- Research Article
- Open Access
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.
- Hilbert Space
- Banach Space
- Variational Inequality
- Iterative Method
- Nonexpansive Mapping
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 ]).
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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