A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces
© Uthai Kamraksa and Rabian Wangkeeree. 2010
Received: 6 April 2010
Accepted: 14 June 2010
Published: 5 July 2010
The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.
From , we know the following property.
The best constant in the above inequality is called the -uniformly smoothness constant of (see  for more details).
It is known that, if is smooth, then is single valued. Recall that the duality mapping is said to be weakly sequentially continuous if, for each sequence with weakly, we have weakly- . We know that, if admits a weakly sequentially continuous duality mapping, then is smooth. For the details, see .
( ) Every -strongly accretive mapping is a -relaxed cocoercive mapping for any positive constant but the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.
( ) Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., ).
( ) The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods . Several classes of relaxed cocoercive variational inequalities have been studied in [5, 6].
Next, we consider a system of quasivariational inclusions as follows.
As special cases of problem (1.11), we have the following.
In 2006, Aoyama et al.  considered the following problem.
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 .
The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 1.3 (see ).
Proposition 1.4 (see ).
It is unclear, in general, what the behavior of is as even if has a fixed point. However, in the case of having a fixed point, Ceng et al.  proved that, if is a Hilbert space, then converges strongly to a fixed point of . Reich  extended Browder's result to the setting of Banach spaces and proved that, if is a uniformly smooth Banach space, then converges strongly to a fixed point of , and the limit defines the (unique) sunny nonexpansive retraction from onto .
Reich  showed that, if is uniformly smooth and is the fixed point set of a nonexpansive mapping from into itself, then there is a unique sunny nonexpansive retraction from onto and it can be constructed as follows.
Proposition 1.6 (see ).
Definition 1.7 (see ).
Recently, many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inequalities (1.11)–(1.14) by using different iterative methods (see, e.g., [7, 14–16]).
Very recently, Qin et al.  considered the problem of finding the solutions of a general system of variational inclusion (1.11) with -inverse strongly accretive mappings. To be more precise, they obtained the following results.
Lemma 1.8 (see ).
Theorem QCCK (see [16, Theorem ]).
On the other hand, we recall the following well-known definitions and results.
where is nonself- -strict pseudo-contraction, is a contraction, and is a strongly positive bounded linear operator on a Hilbert space . They proved, under certain appropriate conditions imposed on the sequences and , that defined by (1.33) converges strongly to a fixed point of , which solves some variational inequality.
In this paper, motivated by Qin et al. , Moudafi , Marino and Xu , and Qin et al. , we introduce a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions (1.11) with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. We prove the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in strictly convex and -uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Qin et al. , Moudafi , Marino and Xu , Qin et al. , and many others.
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Definition 2.7 (see ).
Lemma 2.9 (see [27, Lemma ]).
Lemma 2.10 (see ).
Lemma 2.11 (see ).
Lemma 2.12 (see [17, Lemma ]).
3. Main Results
In this section, we prove that the strong convergence theorem for a countable family of uniformly -strict pseudocontractions in a strictly convex and -uniformly smooth Banach space admits a weakly sequentially continuous duality mapping. Before proving it, we need the following theorem.
Theorem 3.1 (see [17, Lemma ]).
Theorem 3.4 mainly improves Theorem of Qin et al. , in the following respects:
(a)from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings,
As in [27, Theorem ], we can generate a sequence of nonexpansive mappings satisfying AKTT-condition; that is, for any bounded subset of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.
Lemma 3.9 (see ).
Then the following are given.
where satisfies conditions (i)–(iii) of Lemma 3.9, , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions:
The following example appears in  shows that there exists satisfying the conditions of Lemma 3.9.
The first author is supported under grant from the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the second author is supported by the "Centre of Excellence in Mathematics" under the Commission on Higher Education, Ministry of Education, Thailand. Finally, The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
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