New Hybrid Iterative Schemes for an Infinite Family of Nonexpansive Mappings in Hilbert Spaces
© B. Guo and S.Wang. 2010
Received: 20 November 2009
Accepted: 4 February 2010
Published: 14 February 2010
We propose some new iterative schemes for finding common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the proposed schemes. Our results extend and improve ones of Nakajo and Takahashi (2003).
1. Introduction and Preliminaries
Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonlinear mapping of into itself. We use and to denote the set of fixed points of and the metric projection from onto , respectively.
For approximating the fixed point of a nonexpansive mapping in a Hilbert space, Mann  in 1953 introduced a famous iterative scheme as follows:
Nakajo and Takahashi  proposed the following modification of Mann iteration method (1.2) for a single nonexpansive mapping in a Hilbert space :
In this paper, we introduce some new iterative schemes for infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the algorithms. Our results extend and improve the corresponding one of Nakajo and Takahashi .
The following two lemmas will be used for the main results of this paper.
Lemma 1.2 (see ).
2. Main Results
In this paper, we extend result of Nakajo and Takahashi  from a single nonexpansive mapping to an infinite family of nonexpansive mappings.
The iterative schemes introduced in this paper are new and of independent interest.
It is of interest to extend the algorithm (2.25) to certain Banach spaces.
The work was supported by Youth Foundation of North China Electric Power University.
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