- Research Article
- Open Access
Strong Convergence Theorems for Lipschitzian Demicontraction Semigroups in Banach Spaces
© Shih-sen Chang et al. 2011
- Received: 24 November 2010
- Accepted: 9 February 2011
- Published: 2 March 2011
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).
- Banach Space
- Nonexpansive Mapping
- Real Banach Space
- Nonempty Closed Convex Subset
- Affirmative Answer
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.
The one-parameter family of mappings from into itself is called a nonexpansive semigroup if the following conditions are satisfied:
(a) for each ;
for any , the mapping is continuous;
for any , is a nonexpansive mapping on , that is, for any ,
The one-parameter family of mappings from into itself is called a pseudocontraction semigroup if the conditions (a)–(c) and the following condition (e) are satisfied:
A pseudocontraction semigroup of mappings from into itself is said to be Lipschitzian if the conditions (a)–(c), (e), and the following condition (f) are satisfied:
The one-parameter family of mappings from into itself is called a strictly pseudocontractive semigroup if the conditions (a)–(c) and the following condition (g) are satisfied:
for any .
The one-parameter family of mappings from into itself is called a demicontractive semigroup if for all and the conditions (a)–(c) and the following condition (h) are satisfied:
In this case, we also say that is a -demicontractive semigroup.
Every strictly pseudocontractive semigroup with is demi-contractive and Lipschitzian.
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 ).
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  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 ).
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 ).
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  not only extend and improve the corresponding results of Shioji and Takahashi , Suzuki , Xu , and Aleyner and Reich , but also give a partially affirmative answer for the open questions raised by Suzuki  and Xu .
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. , Osilike , Xu and Ori , and Zhou .
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.
for all .
Lemma 1.6 (see ).
for all , where is some nonnegative integer. If , then the limit exists. In addition, if there exists a subsequence of such that , then .
Now, we are ready to give our main results in this paper.
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:
for any bounded subset ,
then we have the following:
(1) exists for all .
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.
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:
for any bounded subset ,
then converges strongly to a common fixed point of the semigroup .
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.
Theorem 2.2 not only extends and improves the corresponding results of Shioji and Takahashi , Suzuki , Xu , and Aleyner and Reich , but also gives an affirmative answer to the open questions raised by Suzuki  and Xu .
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).
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