- Research Article
- Open Access

# Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces

- Shih-sen Chang
^{1}, - YeolJe Cho
^{2}Email author, - HWJoseph Lee
^{3}and - ChiKin Chan
^{3}

**2011**:583423

https://doi.org/10.1155/2011/583423

© Shih-sen Chang et al. 2011

**Received:**24 November 2010**Accepted:**9 February 2011**Published:**2 March 2011

## Abstract

The purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors, but also give an affirmative answer for the open questions raised by Suzuki (2003) and Xu (2005).

## Keywords

- Banach Space
- Nonexpansive Mapping
- Real Banach Space
- Nonempty Closed Convex Subset
- Affirmative Answer

## 1. Introduction and Preliminaries

*normalized duality mapping*defined by

for all . Let be a mapping. We use to denote the set of fixed points of . We also use " " to stand for strong convergence and " " for weak convergence.

In this case, we also say that
is a
*-demicontractive semigroup*.

- (1)

The convergence problems of the implicit or explicit iterative sequences for nonexpansive semigroup to a common fixed has been considered by some authors in the settings of Hilbert or Banach spaces (see, e.g., [1–10]).

Under certain restrictions to the sequence , they proved some strong convergence theorems of to a point .

for each . Under appropriate assumptions imposed upon the sequences and , he proved that the sequence defined by (1.11) converges strongly to a common fixed point of the nonexpansive semigroup. At the same time, he also raised the following open question.

Open Question 1.3 (see [8]).

for each , what conditions to be imposed on and are sufficient to guarantee the strong convergence of to a common fixed point of the nonexpansive semi-group of mapping from into itself?

In 2005, Xu [9] proved that Suzuki's result holds in a uniformly convex Banach space with a weakly continuous duality mapping. At the same time, he also raised the following open question.

Open Question 1.4 (see [9]).

We do not know whether or not the same result holds in a uniformly convex and uniformly smooth Banach space.

for each in a reflexive Banach space with a uniformly Gâteaux differentiable norm such that each nonempty bounded closed and convex subset of has the common fixed point property for nonexpansive mappings (note that all these assumptions are fulfilled whenever is uniformly smooth [11]).

Also, under appropriate assumptions imposed upon the parameter sequences and , they proved that the sequence defined by (1.13) converges strongly to some point in .

for each for the nonexpansive semi-group of mappings from into itself, where is an arbitrary (but fixed) element in and the sequences in , in , in , and proved some strong convergence theorems for the iteration sequence . In fact, the results presented in [3] not only extend and improve the corresponding results of Shioji and Takahashi [7], Suzuki [8], Xu [9], and Aleyner and Reich [1], but also give a partially affirmative answer for the open questions raised by Suzuki [8] and Xu [9].

In order to improve and develop the results mentioned above, recently, Zhang [12, 13], by using the different methods, introduce and study the weak convergence problem of the implicit iteration processes for the Lipschitzian and pseudocontraction semigroups in general Banach spaces. The results given in [12, 13] not only extend the above results, but also extend and improve the corresponding results in Li et al. [6], Osilike [14], Xu and Ori [15], and Zhou [16].

for each for the Lipschitzian and demicontractive semigroup in general Banach spaces. The results presented in this paper improve, extend, and replenish the corresponding results given in [1, 3–10, 12, 13].

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

Lemma 1.5.

Lemma 1.6 (see [17]).

for all , where is some nonnegative integer. If , then the limit exists. In addition, if there exists a subsequence of such that , then .

## 2. Main Results

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

Theorem 2.1.

Let be the sequence defined by (1.15), where is a sequence in and is an increasing sequence in . If the following conditions are satisfied:

then we have the following:

- (2)

which is a contradiction since, by the condition (a), and . Therefore, the conclusion (2.11) is proved.

This completes the proof.

By using Theorem 2.1, we have the following.

Theorem 2.2.

Let be the sequence defined by (1.15), where is a sequence in and is an increasing sequence in . If there exists a compact subset of such that and the following conditions are satisfied:

then converges strongly to a common fixed point of the semigroup .

