Abstract
The main purpose of this paper is to introduce and study a new class of generalized nonlinear set-valued quasi-variational inclusions system involving -accretive mappings in Banach spaces. By using the resolvent operator due to Lan-Cho-Verma associated with -accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point three-step iterative algorithm for this system of set-valued variational inclusions and an existence theorem of solutions for this kind of the variational inclusions system. The results presented in this paper generalize, improve, and unify some recent results in this field.
Highlights
The variational inclusion, which was introduced and studied by Hassouni and Moudafi 1, is a useful and important extension of the variational inequality
Various variational inclusions have been intensively studied in recent years
Ding and Luo 2, Verma 3, 4, Huang 5, Fang et al 6, Fang and Huang 7, Fang et al 8, Lan et al 9, Zhang et al 10 introduced the concepts of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, H, η -monotone operators, A, η -accretive mappings, G, η -monotone operators, and defined resolvent operators associated with them, respectively
Summary
The variational inclusion, which was introduced and studied by Hassouni and Moudafi 1 , is a useful and important extension of the variational inequality. S. Kim introduced a new system of generalized nonlinear quasi-variational inequalities and obtained some existence and uniqueness results on solutions for this system of generalized nonlinear quasi-variational inequalities in Hilbert spaces. Cho et al introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces They proved some existence and uniqueness theorems for solutions for the system of nonlinear variational inequalities. By using the resolvent operator associated with A, η -accretive operator due to Lan, an existence theorem of solution for this class of variational inclusions is proved, and a new hybrid proximal point algorithm is established and suggested, and the convergence of iterative sequences generated by the algorithm is discussed in q-uniformly smooth Banach spaces. The results presented in this paper generalize, and unify some recent results in this field
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