A Class Of Variational-like Inequalities, Vector F-Complementarity Problem, General Vector Variational Inequality

The origin of variational inequality can be traced back to the study concerning partial differential equation by Stampacchia and Hartman in the I960' s. However, variational inequality was developed as a means of studying partial differential equation. Stampacchia and Hartman studied variational inequality as follows:(VIP) x∈K:≥0, y∈K.The variational inequality has close relations with the theory of extreme value theory and partial differential equation, therefore, it has been extended to infinite dimensional space during the progress of relevant studies.With further studies, more different variational inequalities have been discussed, tow of which this paper attempts to deal with.1. Variational-like inequality. By means of the concept of η -subdifferential, this paper employs a new method and constructs resolvent formula of η-subdifferential operators. Using techniques of resolvent formula of η -subdifferential operators and auxiliary principle, we presented the existence and uniqueness of the solution of a class of variational inequalities.2. Vector F-complementarity problem, general vector variational inequality, and least element problem of feasible set. In this paper, several classes of vector F-complementarity problems, which are the vector form of F-complementarity problems(FCP), are introduced and studied. The relations of the vector F-complementarity problems, general vector variational inequalities, and least element problems of feasible set are given under certain conditions.