Generalized set-valued variational inequalities

Abstract

In this paper, we introduce and study a new class of variational inequalities, which is called generalized set-valued variational inequality. The projection technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems and generalized set-valued Wiener-Hopf 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. We also consider the auxiliary principle technique to study the existence of a solution of the generalized set-valued variational inequalities and to suggest a general and novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problem for the generalized set-valued variational inequalities. The results proved in this paper represent a significant refinement and improvement of the previous results.