A System of Random Nonlinear Variational Inclusions Involving Random Fuzzy Mappings and -Monotone Set-Valued Mappings
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Research Article A System of Random Nonlinear Variational Inclusions Involving Random Fuzzy Mappings and H·, ·-Monotone Set-Valued Mappings Xin-kun Wu and Yun-zhi Zou College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Yun-zhi Zou, [email protected] Received 8 June 2010; Accepted 24 July 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2010 X.-k. Wu and Y.-z. Zou. 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. We introduce and study a new system of random nonlinear generalized variational inclusions involving random fuzzy mappings and set-valued mappings with H·, ·-monotonicity in two Hilbert spaces and develop a new algorithm which produces four random iterative sequences. We also discuss the existence of the random solutions to this new kind of system of variational inclusions and the convergence of the random iterative sequences generated by the algorithm.
1. Introduction The classic variational inequality problem VIF, K is to determine a vector x∗ ∈ K ⊂ Rn , such that
Fx∗ T , x − x∗ ≥ 0,
∀x ∈ K,
1.1
where F is a given continuous function from K to Rn and K is a given closed convex subset of the n-dimensional Euclidean space Rn . This is equivalent to find an x∗ ∈ K, such that 0 ∈ Fx∗ N⊥ x∗ ,
1.2
where N⊥ is normal cone operator. Due to its enormous applications in solving problems arising from the fields of economics, mechanics, physical equilibrium analysis, optimization and control, transportation
2
Journal of Inequalities and Applications
equilibrium, and linear or nonlinear programming etcetera, variational inequality and its generalizations have been extensively studied during the past 40 years. For details, we refer readers to 1–7 and the references therein. It is not a surprise that many practical situations occur by chance and so variational inequalities with random variables/mappings have also been widely studied in the past decade. For instance, some random variational inequalities and random quasivariational inequalities problems have been introduced and studied by Chang 8, Chang and Huang 9, 10, Chang and Zhu 11, Huang 12, 13, Husain et al. 14, Tan et al. 15, Tan 16, and Yuan 7. It is well known that one of the most important and interesting problems in the theory of variational inequalities is to develop efficient and implementable algorithms for solving variational inequalities and its generalizations. The monotonic properties of associated operators play essential roles in proving the existence of solutions and the convergence of sequences generated by iterative algorithms. In 2001, Huang and Fang 17 were the first to introduce the generalized m-accretive mapping and give the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. They also showed some properties of the resolvent o
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