Strong Convergence of an Iterative Method for Inverse Strongly Accretive Operators
© Yan Hao. 2008
Received: 12 May 2008
Accepted: 10 July 2008
Published: 14 July 2008
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.
1. Introduction and Preliminaries
for all It is known that if is a nonexpansive mapping of into itself, then is -inverse strongly monotone and , where denotes the set of fixed points of .
for Moreover, is characterized by the properties and for all
One can see that the variational inequality problem (1.1) is equivalent to some fixed-point problem. The element is a solution of the variational inequality (1.1) if and only if satisfies the relation where is a constant.
To find a solution of the variational inequality for an inverse strongly monotone operator, Iiduka et al.  proved the following weak convergence theorem.
for all , where is the metric projection from onto , is a sequence in and is a sequence in . If and are chosen so that for some with and for some with , then the sequence defined by (1.6) converges weakly to some element of .
- (1)for all with ;
- (2)for all and ;
- (3)for all .
where is a function. It is known that is uniformly smooth if and only if . Let be a fixed real number with . A Banach space is said to be -uniformly smooth if there exists a constant such that for all .
- (1)is a uniformly smooth Banach space if and only if is single-valued and uniformly continuous on any bounded subset of ;
all Hilbert spaces, (or ) spaces ( ), and the Sobolev spaces, ( ), are -uniformly smooth, while (or ) and spaces ( ) are -uniformly smooth.
for all .
for all . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator.
whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for all , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . We know the following lemma concerning sunny nonexpansive retraction.
Lemma 1.1 (see ).
for all and .
for all . In order to find a solution of the variational inequality (1.15), the authors proved the following theorem in the framework of Banach spaces.
converges weakly to some element of , where is the -uniformly smoothness constant of .
In this paper, motivated by Aoyama et al. , Iiduka et al. , Takahahsi and Toyoda , we introduce an iterative method to approximate a solution of variational inequality (1.15) for an -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 ).
for all , where is the -uniformly smoothness constant of .
The following lemma is characterized by the set of solutions of variational inequality (1.15) by using sunny nonexpansive retractions.
Lemma 1.3 (see ).
Lemma 1.4 (see ).
Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and let be nonexpansive mapping of into itself. If is a sequence of such that weakly and , then is a fixed point of .
Lemma 1.5 (see ).
Lemma 1.6 (see ).
for all , where is a sequence in and is a sequence in such that
(ii) or .
2. Main Results
Let be a uniformly convex and -uniformly smooth Banach space and a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , an arbitrarily fixed point, and an -inverse strongly accretive operator of into such that . Let and be two sequences in and let a real number sequence in for some satisfying the following conditions:
(i) and ;
converges strongly to , where is a sunny nonexpansive retraction of onto .
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.
Let be a Hilbert space and a nonempty closed convex subset of . Let be a metric projection from onto , an arbitrarily fixed point, and an -inverse strongly monotone operator of into such that . Let and be two sequences in and let be a real number sequence in for some satisfying the following conditions:
(i) and ;
converges strongly to .
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