- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- 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 673680, 11 pages

http://dx.doi.org/10.1155/2012/673680

## A New Hybrid Algorithm for -Strict Asymptotically Pseudocontractions in 2-Uniformly Smooth Banach Spaces

^{1}Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China^{2}College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 29 June 2012; Accepted 27 August 2012

Academic Editor: RuDong Chen

Copyright © 2012 Xin-dong Liu and Shih-sen Chang. 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

A new hybrid projection algorithm is considered for a -strict asymptotically pseudocontractive mapping. Using the metric projection, a strong convergence theorem is obtained in a uniformly convex and 2-uniformly smooth Banach spaces. The result presented in this paper mainly improves and extends the corresponding results of Matsushita and Takahashi (2008), Dehghan (2011) Kang and Wang (2011), and many others.

#### 1. Introduction

Let be a real Banach space and be the dual spaces of . Assume that is the normalized duality mapping from into defined by where is the generalized duality pairing between and .

Let be a nonempty closed convex subset of a real Banach space .

*Definition 1.1. *Let be a mapping:(1)is said to be nonexpansive if for all ,
(2) is said to be asymptotically nonexpansive if there exists a sequence with such that for all ,
(3) is said to be -strictly pseudocontractive in the terminology of Browder-Petryshyn [1] if there exists a constant such that for all ,
(4) is said to be -strict asymptotically pseudocontractive if there exist a constant and a sequence with such that for all and for all ,
(5) is said to be uniformly -Lipschitzian if there exists a constant such that

It is wellknown that the class of ()-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Liu [2]. In the case of Hilbert spaces, it is shown by [2] that (1.5) is equivalent to the inequality
Concerning the convergence problem of iterative sequences for strictly pseudocontractive mappings has been studied by several authors (see [1, 3–20]). Concerning the class of strictly asymptotically pseudocontractive mappings, Liu [2] proved the following results.

Theorem 1.2. *Let be a real Hilbert space, let be a nonempty closed convex and bounded subset of , and let be a completely continuous uniformly -Lipschitzian ()-strictly asymptotically pseudocontractive mapping such that . Let be a sequence satisfying the following condition:
**
Then, the sequence generated from an arbitrary by
**
converges strongly to a fixed point of .*

In 2007, Osilike et al. [21] proved the following theorem.

Theorem 1.3. * Let be a real -uniformly smooth Banach space which is also uniformly convex, let be a nonempty closed convex subset of , let be a ()-strictly asymptotically pseudocontractive mapping such that , and let . Let be a real sequence satisfying the following condition:
**
Let be the sequence defined by (1.9). Then, converges weakly to a fixed point of .*

On the other hand, by using the metric projection, Nakajo and Takahashi [22] introduced the following iterative algorithms for the nonexpansive mapping in the framework of Hilbert spaces: where and is the metric projection from a Hilbert space onto . They proved that generated by (1.11) converges strongly to a fixed point of .

In 2006, Xu [23] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.

In 2008, Matsushita and Takahashi [24] presented the following iterative algorithms for the nonexpansive mapping in the framework of Banach spaces: where denotes the convex closure of the set , is normalized duality mapping, is a sequence in (0, 1) with , and is the metric projection from onto . Then, they proved that generated by (1.12) converges strongly to a fixed point of nonexpansive mapping .

Recently, Dehghan [25] introduced the following hybrid projection algorithm for an asymptotically nonexpansive mapping in the framework of Banach spaces: where denotes the convex closure of the set , is a sequence in (0, 1) with , and is the metric projection from onto . Then, he proved that generated by (1.13) converges strongly to a fixed point of an asymptotically nonexpansive mappings .

Motivated by the research work going on in this direction, the purpose of this paper is to introduce the following iteration for finding a fixed point of -strict asymptotically pseudocontraction in a uniformly convex and 2-uniformly smooth Banach spaces: where denotes the convex closure of the set , is a sequence in (0, 1) with , and is the metric projection from onto . Under suitable conditions some strong convergence theorem for the sequence defined by (1.14) to converge a fixed point of an asymptotically -strictly pseudocontraction. The result presented in the paper extends and improves the main results of Matsushita and Takahashi [24], Dehghan [25], Kang and Wang [26], and others.

