Bregman type regularization of variational inequalities with Mosco approximation of the constraint set

In this paper, we consider a regularized variational inequality for set-valued maps which is defined by means of a Bregman function and prove the existence of its solution by using the equilibrium problem technique. We use regularization technique to prove the existence of solutions of variational inequalities for set-valued maps under premonotonicity assumption which is weaker than the monotonicity of the set-valued map involved in the formulation of the problem. We establish sufficient conditions which guarantee that the sequence of strong solutions of regularized problem admits weak cluster points, and each weak cluster point of this sequence is a strong solution of the variational inequality.