Open Access

Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings

Fixed Point Theory and Applications20072008:284613

https://doi.org/10.1155/2008/284613

Received: 1 June 2007

Accepted: 16 October 2007

Published: 27 November 2007

Abstract

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.

1. Introduction

In 2005, Shin-ya Matsushita and Wataru Takahashi [1] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space :
(1.1)

They proved the following convergence theorem.

Theorem 1.1 (MT).

Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty, closed, and convex subset of , let be a relatively nonexpansive mapping from into itself, and let be a sequence of real numbers such that and . Suppose that is given by (1.1), where is the duality mapping on . If the set of fixed points of is nonempty, then converges strongly to , where is the generalized projection from onto .

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi [1], and some others.

2. Preliminaries

Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by
(2.1)

where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . In this case, is singe valued and also one to one.

Recall that if is a nonempty, closed, and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This is true only when is a real Hilbert space. In this connection, Alber [2] has recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a smooth Banach space. Consider the functional defined as [2, 3] by
(2.2)

Observe that, in a Hilbert space , (2.2) reduces to ,

The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional that is, where is the solution to the minimization problem
(2.3)
existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [24]). In Hilbert space, It is obvious from the definition of the function that
(2.4)

Remark 2.1.

If is a reflexive strict convex and smooth Banach space, then for , if and only if . It is sufficient to show that if , then . From (2.4), we have . This implies From the definition of we have , that is, see [5] for more details.

We refer the interested reader to the [6], where additional information on the duality mapping may be found.

Let be a closed convex subset of , and Let be a mapping from into itself. We denote by the set of fixed points of . is called hemi-relatively nonexpansive if for all and .

A point in is said to be an asymptotic fixed point of [7] if contains a sequence which converges weakly to such that the strong The set of asymptotic fixed points of will be denoted by . A hemi-relatively nonexpansive mapping from into itself is called relatively nonexpansive [1, 7, 8] if .

We need the following lemmas for the proof of our main results.

Lemma 2.2 ({Kamimura and Takahashi [4], [1, Proposition 2.1]}).

Let be a uniformly convex and smooth real Banach space and let be two sequences of . If and either or is bounded, then

Lemma 2.3 ({Alber [2], [1, Proposition 2.2]}).

Let be a nonempty closed convex subset of a smooth real Banach space and . Then, if and only if
(2.5)

Lemma 2.4 ({Alber [2], [1, Proposition 2.3]}).

Let be a reflexive, strict convex, and smooth real Banach space, let be a nonempty closed convex subset of and let Then
(2.6)

By using the similar method as [1, Proposition 2.4], the following lemma is not hard to prove.

Lemma 2.5.

Let be a strictly convex and smooth real Banach space, let be a closed convex subset of , and let be a hemi-relatively nonexpansive mapping from into itself. Then is closed and convex.

Recall that an operator in a Banach space is called closed, if , then .

3. Strong Convergence for Hemi-Relatively Nonexpansive Mappings

Theorem 3.1.

Theorem 3.1 Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty closed convex subset of , let be a closed hemi-relatively nonexpansive mapping such that . Assume that is a sequence in such that . Define a sequence in by the following algorithm:
(3.1)

where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Proof.

We first show that and are closed and convex for each . From the definition of and , it is obvious that is closed and is closed and convex for each . We show that is convex for any . Since
(3.2)
is equivalent to
(3.3)

it follows that is convex.

Next, we show that for all . Indeed, we have for all that
(3.4)

That is, for all .

Next, we show that for all , we prove this by induction. For we have Assume that Since is the projection of onto , by Lemma 2.3, we have
(3.5)

As by the induction assumptions, the last inequality holds, in particular, for all . This together with the definition of implies that .

Since and for all , we have
(3.6)
for all . Therefore, is nondecreasing. In addition, it follows from the definition of and Lemma 2.3 that . Therefore, by Lemma 2.4, we have
(3.7)
for each for all Therefore, is bounded, this together with (3.6) implies that the limit of exists. Put
(3.8)
From Lemma 2.4, we have, for any positive integer , that
(3.9)
for all Therefore,
(3.10)
We claim that is a Cauchy sequence. If not, there exists a positive real number and subsequence such that
(3.11)

for all .

On the other hand, from (3.8) and (3.9) we have

(3.12)
Because from (3.8) we know that is bounded, this and (2.4) imply that is also bounded, so by Lemma 2.2 we obtain
(3.13)

This is a contradiction, so that is a Cauchy sequence, therefore there exists a point such that converges strongly to .

Since , from the definition of , we have
(3.14)
It follows from (3.10), (3.14) that
(3.15)
By using Lemma 2.2, we have
(3.16)
Since is uniformly norm-to-norm continuous on bounded sets, we have
(3.17)
Noticing that
(3.18)
which implies that
(3.19)
This together with (3.17) and implies that
(3.20)
Since is also uniformly norm-to-norm continuous on any bounded sets, we have
(3.21)
Observe that
(3.22)
It follows from (3.16) and (3.21) that
(3.23)

Since is a closed operator and , then is a fixed point of .

Finally, we prove that . From Lemma 2.4, we have
(3.24)
On the other hand, since and , for all , we get from Lemma 2.4 that
(3.25)

By the definition of , it follows that both and , whence . Therefore, it follows from the uniqueness of that . This completes the proof.

Theorem 3.2.

Let be a uniformly convex and uniformly smooth real Banach space, let be a nonempty, closed, and convex subset of , and let be a closed relative nonexpansive mapping such that . Assume that is a sequences in such that . Define a sequence in by the following algorithm:
(3.26)

where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Proof.

Since every relatively nonexpansive mapping is a hemi-relatively one, Theorem 3.2 is implied by Theorem 3.1.

Remaek 3.3.

In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors [1, 811]. In fact, all hybrid iteration methods can be replaced (or modified) by monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial's condition or other methods which make use of the weak topology.

Declarations

Acknowledgments

The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Mathematics, Shijiazhuang University

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© Yongfu Su et al. 2008

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