Set-valued mixed quasi-variational inequalities and implicit resolvent equations

In this paper, we introduce and study a new class of variational inequalities, which is called the set-valued mixed quasi-variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued mixed quasi-variational inequalities, fixed-point problems and the set-valued implicit resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.