Random Variational Inequalities Non-trivial Solution

Topological Degree theory and fixed-point index theory are basic tools to research the theory of nonlinear operator equations' quality. We can obtain many fixed-point theorems by using fixed-point index. Now some single-valued and set-valued fixed-point index have been randomize. The purpose of this paper is to apply random fixed-point index to settle random variational inequalities.First, this paper gives the relationship between random variational inequalities and random fixed-point. Second, we study the existence of random variational inequalities accord with many conditions in R" space and real reflexive Banach space, breaking previous solution for random variational inequalities. This paper is the first in applying random fixed-point index to settle random variational inequalities,assuming that the solution is uncommon. On single-valued and set-valued condition, we conclude the condition of the existence of nonzero solution for random variational inequalities.