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.
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 :
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 .
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 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 ).
The following lemma concerns the sunny nonexpansive retraction.
The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck .
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
3. Main Results
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.
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.
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
Corollary 3.5 (see [5, Theorem 3.1]).
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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