# Strong Convergence Theorems for Strict Pseudocontractions in Uniformly Convex Banach Spaces

- Liang-Gen Hu
^{1}Email author, - Wei-Wei Lin
^{2}and - Jin-Ping Wang
^{1}

**2010**:150539

**DOI: **10.1155/2010/150539

© Liang-Gen Hu et al. 2010

**Received: **20 April 2010

**Accepted: **26 August 2010

**Published: **30 August 2010

## Abstract

The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme to -strict pseudocontractions in -uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. The main result improves and extends many nice results existing in the current literature.

## 1. Introduction

We denote by the set of fixed point of , that is, .

*let*

*be a nonempty convex subset*

*and let*

*be a self-mapping of*

*. For any*

*, the sequence*

*is defined by*

In the last ten years or so, there have been many nice papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive mappings by utilizing the Mann iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and more general class of mappings; see, for example, [1–6] and the references therein for more information about this problem.

In 2007, Marino and Xu [2] showed that the Mann iterative sequence converges weakly to a fixed point of
-strict pseudocontractions in Hilbert spaces. Meanwhile, they have proposed an *open question;* that is,*is the result of*[2, Theorem
]*true in uniformly convex Banach spaces with Fréchet differentiable norm?* In other words, can Reich's theorem [7, Theorem
], with respect to nonexpansive mappings, be extended to
-strict pseudocontractions in uniformly convex Banach spaces?

In 2008, using the Mann iteration and the modified Ishikawa iteration, Zhou [3] obtained some weak and strong convergence theorems for -strict pseudocontractions in Hilbert spaces which extend the corresponding results in [2].

Recently, Hu and Wang [4] obtained that the Mann iterative sequence converges weakly to a fixed point of -strict pseudocontractions with respect to in -uniformly convex Banach spaces.

*Let*

*be a nonempty closed convex subset of*

*and let*

*be a*

*-contraction. For any*

*, the sequence*

*is defined by*

*where*
, *for all*
,
, *and*
*in*
. The iterative sequence (1.7) is a natural generalization of the Mann iterative sequences (1.6). If we take
, for all
, in (1.7), then (1.7) is reduced to the Mann iteration.

The purpose in this paper is to show strong convergence theorems of the modified Mann iteration scheme for -strict pseudocontractions with respect to in -uniformly convex Banach spaces with uniformly differentiable norm by using viscosity approximation methods. Our theorems improve and extend the comparable results in the following four aspects: in contrast to weak convergence results in [2–4], strong convergence theorems of the modified Mann iterative sequence are obtained in -uniformly convex Banach spaces; in contrast to the results in [7, 8], these results with respect to nonexpansive mappings are extended to -strict pseudocontractions with respect to ; the restrictions and in [8, Theorem ] are removed; our results partially answer the open question.

## 2. Preliminaries

*uniformly convex*if and only if, for all such that . is said to be

*-uniformly convex*if there exists a constant such that . Hilbert spaces, (or ) spaces ( ) and Sobolev spaces ( ) are -uniformly convex. Hilbert spaces are -uniformly convex, while

exists for each
, where
. The norm of
is *a uniformly*
*differentiable* if for each
, the limit is attained uniformly for
. It is well known that if
is a uniformly
differentiable norm, then the duality mapping
is single valued and norm-to-weak
uniformly continuous on each bounded subset of
.

Lemma 2.1 (see [4]).

where is a constant in [9, Theorem ]. In addition, if , , and , then , for all .

Lemma 2.2 (see [10]).

Lemma 2.3.

Lemma 2.4 (see [11]).

where is a sequence in and is a sequence in satisfying the following conditions: ; or . Then, .

## 3. Main Results

Theorem 3.1.

Let be a real -uniformly convex Banach space with a uniformly G teaux differentiable norm, and let be a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudocontraction with respect to , and . Let be a -contraction with . Assume that real sequences , , and in satisfy the following conditions:

where , for all . Then, the sequence converges strongly to a fixed point of .

Proof.

From (3.18), (3.19), and the conditions it follows that and . Consequently, applying Lemma 2.4 to (3.21), we conclude that .

Corollary 3.2.

where , for all . Then the sequence converges strongly to a fixed point of .

Remark 3.3.

Theorem 3.1 and Corollary 3.2 improve and extend the corresponding results in [2–4, 7, 8] essentially since the following facts hold.

Theorem 3.1 and Corollary 3.2 give strong convergence results in -uniformly convex Banach spaces for the modification of Mann iteration scheme in contrast to the weak convergence result in [2, Theorem ], [3, Theorem and Corollary ], and [4, Theorems and ].

In contrast to the results in [7, Theorem ], and [8, Theorem ], these results with respect to nonexpansive mappings are extended to -strict pseudocontraction in -uniformly convex Banach spaces.

In contrast to the results in [8, Theorem ], the restrictions and are removed.

## Declarations

### Acknowledgments

The authors would like to thank the referees for the helpful suggestions. Liang-Gen Hu was supported partly by Ningbo Natural Science Foundation (2010A610100), the NNSFC(60872095), the K. C. Wong Magna Fund of Ningbo University and the Scientific Research Fund of Zhejiang Provincial Education Department (Y200906210). Wei-Wei Lin was supported partly by the Fundamental Research Funds for the Central Universities, SCUT(20092M0103). Jin-Ping Wang were supported partly by the NNSFC(60872095) and Ningbo Natural Science Foundation (2008A610018).

## Authors’ Affiliations

## References

- Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):336–346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**343**(1):546–556. 10.1016/j.jmaa.2008.01.045MathSciNetView ArticleMATHGoogle Scholar - Hu L-G, Wang J-P:
**Mann iteration of weak convergence theorems in Banach space.***Acta Mathematicae Applicatae Sinica. English Series*2009,**25**(2):217–224. 10.1007/s10255-007-7054-1MathSciNetView ArticleMATHGoogle Scholar - Liu L:
**Approximation of fixed points of a strictly pseudocontractive mapping.***Proceedings of the American Mathematical Society*1997,**125**(5):1363–1366. 10.1090/S0002-9939-97-03858-6MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Shehu Y:
**Cyclic algorithm for common fixed points of finite family of strictly pseudocontractive mappings of Browder-Petryshyn type.***Nonlinear Analysis*2009,**70**(10):3575–3583. 10.1016/j.na.2008.07.015MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations.***Nonlinear Analysis*2005,**61**(1–2):51–60. 10.1016/j.na.2004.11.011MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar - Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory and Applications 2005, (1):103–123.Google Scholar
- Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society. Second Series*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.