- Research Article
- Open Access
Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces
© Yonghong Yao et al. 2009
- Received: 6 April 2009
- Accepted: 12 September 2009
- Published: 12 October 2009
- Hilbert Space
- Partial Differential Equation
- Inverse Problem
- Iterative Method
- Nonlinear Mapping
Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative algorithms for finding fixed points of nonexpansive mappings have received vast investigation (cf. [1, 2]) since these algorithms find applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing see; [3–8]. Iterative methods for nonexpansive mappings have been extensively investigated in the literature; see [1–7, 9–21].
It is our purpose in this paper to introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of nonexpansive mapping .
In order to prove our main results, we need the following well-known lemmas.
Lemma 2.1 (see , Demiclosed principle).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . For each , let the net be generated by (3.2). Then, as , the net converges strongly to a fixed point of .
Suppose that the following conditions are satisfied:
The second author was partially supposed by the Grant NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.
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