Regularization of Brezis pseudomonotone variational inequalities

In this paper we prove the existence of solutions of regularized set-valued variational\r\ninequalities involving Br´ezis pseudomonotone operators in reflexive and locally uniformly\r\nconvex Banach spaces. By taking advantage of this result, we approximate a general setvalued\r\nvariational inequality with suitable regularized set-valued variational inequalities,\r\nand we show that their solutions weakly converge to a solution of the original one. Furthermore,\r\nby strengthening the coercivity conditions, we can prove the strong convergence of\r\nthe approximate solutions.