Proof.

for some point . Hence it follows from (2.21) that as .

Since as and the limit exists by Theorem 2.1 (1), which implies that as . This completes the proof.

Remark 2.3.

Theorem 2.2 not only extends and improves the corresponding results of Shioji and Takahashi [7], Suzuki [8], Xu [9], and Aleyner and Reich [1], but also gives an affirmative answer to the open questions raised by Suzuki [8] and Xu [9].

## Declarations

### Acknowledgment

The first author was supported by the Natural Science Foundation of Yibin University (No. 2009Z3), and the second author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

## References

- Aleyner A, Reich S:
**An explicit construction of sunny nonexpansive retractions in Banach spaces.***Fixed Point Theory and Applications*2005, (3):295–305.Google Scholar - Boonchari D, Saejung S:
**Construction of common fixed points of a countable family of -demicontractive mappings in arbitrary Banach spaces.***Applied Mathematics and Computation*2010,**216**(1):173–178. 10.1016/j.amc.2010.01.027MathSciNetView ArticleMATHGoogle Scholar - Zhang S-S, Yang L, L.u J-A:
**Strong convergence theorems for nonexpansive semi-groups in Banach spaces.***Applied Mathematics and Mechanics*2007,**28**(10):1287–1297. 10.1007/s10483-007-1002-xMathSciNetView ArticleMATHGoogle Scholar - Chang SS, Chan CK, Joseph Lee HW, Yang L:
**A system of mixed equilibrium problems, fixed point problems of strictly pseudo-contractive mappings and nonexpansive semi-groups.***Applied Mathematics and Computation*2010,**216**(1):51–60. 10.1016/j.amc.2009.12.060MathSciNetView ArticleMATHGoogle Scholar - Chen R, Song Y, Zhou H:
**Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings.***Journal of Mathematical Analysis and Applications*2006,**314**(2):701–709. 10.1016/j.jmaa.2005.04.018MathSciNetView ArticleMATHGoogle Scholar - Li S, Li LH, Su F:
**General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(9):3065–3071. 10.1016/j.na.2008.04.007MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*1998,**34**(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.***Proceedings of the American Mathematical Society*2003,**131**(7):2133–2136. 10.1090/S0002-9939-02-06844-2MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**A strong convergence theorem for contraction semigroups in Banach spaces.***Bulletin of the Australian Mathematical Society*2005,**72**(3):371–379. 10.1017/S000497270003519XMathSciNetView ArticleMATHGoogle Scholar - Zhang SS, Yang L, Lee HWJ, Chan CK:
**Strong convergence theorems for nonexpansive semigroups in Hilbert spaces.***Acta Mathematica Sinica*2009,**52**(2):337–342.MathSciNetMATHGoogle Scholar - Browder FE:
**Nonlinear operators and nonlinear equations of evolution in Banach spaces.**In*Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968)*. American Mathematical Society, Providence, RI, USA; 1976:1–308.Google Scholar - Zhang S-S:
**Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces.***Applied Mathematics and Mechanics*2009,**30**(2):145–152. 10.1007/s10483-009-0202-yMathSciNetView ArticleMATHGoogle Scholar - Zhang SS:
**Weak convergence theorem for Lipschizian pseudocontraction semigroups in Banach spaces.***Acta Mathematica Sinica*2010,**26**(2):337–344. 10.1007/s10114-010-7610-0MathSciNetView ArticleMATHGoogle Scholar - Osilike MO:
**Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps.***Journal of Mathematical Analysis and Applications*2004,**294**(1):73–81. 10.1016/j.jmaa.2004.01.038MathSciNetView ArticleMATHGoogle Scholar - Xu H-K, Ori RG:
**An implicit iteration process for nonexpansive mappings.***Numerical Functional Analysis and Optimization*2001,**22**(5–6):767–773. 10.1081/NFA-100105317MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(10):2977–2983. 10.1016/j.na.2007.02.041MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis: Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView 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.