- Research Article
- Open Access
Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces
© S. Imnang and S. Suantai. 2010
- Received: 26 July 2010
- Accepted: 30 December 2010
- Published: 4 January 2011
We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces. Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob.
- Banach Space
- Variational Inequality
- Nonexpansive Mapping
- Variational Inequality Problem
- Real Banach Space
In particular, if , the mapping is called the normalized duality mapping and usually, we write . If is a Hilbert space, then . Further, we have the following properties of the generalized duality mapping :
(1) for all with ,
(2) for all and ,
(3) for all .
It is known that if is smooth, then is single-valued, which is denoted by . Recall that the duality mapping is said to be weakly sequentially continuous if for each with weakly, we have weakly- . We know that if admits a weakly sequentially continuous duality mapping, then is smooth. For the details, see the work of Gossez and Lami Dozo in .
for all . A mapping is said to be nonexpansive if for all . The fixed point set of is denoted by .
whenever for and . A mapping is called a retraction if for all . Furthermore, is a sunny nonexpansive retraction from onto if is a retraction from onto which is also sunny and nonexpansive.
A subset of is called a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction from onto . It is well known that if is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto .
where is a Hilbert space and is a mapping from into . The variational inequality has been widely studied in the literature; see, for example, the work of Chang et al. in , Zhao and He , Plubtieng and Punpaeng , Yao et al.  and the references therein.
which is defined by Verma  and is called the new system of variational inequalities. Further, if we add up the requirement that , then problem (1.11) reduces to the classical variational inequality .
The problem (1.12) is very interesting as it is connected with the fixed point problem for nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces, see [9–11] and the references therein.
where , are two sequences in and .
where , , and are three sequences in and .
where for all and , are two sequences in . Using the demiclosedness principle for nonexpansive mapping, we will show that the sequence converges strongly to a solution of a new general system of variational inequalities in Banach spaces under some control conditions.
In this section, we recall the well known results and give some useful lemmas that will be used in the next section.
Lemma 2.1 (see ).
for all , where is the -uniformly smooth constant of .
The following lemma concerns the sunny nonexpansive retraction.
for all and .
The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck .
If is strictly convex and uniformly smooth and if is a nonexpansive mapping having a nonempty fixed point set , then there exists a sunny nonexpansive retraction of onto .
Lemma 2.4 (see ).
where is a sequence in and is a sequence such that
(ii) or .
Lemma 2.5 (see ).
Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .
Lemma 2.6 (see ).
Let be a uniformly convex Banach space, a nonempty closed convex subset of , and be an nonexpansive mapping. Then is demiclosed at 0, that is, if weakly and strongly, then .
In this section, we establish the equivalence between the new general system of variational inequalities (1.17) and some fixed point problem involving a nonexpansive mapping. Using the demiclosedness principle for nonexpansive mapping, we prove that the iterative scheme (1.18) converges strongly to a solution of a new general system of variational inequalities (1.17) in a Banach space under some control conditions. In order to prove our main result, the following lemmas are needed.
The next lemmas are crucial for proving the main theorem.
where is the 2-uniformly smooth constant of . In particular, if , then is a nonexpansive mapping.
It is clear that, if , then is a nonexpansive mapping.
If for all , then is nonexpansive.
From Lemma 3.1, we have is nonexpansive which implies by (3.4) that is nonexpansive.
Let be a nonempty closed convex subset of a real smooth Banach space . Let be the sunny nonexpansive retraction from onto . Let be three nonlinear mappings. For given , is a solution of problem (1.17) if and only if , and , where is the mapping defined as in Lemma 3.2.
It is easy to see that (3.6) is equivalent to , and .
From now on we denote by the set of all fixed points of the mapping . Now we prove the strong convergence theorem of algorithm (1.18) for solving problem (1.17).
Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping. Let be the sunny nonexpansive retraction from onto . Let the mappings be -inverse strongly accretive with , for all and . For given , let the sequence be generated iteratively by (1.18). Suppose the sequences and are two sequences in such that
(C1) and ;
Then converges strongly to where is the sunny nonexpansive retraction of onto .
which implies as .
It follows from Lemma 2.4, (3.24), and (3.26) that converges strongly to . This completes the proof.
Letting and for in Theorem 3.4, we obtain the following result.
Corollary 3.5 (see [5, Theorem 3.1]).
where , are two sequences in such that
(C1) and ;
Then converges strongly to where is the sunny nonexpansive retraction of onto .
The authors wish to express their gratitude to the referees for careful reading of the manuscript and helpful suggestions. The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin university, the Centre of Excellence in Mathematics, and the Graduate School of Chiang Mai University, Thailand for their financial support.
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