A kind of system of multivariate variational inequalities and the existence theorem of solutions

摘要

Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let A1, A2, …, AN be N-variables monotone demi-continuous mappings from K^N into X. Then: (1) the system of multivariate variational inequalities {〈A1(x1,x2,…,xN),y1−x1〉≥0,∀y1∈K,〈A2(x1,x2,…,xN),y2−x2〉≥0,∀y2∈K,⋯〈AN(x1,x2,…,xN),yN−xN〉≥0,∀yN∈K, has a solution (x1∗,x2∗,…,xN∗)∈KN; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in K^N; (3) if A1, A2, …, AN are also strictly monotone, this system of multivariate variational inequalities has a unique solution.