#### 2. Preliminaries

In this section, we recall the well-known concepts and results which will be needed to prove our main results. Throughout this paper, we assume that is a real Banach space and is a nonempty subset of . When is a sequence in , we denote strong convergence of to by and weak convergence by .

A Banach space is said to be *strictly convex* if for all with . is said to be *uniformly convex* if for each there is a such that for with and holds. The modulus of convexity of is defined by
is said to be *smooth* if the limit
exists for all . The modulus of smoothness of is defined by
A Banach space is said to be *uniformly smooth* if as . A Banach space is said to be *-uniformly smooth*, if there exists a fixed constant such that .

If is a reflexive, strictly convex and smooth Banach space, then for any , there exists a unique point such that
The mapping defined by is called the *metric projection* from onto . Let and . Then it is known that if and only if
For the details on the metric projection, refer to [27–30].

In the sequel, we make use of the following lemmas for our main results.

Lemma 2.1 (see [31]). * Let be a real Banach space, a nonempty subset of , and a ()-strictly asymptotically pseudocontractive mapping. Then is uniformly -Lipschitzian.*

Lemma 2.2 (see [32]). *Let be a real 2-uniformly smooth Banach spaces with the best smooth constant . Then the following inequality holds:
**
for any .*

Lemma 2.3 (demiclosed principle [21]). *Let be a real -uniformly smooth Banach space which is also uniformly convex. Let be a nonempty closed convex subset of and a ()-strictly asymptotically pseudocontractive mapping with a nonempty fixed point set. Then is demiclosed at zero, where is the identical mapping.*

Lemma 2.4 (see [33]). *Let be a closed convex subset of a uniformly convex Banach space. Then for each , there exists a strictly increasing convex continuous function such that and
**
for all , , and , where and , and is the set of all nonexpansive mappings from into . *

#### 3. Main Results

Now we are ready to give our main results in this paper.

Lemma 3.1. *Let be a nonempty subset of a real 2-uniformly smooth Banach space with the best smooth constant , and be a -strict asymptotically pseudocontraction. For , one defines
**
for all and each . Then is a nonexpansive. *

*Proof. *For any , put , we compute
which shows that is a nonexpansive mapping. This completes the proof.

Theorem 3.2. *Let be a nonempty bounded and closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaces with the best smooth constant , assume that is a -strict asymptotically pseudocontraction such that . Let be a sequence in (0, 1) with . Let be a sequence generated by (1.14), where
**
denotes the convex closure of the set and is the metric projection from onto . Then converges strongly to . *

*Proof. *(I) First we prove that is well defined and bounded.

It is easy to check that is closed and convex and for all . Therefore, is well defined.

Put . Since and , we have that
for all . Hence, is bounded.

(II) Now we prove that as for any ( denotes the set of all positive integers).

Fix and put . Since , we have . Then there exist some positive integer , and such that
for all . Take . Put , and . we define
for all and each , then . It follows from Lemma 3.1 and (3.5) that
Moreover, from Lemmas 2.4 and 3.1, we have
Observe that as , it follows from (3.5)–(3.9) that
This shows that
(III) we prove that as .

Since is a uniformly -Lipschitzian, we have
(IV) Finally, we prove that .

It follows from the boundedness of that for each subsequence there exists a subsequence (without loss of generality we can still denote it by) such that as . Since is a uniformly -Lipschitzian and -strict asymptotically pseudocontraction, from Lemma 2.3, we know that is demiclosed. Hence we have .

From the weakly lower semicontinuity of the norm and (3.4), it follows that
This shows and hence as . By the arbitrariness of , we obtain . Further, it follows from (3.13) that
Since is uniformly convex, it has the Kadec-Klee property. Hence, we have , that is, . This completes the proof.

#### Acknowledgment

The authors would like to express their thanks to the referees for their helpful comments and suggestions. Supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA221) and the Scientic Research Fund of Science Technology Department of Sichuan Province 2011JYZ010.

