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
We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .
Let be a nonempty closed convex subset of a real Hilbert space . Recall that a mapping is said to be nonexpansive if
for all . We use to denote the set of fixed points of .
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 .
Let be a nonempty closed convex subset of . For every point , there exists a unique nearest point in , denoted by such that
The mapping is called the metric projection of onto . It is well known that is a nonexpansive mapping.
In order to prove our main results, we need the following well-known lemmas.
Lemma 2.1 (see , Demiclosed principle).
Let be a nonempty closed convex of a real Hilbert space . Let be a nonexpansive mapping. Then is demiclosed at , that is, if and , then .
Lemma 2.2 (see ).
Let , be bounded sequences in a Banach space , and let be a sequence in which satisfies the following condition: . Suppose that for all and , then .
Lemma 2.3 (see ).
Assume, that is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence in such that
(ii) or ,
3. Main Results
Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping. For each , we consider the following mapping given by
It is easy to check that which implies that is a contraction. Using the Banach contraction principle, there exists a unique fixed point of in , that is,
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 .
Hence, is bounded.
Hence, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .
which implies that . This completes the proof.
Suppose that the following conditions are satisfied:
(i) and ,
then the sequence generated by (3.15) strongly converges to a fixed point of .
Hence, is bounded and so is .
We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, . This completes the proof.
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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