Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces

Abstract

In this paper, utilizing the properties of the generalized f -projection operator and the well-known KKM and Kakutani---Fan---Glicksberg theorems, under quite mide assumptions, we derive some new existence theorems for the generalized set-valued mixed variational inequality and the generalized set-valued mixed quasi-variational inequality in reflexive and smooth Banach spaces, respectively. The results presented in this paper can be viewed as the supplement, improvement and extension of recent results in Wu and Huang (Nonlinear Anal 71:2481---2490, 2009).