A kind of system of multivariate variational inequalities and the existence theorem of solutions
Yanxia Tang, Jinyu Guan, Yongchun Xu, Yongfu Su · Journal of Inequalities and Applications · 2017
Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let $A_{1}, A_{2},\ldots,A_{N}$ be N-variables monotone demi-continuous mappings from $K^{N}$ into X. Then: (1) the system of multivariate variational inequalities $$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$ has a solution $(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$ ; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in $K^{N}$ ; (3) if $A_{1}, A_{2},\ldots,A_{N}$ are also strictly monotone, this system of multivariate variational inequalities has a unique solution.