© J.-W. Peng and L.-J. Zhao. 2009
Received: 25 September 2009
Accepted: 2 November 2009
Published: 3 November 2009
We introduce and study a new system of nonlinear variational inclusions involving a combination of -Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of the -monotone operators, we prove the existence and uniqueness of solution and the convergence of a new multistep iterative algorithm for this system of variational inclusions. The results in this paper unify, extend, and improve some known results in literature.
Recently, Fang and Huang  introduced a new class of -monotone mappings in the context of solving a system of variational inclusions involving a combianation of -monotone and strongly monotone mappings based on the resolvent operator techniques. The notion of the -monotonicity has revitalized the theory of maximal monotone mappings in several directions, especially in the domain of applications. Verma  introduced the notion of -monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of -monotone and strongly monotone mappings. As Verma point out "the class of -monotone mappings generalizes -monotone mappings. On the top of that, -monotonicity originates from hemivariational inequalities, and emerges as a major contributor to the solvability of nonlinear variational problems on nonconvex settings." and as a matter of fact, some nice examples on -monotone (or generalized maximal monotone) mappings can be found in Naniewicz and Panagiotopoulos  and Verma . Hemivariational inequalities—initiated and developed by Panagiotopoulos —are connected with nonconvex energy functions and turned out to be useful tools proving the existence of solutions of nonconvex constrained problems. It is worthy noting that -monotonicity is defined in terms of relaxed monotone mappings—a more general notion than the monotonicity or strong monotonocity—which gives a significant edge over the -monotonocity. Very recently, Verma  studied the solvability of a system of variational inclusions involving a combination of -monotone and relaxed cocoercive mappings using resolvent operator techniques of -monotone mappings. Since relaxed cocoercive mapping is a generalization of strong monotone mappings, the main result in  is more general than the corresponding results in [1, 2].
Inspired and motivated by recent works in [1, 2, 6], the purpose of this paper is to introduce a new mathematical model, which is called a general system of -monotone nonlinear variational inclusion problems, that is, a family of -monotone nonlinear variational inclusion problems defined on a product set. This new mathematical model contains the system of inclusions in [1, 2, 6], the variational inclusions in [7, 8], and some variational inequalities in literature as special cases. By using the resolvent technique for the -monotone operators, we prove the existence and uniqueness of solution for this system of variational inclusions. We also prove the convergence of a multistep iterative algorithm approximating the solution for this system of variational inclusions. The result in this paper unifies, extends, and improves some results in [1, 2, 6–8] and the references therein.
We suppose that is a real Hilbert space with norm and inner product denoted by and , respectively, denotes the family of all the nonempty subsets of . If be a set-valued operator, then we denote the effective domain of as follows:
Now we recall some definitions needed later.
If , , then the definition of -monotonicity is that of -monotonicity in [1, 8]. It is easy to know that if ( the identity map on ), then the definition of -monotone operators is that of maximal monotone operators. Hence, the class of -monotone operators provides a unifying frameworks for classes of maximal monotone operators, -monotone operators. For more details about the above definitions, please refer to [1–8] and the references therein.
It follow from [3, Lemma ] we know that if is a reflexive Banach space with its dual, and be -strongly monotone and is a locally Lipschitz such that is -relaxed monotone, then is -monotone with a constant .
Definition 2.4 . (see ).
We also need the following result obtained by Verma .
One needs the following new notions.
3. A System of Set-Valued Variational Inclusions
Below are some special cases of (3.1).
4. Existence of Solutions and Convergence of an Iterative Algorithm
In this section, we will prove existence and uniqueness of solution for (3.1). For our main results, we give a characterization of the solution of (3.1) as follows.
Then, (3.1) admits a unique solution.
Then, problem (3.1) admits a unique solution.
5. Iterative Algorithm and Convergence
In this section, we will construct some multistep iterative algorithm for approximating the unique solution of (3.1) and discuss the convergence analysis of these Algorithms.
This study was supported by grants from National Natural Science Foundation of China (project no. 70673012, no. 70741028 and no. 90924030), China National Social Science Foundation (project no. 08CJY026).
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