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Let  K  be a nonempty closed convex and bounded subset of a reflexive Banach space  X . Let    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}    A  1   ,   A  2   ,  …  ,   A  Nbe  N -variables monotone demi-continuous mappings from    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$K^{N}$\end{document}    K  Ninto  X . Then: (1) the system of multivariate variational inequalities    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$\end{document}    {〈   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  ,has a solution    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$\end{document}   (   x  1  ∗   ,   x  2  ∗   ,  …  ,   x  N  ∗   )  ∈   K  N; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$K^{N}$\end{document}    K  N; (3) if    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}    A  1   ,   A  2   ,  …  ,   A  Nare also strictly monotone, this system of multivariate variational inequalities has a unique solution.

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