Iterative approximation of solution of generalized mixed set-valued variational inequality problem

Abstract

Abstract. In this paper, we consider a generalized mixed set-valued variational inequality problem which includes many important known variational inequality problems and equilibrium problem, and its related some auxiliary variational inequality problems. We prove the existence of solutions of the auxiliary variational inequality problems and suggest a two-step iterative algorithm and an inertial proximal iterative algorithm. Further, we discuss the convergence analysis of iterative algorithms. The theorems presented in this paper improve and generalize many known results for solving equilibrium problems, variational inequality and complementarity problems in the literature.

Existence Results for Vector Mixed Quasi-Complementarity Problems

“…Every monotone with respect to is pseudomonotone with respect to but converse does not hold in general. Definition 3 is vector version of -pseudomonotonicity studied by Kazmi et al in [23,24].…”

A viscosity Cesàro mean approximation method for split generalized vector equilibrium problem and fixed point problem

“…We denote the solution set of EP(1.1) by sol(EP(1.1)). In the last two decades, EP(1.1) has been generalized and extensively studied in many directions due to its importance; see for example [2][3][4][5][6][7][8][9][10] for the literature on the existence and iterative approximation of solution of the various generalizations of EP(1.1). Recently, Kazmi and Rizvi [11] considered the following pair of equilibrium problems in different spaces, which is called split equilibrium problem (in short, SEP): Let F 1 : C Â C !…”

Two Step Algorithm for Solving Regularized Generalized Mixed Variational Inequality Problem

“…(2) Regularized γ-partially relaxed strongly θ-psedomonotonicity of F generalize the concepts of partially relaxed strongly jointly pseudomonotonicity of F given by Xia and Ding [19] and θ-pseudomonotonicity of F given by Kazmi et al [9]. …”

“…(ii) [9] N is said to be µ-partially relaxed strongly mixed monotone with respect to A and B if there exists a constant η > 0 such that…”

“…We give the concepts of partially relaxed strongly mixed monotone and regularized partially relaxed strongly θ-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al [9]. Further, we use the auxiliary principle technique to suggest a twostep iterative algorithm for approximating the solution of RGMVIP.…”