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

Abstract

Abstract Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let A 1 , A 2 , … , A N $A_{1}, A_{2},\ldots,A_{N}$ be N-variables monotone demi-continuous mappings from K N $K^{N}$ into X. Then: (1) the system of multivariate variational inequalities { 〈 A 1 ( x 1 , x 2 , … , x N ) , y 1 − x 1 〉 ≥ 0 , ∀ y 1 ∈ K , 〈 A 2 ( x 1 , x 2 , … , x N ) , y 2 − x 2 〉 ≥ 0 , ∀ y 2 ∈ K , ⋯ 〈 A N ( x 1 , x 2 , … , x N ) , y N − x N 〉 ≥ 0 , ∀ y N ∈ K , $$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$ has a solution ( x 1 ∗ , x 2 ∗ , … , x N ∗ ) ∈ K N $(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$ ; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in K N $K^{N}$ ; (3) if A 1 , A 2 , … , A N $A_{1}, A_{2},\ldots,A_{N}$ are also strictly monotone, this system of multivariate variational inequalities has a unique solution.

Keywords

• variational inequality
• system of variational inequalities
• reflexive Banach space
• dual space
• solution