General System of -Monotone Nonlinear Variational Inclusions Problems with Applications
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Research Article General System of A-Monotone Nonlinear Variational Inclusions Problems with Applications Jian-Wen Peng1 and Lai-Jun Zhao2 1 2
College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China Management School, Shanghai University, Shanghai 200444, China
Correspondence should be addressed to Lai-Jun Zhao, zhao [email protected] Received 25 September 2009; Accepted 2 November 2009 Recommended by Vy Khoi Le We introduce and study a new system of nonlinear variational inclusions involving a combination of A-Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of the A-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. Copyright q 2009 J.-W. Peng and L.-J. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Recently, Fang and Huang 1 introduced a new class of H-monotone mappings in the context of solving a system of variational inclusions involving a combianation of Hmonotone and strongly monotone mappings based on the resolvent operator techniques. The notion of the H-monotonicity has revitalized the theory of maximal monotone mappings in several directions, especially in the domain of applications. Verma 2 introduced the notion of A-monotone mappings and its applications to the solvability of a system of variational inclusions involving a combination of A-monotone and strongly monotone mappings. As Verma point out “the class of A-monotone mappings generalizes H-monotone mappings. On the top of that, A-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 A-monotone or generalized maximal monotone mappings can be found in Naniewicz and Panagiotopoulos 3 and Verma 4. Hemivariational inequalities—initiated and developed by Panagiotopoulos 5— 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
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Journal of Inequalities and Applications
A-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 Hmonotonocity. Very recently, Verma 6 studied the solvability of a system of variational inclusions involving a combination of A-monotone and relaxed cocoercive mappings using resolvent operator techniques of A-monotone mappings. Since relaxed cocoercive mapping is a generalization of strong monotone mappings, the main result in 6 is more general than the
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