Abstract
We introduce and study a new system of random nonlinear generalized variational inclusions involving random fuzzy mappings and set-valued mappings with -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.
Highlights
The classic variational inequality problem VI F, K is to determine a vector x∗ ∈ K ⊂ Rn, such thatF x∗ 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
Due to its enormous applications in solving problems arising from the fields of economics, mechanics, physical equilibrium analysis, optimization and control, transportation 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
Motivated and inspired by recent research work mentioned above in this field, in this paper, we try to inject some new energy into this interesting field by studying on a new kind of random nonlinear variational inclusions in two Hilbert spaces
Summary
The classic variational inequality problem VI F, K is to determine a vector x∗ ∈ K ⊂ Rn, such that. The monotonic properties of associated operators play essential roles in proving the existence of solutions and the convergence of sequences generated by iterative algorithms. Fang and Huang, Verma, and Cho and Lan investigated many generalized operators such as Hmonotone, H-accretive, H, η -monotone, H, η -accretive, and A, η -accretive mappings. Huang studied the random generalized nonlinear variational inclusions for random fuzzy mappings. In 2005, Ahmad and Bazan studied a class of random generalized nonlinear mixed variational inclusions for random fuzzy mappings and constructed an iterative algorithm for solving such random problems. Motivated and inspired by recent research work mentioned above in this field, in this paper, we try to inject some new energy into this interesting field by studying on a new kind of random nonlinear variational inclusions in two Hilbert spaces. For a suitable choice of some mappings, we can obtain several known results 10, 11, 21, 23, 31, 34 as special cases of the main results of this paper
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