Abstract
In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.
Highlights
In the deterministic cases, there have been a lot of research to describe the properties of different types of variational inequalities [1,2,3,4], including the sufficient conditions of the existence and the uniqueness of the solutions, and the compactness and connectivity of the solution sets
The above study on the compactness and convexity of the solution set for generalized stochastic Browder mixed variational inequalities (GS-BMVI) (1) can help us deal with general infinite-dimensional problems as with the finite-dimensional problems we considered
4 Conclusion In this paper we introduced a class of generalized stochastic Browder mixed variational inequalities
Summary
There have been a lot of research to describe the properties of different types of variational inequalities [1,2,3,4], including the sufficient conditions of the existence and the uniqueness of the solutions, and the compactness and connectivity of the solution sets. For the bounded, closed, and convex subset K in the reflexive Banach space E, if we weaken the conditions which are presented in conditions (ii) and (iii), the existence of solution for GS-BMVI (1) could be reached.
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