Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups
© Kriengsak Wattanawitoon and Poom Kumam. 2010
Received: 15 April 2010
Accepted: 11 October 2010
Published: 12 October 2010
We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.
Recall also that a one-parameter family of self-mappings of a nonempty closed convex subset of a Hilbert space is said to be a (continuous) Lipschitzian semigroup on if the following conditions are satisfied:
Recently, Su and Qin  modified the hybrid iteration method of Nakajo and Takahashi through the monotone hybrid method, and to prove strong convergence theorems.
In this paper, motivated and inspired by the above results, we modify iteration process (1.4)–(1.11) by the new hybrid methods for countable families of asymptotically nonexpansive mappings and semigroups in a Hilbert space, and to prove strong convergence theorems. Our results presented are improvement and extension of the corresponding results in [3, 5–8] and many authors.
This section collects some lemmas which will be used in the proofs for the main results in the next section.
we get that a Hilbert space has the Kadec-Klee property.
Let be a nonempty closed convex subset of a Hilbert space . Motivated by Nakajo et al. , we give the following definitions: Let and be families of nonexpansive mappings of into itself such that , where is the set of all fixed points of and is the set of all common fixed points of . We consider the following conditions of and (see ):
Lemma 2.3 (Opial ).
Lemma 2.4 (Lin et al. ).
Lemma 2.5 (Nakajo and Takahashi ).
Lemma 2.6 (Kim and Xu ).
Lemma 2.7 (Kim and Xu ).
3. Strong Convergence for a Family of Asymptotically Nonexpansive Mappings
Since every family's nonexpansive mapping is family's asymptotically nonexpansive mapping we obtain the following result.
We have the following corollary for nonexpansive mappings by Lemma 2.2(i) and Theorem 3.1.
Corollary 3.4 (Takahashi et al. [7, Theorem ]).
4. Strong Convergence for a Family of Asymptotically Nonexpansive Semigroups
Furthermore, from (4.9) and Lemma 2.6 and the boundedness of we obtain that . By the fact that for any , where and the weak lower semi-continuity of the norm, we have for all . However, since , we must have for all . Thus and then converges weakly to . Moreover, following the method of Theorem 3.1, . This completes the proof.
Corollary 4.3 (Takahashi et al. [7, Theorem ]).
The authors would like to thank professor Somyot Plubtieng for drawing my attention to the subject and for many useful discussions and the referees for helpful suggestions that improved the contents of the paper. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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