Fuzzy resolution on the infeasibility of variational inequality

In this study, we discuss variational inequality problems when there is no solution. By applying fuzzy set theory, we ‘soften’ the original problem with fuzzy relation and denote it by VI(X, f). Then the solutions which satisfy the variation inequality to some degree μ, 0 ⩽ μ ⩽ 1 are searched with minimum relaxation. According to the proposed existence theorem, an approach derived from the concepts of single objective mathematical programming problems for solving the VI(X,f) is proposed to find its fuzzy solution set.