On the convergence of some iteration processes for $J$-pseudomonotone mixed variational inequalities in uniformly smooth Banach spaces

Abstract

This paper aims at studying the convergence of some iteration processes for mixed variational inequalities with convex nondifferentiable functionals and $J$-potential, $J$-coercive, $J$-pseudomonotone or $J$-strongly inverse monotone operators in uniformly smooth Banach spaces. Our results generalize corresponding theorems of [I.B. Badriev, O.A. Zadvornov, A decomposition method for variational inequalities of the second kind with strongly inverse-monotone operators, Differ. Equ. 39 (7) (2003) 936–944; I.B. Badriev, O.A. Zadvornov, A.D. Lyashko, A study of variable step iterative methods for variational inequalities of the second kind, Differ. Equ. 40 (7) (2004) 971–983] and [I.B. Badriev, O.A. Zadvornov, A.M. Saddeek, On the iterative methods for solving some variational inequalities of the second kind, in: Contemporary Problems of Mathematical Modeling (Materials of the IX All-Russian School-Seminar, 8–13 September 2001, Abrau-Dyrso), Rostov University Publishers, Rostov-Un-Don, 2001, pp. 36–41].