Modified Iterative Algorithms for Nonexpansive Mappings

Abstract

Let be a real Hilbert space, let , be two nonexpansive mappings such that , let be a contractive mapping, and let be a strongly positive linear bounded operator on . In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as , , is a real number and , are two sequences in satisfying the following control conditions: (C1) , (C3) , then . We also discuss several special cases of this iterative algorithm.

1. Introduction

Let be a real Hilbert space. Recall that a mapping is a contractive mapping on if there exists a constant such that

We denote by the collection of all contractive mappings on , that is,

Let be a nonexpansive mapping, namely,

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems (see [1–4] and the references therein).

A typical problem is to minimize a quadratic function over the closed convex set of the fixed points of a nonexpansive mapping on a real Hilbert space :

where is a closed convex set of the fixed points a nonexpansive mapping on , is a given point in and is a linear, symmetric and positive operator.

In [5] (see also [6]), the author proved that the sequence defined by the iterative method below with the initial point chosen arbitrarily

converges strongly to the unique solution of the minimization problem (1.4) provided the sequence satisfies certain control conditions.

On the other hand, Moudafi [3] introduced the viscosity approximation method for nonexpansive mappings (see also [7] for further developments in both Hilbert and Banach spaces). Let be a contractive mapping on . Starting with an arbitrary initial point , define a sequence in recursively by

where is a sequence in which satisfies some suitable control conditions.

Recently, Marino and Xu [8] combined the iterative algorithm (1.5) with the viscosity approximation algorithm (1.6), considering the following general iterative algorithm:

where

In this paper, we suggest a new iterative method for finding the pair of nonexpansive mappings. As an application and as special cases, we also obtain some new iterative algorithms which can be viewed as an improvement of the algorithm of Xu [7] and Marino and Xu [8]. Also we show that the convergence of the proposed algorithms can be proved under weaker conditions on the parameter . In this respect, our results can be considered as an improvement of the many known results.

2. Preliminaries

In the sequel, we will make use of the following for our main results:

Lemma 2.1 (see [4]). Let be a sequence of nonnegative numbers satisfying the condition where , are sequences of real numbers such that (i) and ,(ii) or is convergent.Then .

Lemma 2.2 (see [9, 10]). Let and be bounded sequences in a Banach space and be a sequence in with Suppose that for all and . Then .

Lemma 2.3 (see [2] (demiclosedness Principle)). Assume that is a nonexpansive self-mapping of a closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed, that is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that , where is the identity operator of .

Lemma 2.4 (see [8]). Let be generated by the algorithm . Then converges strongly as to a fixed point of which solves the variational inequality

Lemma 2.5 (see [8]). Assume is a strong positive linear bounded operator on a Hilbert space with coefficient and . Then .

3. Main Results

Let be a real Hilbert space, let be a bounded linear operator on , and let , be two nonexpansive mappings on such that . Throughout the rest of this paper, we always assume that is strongly positive.

Now, let with the contraction coefficient and let be a strongly positive linear bounded operator with coefficient satisfying . We consider the following modified iterative algorithm:

where is a real number and , are two sequences in .

First, we prove a useful result concerning iterative algorithm (3.1) as follows.

Lemma 3.1. Let be a sequence in generated by the algorithm (3.1) with the sequences and satisfying the following control conditions: (C1),(C3).Then .

Proof. From the control condition (C1), without loss of generality, we may assume that . First observe that by Lemma 2.5.
Now we show that is bounded. Indeed, for any , At the same time, It follows from (3.2) and (3.3) that which implies that Hence is bounded and so are and .
From (3.1), we observe that It follows that which implies, from (C1) and the boundedness of , , and , that Hence, by Lemma 2.2, we have Consequently, it follows from (3.1) that This completes the proof.

Remark 3.2. The conclusion is important to prove the strong convergence of the iterative algorithms which have been extensively studied by many authors, see, for example, [3, 6, 7].

If we take in (3.1), we have the following iterative algorithm:

Now we state and prove the strong convergence of iterative scheme (3.11).

Theorem 3.3. Let be a sequence in generated by the algorithm (3.11) with the sequences and satisfying the following control conditions: (C1),(C2),(C3).Then converges strongly to a fixed point of which solves the variational inequality

Proof. From Lemma 3.1, we have
On the other hand, we have that is, this together with (C1), (C3), and (3.13), we obtain
Next, we show that, for any ,
In fact, we take a subsequence of such that Since is bounded, we may assume that , where “” denotes the weak convergence. Note that by virtue of Lemma 2.3 and (3.16). It follows from the variational inequality (2.3) in Lemma 2.4 that By Lemma 3.1 (noting ), we have Hence, we get
Finally, we prove that converges to the point . In fact, from (3.2) we have Therefore, from (3.16), we have Since , and are all bounded, we can choose a constant such that It follows from (3.23) that where By (C1) and (3.17), we get Now, applying Lemma 2.1 and (3.25), we conclude that . This completes the proof.

Taking in (3.1), we have the following iterative algorithm:

Now we state and prove the strong convergence of iterative scheme (3.28).

Theorem 3.4. Let be a sequence in generated by the algorithm (3.28) with the sequences and satisfying the following control conditions: (C1),(C2),(C3).Then converges strongly to a fixed point of which solves the variational inequality

Proof. From Lemma 3.1, we have Thus, we have By the similar argument as (3.17), we also can prove that From (3.28), we obtain The remainder of proof follows from the similar argument of Theorem 3.3. This completes the proof.

From the above results, we have the following corollaries.

Corollary 3.5. Let be a sequence in generated by the following algorithm where the sequences and satisfy the following control conditions: (C1), (C2), (C3).Then converges strongly to a fixed point of which solves the variational inequality

Corollary 3.6. Let be a sequence in generated by the following algorithm where the sequences and satisfy the following control conditions: (C1), (C2), (C3).Then converges strongly to a fixed point of which solves the variational inequality

Remark 3.7. Theorems 3.3 and 3.4 provide the strong convergence results of the algorithms (3.11) and (3.28) by using the control conditions (C1) and (C2), which are weaker conditions than the previous known ones. In this respect, our results can be considered as an improvement of the many known results.

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Copyright

Copyright © 2009 Yonghong Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.