# Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces

- Yonghong Yao
^{1}Email author, - Yeong Cheng Liou
^{2}and - Giuseppe Marino
^{3}

**2009**:279058

**DOI: **10.1155/2009/279058

© Yonghong Yao et al. 2009

**Received: **6 April 2009

**Accepted: **12 September 2009

**Published: **12 October 2009

## Abstract

## 1. Introduction

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 .

## 2. Preliminaries

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 [22], 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 [20]).

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 [22]).

Assume, that is a sequence of nonnegative real numbers such that , where is a sequence in and is a sequence in such that

## 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,

Theorem 3.1.

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 .

Proof.

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.

Theorem 3.2.

Suppose that the following conditions are satisfied:

then the sequence generated by (3.15) strongly converges to a fixed point of .

Proof.

We can check that all assumptions of Lemma 2.3 are satisfied. Therefore, . This completes the proof.

## Declarations

### Acknowledgment

The second author was partially supposed by the Grant NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.

## Authors’ Affiliations

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