Weighted Variational Inequalities with Set-valued Mappings

Because of their applications in economics, game theory, mathematical physics, operations research and other areas, many classes of vector variational inequalities were intensively studied. For existence of solutions, resolution methods or equivalence with equilibrium and optimization problems see, for example, [11], [14], [15], [19], [17] and the references therein. For the study of variational inequalities systems in the single valued case, see [16] and [18]. Ansari, Schaible and Yao [4] introduced a system of vector equilibrium problems with vector valued bifunction defined over a product set, which includes as particular cases systems of generalized vector variational inequalities and optimization problems. The same authors [5] defined another systems of vector equilibrium problems and obtained an existence theorem based on a known maximal element theorem. From this, they derived existence results for some systems of vector variational-like inequalities and systems of implicit vector variational inequalities. Ansari and Khan [2] proved the existence of solution for a variational inequality over product sets, which is equivalent to a system of vector variational inequalities, under generalized pseudo-monotonicity assumptions, in the meaning of Brezis [8]. They used a fixed point theorem of Chowdury and Tan [9].