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

Abstract

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 .

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . Recall that a mapping is said to be nonexpansive if

(1.1)

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

(2.1)

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

(i),

(ii) or ,

then .

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

(3.1)

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,

(3.2)

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.

First, we prove that is bounded. Take . From (3.2), we have

(3.3)

that is,

(3.4)

Hence, is bounded.

Again from (3.2), we obtain

(3.5)

Next we show that is relatively norm compact as . Let be a sequence such that as . Put . From (3.5), we have

(3.6)

From (3.2), we get, for ,

(3.7)

Hence,

(3.8)

where is a constant such that . In particular,

(3.9)

Since is bounded, without loss of generality, we may assume that converges weakly to a point . Noticing (3.6) we can use Lemma 2.1 to get . Therefore we can substitute for in (3.9) to get

(3.10)

Hence, the weak convergence of to actually implies that strongly. This has proved the relative norm compactness of the net as .

To show that the entire net converges to , assume , where . Put . Similarly we have

(3.11)

Therefore,

(3.12)

Interchange and to obtain

(3.13)

Adding up (3.12) and (3.13) yields

(3.14)

which implies that . This completes the proof.

Theorem 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Let and be two real sequences in . For given arbitrarily, let the sequence , , be generated iteratively by

(3.15)

Suppose that the following conditions are satisfied:

(i) and ,

(ii),

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

Proof.

First, we prove that the sequence is bounded. Take . From (3.15), we have

(3.16)

Hence, is bounded and so is .

Set . It follows that

(3.17)

Hence,

(3.18)

This together with Lemma 2.2 implies that

(3.19)

Therefore,

(3.20)

We observe that

(3.21)

that is,

(3.22)

Let the net be defined by (3.2). By Theorem 3.1, we have as . Next we prove . Indeed,

(3.23)

where such that . It follows that

(3.24)

Therefore,

(3.25)

We note that

(3.26)

This together with and (3.25) implies that

(3.27)

Finally we show that . From (3.15), we have

(3.28)

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