Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators

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Research Article

Abstract

We study the strong convergence of an iterative method for inverse strongly accretive operators in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.

Keywords

Banach Space Variational Inequality Convex Subset Nonexpansive Mapping Strong Convergence 

1. Introduction and Preliminaries

Let Open image in new window be a real Hilbert space with norm Open image in new window and inner product Open image in new window , Open image in new window a nonempty closed convex subset of Open image in new window and Open image in new window a monotone operator of Open image in new window into Open image in new window . The classical variational inequality problem is formulated as finding a point Open image in new window such that
for all Open image in new window . Such a point Open image in new window is called a solution of the variational inequality (1.1). Next, the set of solutions of the variational inequality (1.1) is denoted by Open image in new window . In the case when Open image in new window , Open image in new window holds, where
Recall that an operator Open image in new window of Open image in new window into Open image in new window is said to be inverse strongly monotone if there exists a positive real number Open image in new window such that

for all Open image in new window (see [1, 2, 3, 4]). For such a case, Open image in new window is said to be Open image in new window -inverse strongly monotone.

Recall that Open image in new window is nonexpansive if

for all Open image in new window It is known that if Open image in new window is a nonexpansive mapping of Open image in new window into itself, then Open image in new window is Open image in new window -inverse strongly monotone and Open image in new window , where Open image in new window denotes the set of fixed points of Open image in new window .

Let Open image in new window be the projection of Open image in new window onto the convex subset Open image in new window . It is known that projection operator Open image in new window is nonexpansive. It is also known that Open image in new window satisfies

for Open image in new window Moreover, Open image in new window is characterized by the properties Open image in new window and Open image in new window for all Open image in new window

One can see that the variational inequality problem (1.1) is equivalent to some fixed-point problem. The element Open image in new window is a solution of the variational inequality (1.1) if and only if Open image in new window satisfies the relation Open image in new window where Open image in new window is a constant.

To find a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al. [2] proved the following weak convergence theorem.

Theorem 1.

Let Open image in new window be a nonempty closed convex subset of a real Hilbert space Open image in new window and let Open image in new window be an Open image in new window -inverse strongly monotone operator of Open image in new window into Open image in new window with Open image in new window . Let Open image in new window be a sequence defined as follows:

for all Open image in new window , where Open image in new window is the metric projection from Open image in new window onto Open image in new window , Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window . If Open image in new window and Open image in new window are chosen so that Open image in new window for some Open image in new window Open image in new window with Open image in new window and Open image in new window for some Open image in new window Open image in new window with Open image in new window , then the sequence Open image in new window defined by (1.6) converges weakly to some element of Open image in new window .

Next, we assume that Open image in new window is a nonempty closed and convex subset of a Banach space Open image in new window . Let Open image in new window be the dual space of Open image in new window and let Open image in new window denote the pairing between Open image in new window and Open image in new window . For Open image in new window , the generalized duality mapping Open image in new window is defined by
for all Open image in new window . In particular, Open image in new window is called the normalized duality mapping. It is known that Open image in new window for all Open image in new window . If Open image in new window is a Hilbert space, then Open image in new window . Further, we have the following properties of the generalized duality mapping Open image in new window :
Let Open image in new window . A Banach space Open image in new window is said to be uniformly convex if, for any Open image in new window , there exists Open image in new window such that, for any Open image in new window ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space Open image in new window is said to be smooth if the limit
exists for all Open image in new window . It is also said to be uniformly smooth if the limit (1.9) is attained uniformly for Open image in new window . The norm of Open image in new window is said to be Fréchet differentiable if, for any Open image in new window , the limit (1.9) is attained uniformly for all Open image in new window . The modulus of smoothness of Open image in new window is defined by

where Open image in new window is a function. It is known that Open image in new window is uniformly smooth if and only if Open image in new window . Let Open image in new window be a fixed real number with Open image in new window . A Banach space Open image in new window is said to be Open image in new window -uniformly smooth if there exists a constant Open image in new window such that Open image in new window for all Open image in new window .

Note that
  1. (1)
    is a uniformly smooth Banach space if and only if Open image in new window is single-valued and uniformly continuous on any bounded subset of Open image in new window ;
     
  2. (2)
     
Recall that an operator Open image in new window of Open image in new window into Open image in new window is said to be accretive if there exists Open image in new window such that

for all Open image in new window .

for all Open image in new window . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.

whenever Open image in new window for Open image in new window and Open image in new window . A mapping Open image in new window of Open image in new window into itself is called a retraction if Open image in new window . If a mapping Open image in new window of Open image in new window into itself is a retraction, then Open image in new window for all Open image in new window , where Open image in new window is the range of Open image in new window . A subset Open image in new window of Open image in new window is called a sunny nonexpansive retract of Open image in new window if there exists a sunny nonexpansive retraction from Open image in new window onto Open image in new window . We know the following lemma concerning sunny nonexpansive retraction.

Lemma 1.1 (see [5]).

Let Open image in new window be a closed convex subset of a smooth Banach space Open image in new window , let Open image in new window be a nonempty subset of Open image in new window , and let Open image in new window be a retraction from Open image in new window onto Open image in new window . Then Open image in new window is sunny and nonexpansive if and only if

for all Open image in new window and Open image in new window .

Recently, Aoyama et al. [6] first considered the following generalized variational inequality problem in a smooth Banach space. Let Open image in new window be an accretive operator of Open image in new window into Open image in new window . Find a point Open image in new window such that

for all Open image in new window . In order to find a solution of the variational inequality (1.15), the authors proved the following theorem in the framework of Banach spaces.

