Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces
© S.Wang and C. Hu. 2010
Received: 6 August 2010
Accepted: 5 October 2010
Published: 11 October 2010
We consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces. We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality. Our results improve and extend the corresponding ones announced by many others.
In this paper, motivated and inspired by the above results, we introduce two new algorithms (3.3) and (3.13) for a countable family of nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to which solves the variational inequality: , .
In order to prove our main results, we need the following lemmas.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.4 (see [17, Lemma ]).
3. Main Results
The Corollary 3.3 complements the results of Theorem in Yao et al. , that is, is the solution of the variational inequality: .
Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that . Let be a -Lipschitzian and -strongly monotone operator on with . Let and be two real sequences in and satisfy the conditions:
We proceed with the following steps.
From Remark of Peng and Yao , we obtain that is a sequence of nonexpansive mappings satisfying condition for any bounded subset B of H. Moreover, let be the mapping; we know that for all and that . If we replace by in the recursion formula (3.19), we can obtain the corresponding results of the so-called mapping.
The Corollary 3.8 complements the results of Theorem in Yao et al. , that is, is the solution of the variational inequality: .
This paper is supported by the National Science Foundation of China under Grant (10771175).
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