Mathematics and Computation*, vol. 218, no. 3, pp. 1112–1123, 2011. View at Publisher · View at Google Scholar - Y. H. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extragradient method to the minium-norm solution of variational inequalities,”
*Abstract and Applied Analysis*, vol. 2012, Article ID Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar - Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 68, no. 6, pp. 1687–1693, 2008. View at Publisher · View at Google Scholar - Y. L. Song, H. Y. Hu, Y. Q. Wang, et al., “Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces,”
*Fixed Point Theory and Applications*, vol. 2012, 46 pages, 2012. View at Publisher · View at Google Scholar - Y. H. Yao, Y. J. Cho, and Y.-C. Liou, “Iterative algorithms for hierarchical fixed points problems and variational inequalities,”
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1697–1705, 2010. View at Publisher · View at Google Scholar - Y. H. Wang and Y. H. Xia, “Strong convergence for asymptotically pseudocontractions with the demiclosedness
principle in Banach spaces,”
*Fixed Point Theory and Applications*, vol. 2012, 45 pages, 2012. View at Publisher · View at Google Scholar - S. S. Chang, Y. J. Cho, and H. Y. Zhou,
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*, Nova Science Publisher Inc., Huntington, NY, USA, 2002. - H. Y. Zhou, “Convergence theorems of fixed points for
*k*-strict pseudo-contractions in Hilbert spaces,”*Nonlinear Analysis. Theory, Methods & Applications*, vol. 69, no. 2, pp. 456–462, 2008. View at Publisher · View at Google Scholar - 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 Google Scholar - Y. H. Wang and L. Yang, “Modified relaxed extragradient method for a general system of variational inequalities and nonexpansive mappings in Banach spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 818970, 14 pages, 2012. View at Publisher · View at Google Scholar