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).
1. Introduction and Preliminaries
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]).
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 ).
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 ).
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.
Lemma 1.6 (see ).
2. Main Results
Now, we are ready to give our main results in this paper.
then we have the following:
This completes the proof.
By using Theorem 2.1, we have the following.
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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