Theorem 1.

Let Open image in new window be a uniformly convex and Open image in new window -uniformly smooth Banach space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a sunny nonexpansive retraction from Open image in new window onto Open image in new window , Open image in new window and Open image in new window an Open image in new window -inverse strongly accretive operator of Open image in new window into Open image in new window with Open image in new window , where

converges weakly to some element Open image in new window of Open image in new window , where Open image in new window is the Open image in new window -uniformly smoothness constant of Open image in new window .

In this paper, motivated by Aoyama et al. [6], Iiduka et al. [2], Takahahsi and Toyoda [4], we introduce an iterative method to approximate a solution of variational inequality (1.15) for an Open image in new window -inverse strongly accretive operators. Strong convergence theorems are obtained in the framework of Banach spaces under appropriate conditions on parameters.

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

Lemma 1.2 (see [7]).

Let Open image in new window be a given real number with Open image in new window and let Open image in new window be a Open image in new window -uniformly smooth Banach space. Then

for all Open image in new window , where Open image in new window is the Open image in new window -uniformly smoothness constant of Open image in new window .

The following lemma is characterized by the set of solutions of variational inequality (1.15) by using sunny nonexpansive retractions.

Lemma 1.3 (see [6]).

Let Open image in new window be a nonempty closed convex subset of a smooth Banach space Open image in new window . Let Open image in new window be a sunny nonexpansive retraction from Open image in new window onto Open image in new window and let Open image in new window be an accretive operator of Open image in new window into Open image in new window . Then, for all Open image in new window ,

Lemma 1.4 (see [8]).

Let Open image in new window be a nonempty bounded closed convex subset of a uniformly convex Banach space Open image in new window and let Open image in new window be nonexpansive mapping of Open image in new window into itself. If Open image in new window is a sequence of Open image in new window such that Open image in new window weakly and Open image in new window , then Open image in new window is a fixed point of Open image in new window .

Lemma 1.5 (see [9]).

Let Open image in new window , Open image in new window be bounded sequences in a Banach space Open image in new window and let Open image in new window be a sequence in Open image in new window which satisfies the following condition:
Suppose that

Then Open image in new window .

Lemma 1.6 (see [10]).

Assume that Open image in new window is a sequence of nonnegative real numbers such that

for all Open image in new window , where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(i) Open image in new window ;

(ii) Open image in new window or Open image in new window .

Then Open image in new window .

2. Main Results

Theorem 2.1.

Let Open image in new window be a uniformly convex and Open image in new window -uniformly smooth Banach space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a sunny nonexpansive retraction from Open image in new window onto Open image in new window , Open image in new window an arbitrarily fixed point, and Open image in new window an Open image in new window -inverse strongly accretive operator of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window and let Open image in new window a real number sequence in Open image in new window for some Open image in new window satisfying the following conditions:

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

Then the sequence Open image in new window defined by

converges strongly to Open image in new window , where Open image in new window is a sunny nonexpansive retraction of Open image in new window onto Open image in new window .

Proof.

First, we show that Open image in new window is nonexpansive for all Open image in new window . Indeed, for all Open image in new window and Open image in new window , from Lemma 1.2, one has
Therefore, one obtains that Open image in new window is a nonexpansive mapping for all Open image in new window . For all Open image in new window , it follows from Lemma 1.3 that Open image in new window . Put Open image in new window . Noticing that
from which it follows that
Now, an induction yields
Hence, Open image in new window is bounded, and so is Open image in new window . On the other hand, one has
Next, we compute Open image in new window Observing that
Combining (2.7) with (2.10), one obtains
It follows that
Hence, from Lemma 1.5, we obtain Open image in new window . From (2.7) and the condition (ii), one arrives at
On the other hand, from (2.1), one has
which combines with (2.13), and from the conditions (i), (ii), one sees that
Next, we show that
To show (2.16), we choose a sequence Open image in new window of Open image in new window that converges weakly to Open image in new window such that
Next, we prove that Open image in new window . Since Open image in new window for some Open image in new window , it follows that Open image in new window is bounded and so there exists a subsequence Open image in new window of Open image in new window which converges to Open image in new window . We may assume, without loss of generality, that Open image in new window . Since Open image in new window is nonexpansive, it follows from Open image in new window that
It follows from (2.15) that
From Lemma 1.4, we have Open image in new window . It follows from Lemma 1.3 that Open image in new window . Now, from (2.17) and Lemma 1.1, we have
From (2.1), we have
It follows that

Applying Lemma 1.6 to (2.22), we can conclude the desired conclusion. This completes the proof.

As an application of Theorem 2.1, we have the following results in the framework of Hilbert spaces.

Corollary 2.2.

Let Open image in new window be a Hilbert space and Open image in new window a nonempty closed convex subset of Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window , Open image in new window an arbitrarily fixed point, and Open image in new window an Open image in new window -inverse strongly monotone operator of Open image in new window into Open image in new window such that Open image in new window . Let Open image in new window and Open image in new window be two sequences in Open image in new window and let Open image in new window be a real number sequence in Open image in new window for some Open image in new window satisfying the following conditions:

(i) Open image in new window and Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

Then the sequence Open image in new window defined by

converges strongly to Open image in new window .

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Copyright information

© Yan Hao. 2008

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.School of Mathematics, Physics and Information ScienceZhejiang Ocean UniversityZhoushanChina

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