#### 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 - Q. Liu, “Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings,”
*Nonlinear Analysis*, vol. 26, no. 11, pp. 1835–1842, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Gu, “The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 2, pp. 766–776, 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 - Y. Su and S. Li, “Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,”
*Journal of Mathematical Analysis and Applications*, vol. 320, no. 2, pp. 882–891, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,”
*Numerical Functional Analysis and Optimization*, vol. 22, no. 5-6, pp. 767–773, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Su, M. Li, and H. Zhang, “New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5458–5465, 2011. View at Publisher · View at Google Scholar - S. S. Chang, H. W. J. Lee, C. K. Chan, and J. K. Kim, “Approximating solutions of variational inequalities for asymptotically nonexpansive mappings,”
*Applied Mathematics and Computation*, vol. 212, no. 1, pp. 51–59, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Zegeye and N. Shahzad, “Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings,”
*Computers & Mathematics with Applications*, vol. 62, no. 11, pp. 4007–4014, 2011. View at Publisher · View at Google Scholar - X. Qin, S. Huang, and T. Wang, “On the convergence of hybrid projection algorithms for asymptotically quasi-$\phi $-nonexpansive mappings,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 851–859, 2011. View at Publisher · View at Google Scholar - S. S. Chang, L. Wang, Y.-K. Tang, B. Wang, and L.-J. Qin, “Strong convergence theorems for a countable family of quasi-$\phi $-asymptotically nonexpansive nonself mappings,”
*Applied Mathematics and Computation*, vol. 218, no. 15, pp. 7864–7870, 2012. View at Publisher · View at Google Scholar - W. Nilsrakoo, “Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 12, pp. 4656–4666, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, H. W. J. Lee, C. K. Chan, and J. ai Liu, “Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3187–3198, 2011. View at Publisher · View at Google Scholar - W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 14, pp. 6577–6586, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Suantai, W. Cholamjiak, and P. Cholamjiak, “An implicit iteration process for solving a fixed point problem of a finite family of multi-valued mappings in Banach spaces,”
*Applied Mathematics Letters*, vol. 25, no. 11, pp. 1656–1660, 2012. View at Publisher · View at Google Scholar - S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 313, no. 1, pp. 273–283, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, H. W. J. Lee, C. K. Chan, and W. B. Zhang, “A modified halpern-type iteration algorithm for totally quasi-$\phi $-asymptotically nonexpansive mappings with applications,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6489–6497, 2012. View at Publisher · View at Google Scholar - X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-$\phi $-nonexpansive mappings and equilibrium problems,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 750–760, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Wu, S. S. Chang, and G. X. Yuan, “Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space,”
*Nonlinear Analysis*, vol. 63, no. 5-7, pp. 987–999, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Zhang, J. Li, and B. Liu, “Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,”
*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 262–276, 2011. View at Publisher · 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 - K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”
*Journal of Mathematical Analysis and Applications*, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu, “Strong convergence of approximating fixed point sequences for nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 74, no. 1, pp. 143–151, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-Y. Matsushita and W. Takahashi, “Approximating fixed points of nonexpansive mappings in a Banach space by metric projections,”
*Applied Mathematics and Computation*, vol. 196, no. 1, pp. 422–425, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Dehghan, “Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections,”
*Applied Mathematics Letters*, vol. 24, no. 9, pp. 1584–1587, 2011. View at Publisher · View at Google Scholar - S. M. Kang and S. Wang, “New hybrid algorithms for nonexpansive mappings in Banach spaces,”
*International Journal of Mathematical Analysis*, vol. 5, no. 9-12, pp. 433–440, 2011. - Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*, vol. 178 of*Lecture Notes in Pure and Applied Mathematics*, pp. 15–50, Dekker, New York, NY, USA, 1996. View at Zentralblatt MATH - W. Takahashi,
*Nonlinear Function Analysis. Fixed Points Theory and Its Applications*, Yokohama Publishers, Yokohama, 2000. - F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”
*Mathematische Zeitschrift*, vol. 100, pp. 201–225, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Pascali and S. Sburlan,
*Nonlinear Mappings of Monotone Type*, Noordhoff, Leyden, Mass, USA, 1978. - M. O. Osilike, S. C. Aniagbosor, and B. G. Akuchu, “Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces,”
*Panamerican Mathematical Journal*, vol. 12, no. 2, pp. 77–88, 2002. View at Zentralblatt MATH - H. K. Xu, “Inequalities in Banach spaces with applications,”
*Nonlinear Analysis*, vol. 16, no. 12, pp. 1127–1138, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. E. Bruck, “On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces,”
*Israel Journal of Mathematics*, vol. 38, no. 4, pp. 304–314, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH