The Solvability of a New System of Nonlinear Variational-Like Inclusions
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Research Article The Solvability of a New System of Nonlinear Variational-Like Inclusions Zeqing Liu,1 Min Liu,1 Jeong Sheok Ume,2 and Shin Min Kang3 1
Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian Liaoning 116029, China Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea 3 Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, South Korea 2
Correspondence should be addressed to Jeong Sheok Ume, [email protected] Received 23 November 2008; Accepted 1 April 2009 Recommended by Marlene Frigon We introduce and study a new system of nonlinear variational-like inclusions involving s-G, ηmaximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, λ, ξ-relaxed cocoercive operators, ζ, ϕ, g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with s-G, η-maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature. Copyright q 2009 Zeqing Liu et al. 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 It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions 1–20. In particular, the generalized resolvent operator technique has been applied more and more and has also been improved intensively. For instance, Fang and Huang 5 introduced the class of H-monotone operators and defined the associated resolvent operators, which extended the resolvent operators associated with ηsubdifferential operators of Ding and Luo 3 and maximal η-monotone operators of Huang and Fang 6, respectively. Later, Liu et al. 17 researched a class of general nonlinear implicit variational inequalities including the H-monotone operators. Fang and Huang 4 created a class of H, η-monotone operators, which offered a unifying framework for the classes of maximal monotone operators, maximal η-monotone operators and H-monotone operators. Recently, Lan 8 introduced a class of A, η-accretive operators which further
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Fixed Point Theory and Applications
enriched and improved the class of generalized resolvent operators. Lan 10 studied a system of general mixed quasivariational inclusions involving A, η-accretive mappings in q-uniformly smooth Banach spaces. Lan et al. 14 constructed some iterative algorithms for solving a class of nonlinear A, η-monotone operator inclusion systems involving nonmonotone set-valued mappings in Hilbert spaces. Lan 9 investigated the existence of so
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