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

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}A1,A2,…,AN be 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}KN into 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}{〈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 \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}(x1∗,x2∗,…,xN∗)∈KN; (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}KN; (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}A1,A2,…,AN are also strictly monotone, this system of multivariate variational inequalities has a unique solution.


Introduction
Let X be a Banach space with the dual space X * and let ·, · denote the duality pairing of X and X * . Let K be a nonempty closed convex subset of X, A : K → X * a mapping. The classical variational inequality problem is to find x ∈ K such that Ax, yx ≥ , ∀y ∈ K. (.) The variational inequality problem has been recognized as one of the suitable mathematical models for dealing with many problems arising in different fields, such as optimization theory, game theory, economic equilibrium, mechanics. In the last four decades, since the time of the celebrated Hartman-Stampacchia theorem (see [, ]), the existence of a solution of a variational inequality and other related problems has become a basic research topic, which continues to attract attention of researchers in applied mathematics (see e.g., [-] and the references therein). In , Hartman and Stampacchia [] proved the following result.

Theorem . ([]) Let K be a nonempty closed convex and bounded subset of R n . Let A :
K → R n be a continuous mapping. Then the variational inequality (.) has a solution x * ∈ K .
In , Browder proved the following more general result (see []).
Theorem . ([]) Let K be a nonempty compact convex subset of a locally convex topological vector space X. Let A : K → X * be a continuous mapping. Then the variational inequality (.) has a solution x * ∈ K .
The variational inequality (.) is called the Hartman-Stampacchia variational inequality. It is an important classical variational inequality which is also a classical and powerful tool in nonlinear analysis and other mathematical fields.
A is said to be demi-continuous at x  , if for any given y ∈ X, A(x  + t n y) weak * converges to A(x  ) wherever t n → , t n ≥ .
In , Chang [] proved the following result in reflexive Banach spaces.

Theorem . ([]) Let K be a nonempty closed convex and bounded subset of a reflexive
Banach space X. Let A : K → X * be a monotone demi-continuous mapping. Then () the variational inequality (.) has a solution x * ∈ K ; () the set of solutions of (.) is closed convex; () if A is strictly monotone, then (.) has a unique solution.
In , Plubtieng and Sombut [] proved the following result.
Theorem . Let X be a reflexive Banach space, let K be a compact convex subset of X, and let A, B : K → X * be two continuous mappings. Then the system of variational inequalities has a solution (x, y) ∈ K × K and the set of solutions of (.) is closed.
Multivariate calculus is a more general mathematical branch which paly a more important role in mathematical and applied fields. In recently, multivariate fixed point theorems and the system of N -variables nonlinear operators have been studied by some authors. Many interesting results and the applications have also been given. In , Su et al. The purpose of this paper is to study a kind of system of multivariate variational inequalities and to prove the existence theorem of solutions. The results of this paper improve and extend the results of [, ] in reflexive Banach spaces. In order to get the expected results, an ingenious mathematical method is used in this paper.

Preliminaries
Let us introduce some conclusions which will be useful for our main results.

Lemma . ([]
) Let X be a Banach space with the norm · . We consider on the Cartesian product space X N = X × X × · · · × X the following functional: Then (X N , · * ) is a reflexive Banach space.

Main results
Let K be a nonempty subset of a normed space X, A i : K N → X * a N -variables mapping for all i = , , . . . , N . We consider the following system of multivariate variational inequalities: for all (x  , x  , . . . , x N ), (y  , y  , . . . , y N ) ∈ K N . A N -variables monotone mapping is said to be strictly monotone, if implies (x  , x  , . . . , x N ) = (y  , y  , . . . , y N ).
The following is the main result of this paper.
Theorem . Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let A i : K N → X * be a N -variables monotone demi-continuous mapping for all i = , , . . . , N . Then: From Lemma . and Lemma ., we know (X * ) N = (X N ) * and hence A * is a mapping from K N into (X N , · * ) * .
Next, we prove that A * is a monotone mapping from K N into (X N , · * ) * . Since A i is a monotone mapping from K into (X, · ) * for all i = , , . . . , N , We also need to prove A * is demi-continuous on K N . For any given x  ∈ K N and any given y = (y  , y  , . . . , y N ) ∈ K N such that x  + t n y ∈ K N , we have Then A * is demi-continuous on K N .
It is easy to see that K N is a nonempty closed convex and bounded subset of Banach space (X N , · * ). By using Theorem ., we know that the following variational inequality: We rewrite (.) as follows: From (.)-(.), we know that x * = (x *  , x *  , . . . , x * N ) is a solution of (.). This completes the proof of conclusion ().
On the other hand, let x = (x  , x  , . . . , x N ) be an arbitrary solution of (.). We have which implies Then x = (x  , x  , . . . , x N ) is a solution of the variational inequality (.) in reflexive Banach space (X N , · * ). From the above, we claim that the system of multivariate variational inequalities (.) is equivalent to the variational inequality (.). By using Theorem ., we know that the set of solutions of the variational inequality (.) is closed convex. This completes the proof of conclusion ().

Finally, if A is strictly monotone, then
Hence A * is also strictly monotone. By using Theorem ., the variational inequality (.) has a unique solution and hence the multivariate variational inequalities (.) has a unique solution. This completes the proof.
Corollary . Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let A : K → X * be a N -variables monotone semi-continuous mapping. Then: () the multivariate variational inequalities Proof Let A i = A for all i = , , . . . , N in Theorem ., we can get the conclusion.

Corollary . Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let A : K → X * be a N -variables monotone semi-continuous mapping. Then () the multivariate variational inequalities
has a solution (x *  , x *  , . . . , x * N ) ∈ K N ; () the set of solutions of (.) is closed convex in K N ; () if A is strictly monotone, then (.) has a unique solution.
That is, This completes the proof.
Next, we prove an existence theorem of solutions for the system of variational inequalities (.) in normed spaces.
Theorem . Let X be a normed space, let K be a compact convex subset of X, and let A, B : K → X * be two continuous mappings. Then the system of variational inequalities (.) has a solution (x, y) ∈ K × K and the set of solutions of (.) is closed.
Proof Let A(x, y) = A(y), B(x, y) = B(x) for all (x, y) ∈ K × K , then the system of variational inequalities (.) is equivalent to for all (x, y) ∈ K × K . It is easy to see that C * is a continuous mapping from the nonempty compact convex subset K × K into the dual space (X × X) * of normed space X × X. By using Theorem ., there exists an element (x * , y * ) ∈ K × K such that where ·, · * denotes the duality pairing of X × X and X * × X * = (X × X) * . This implies From the definition of ·, · * , we have Let z  = y * and z  = x * in (.), respectively, we get Then (x * , y * ) ∈ K × K is a solution of the system of variational inequalities (.). Since A, B are continuous, so the set of solutions of (.) is closed. This completes the proof.
It is obvious that Theorem . is a special form of Theorem . in reflexive Banach spaces.
Corollary . (Theorem .) Let X be a reflexive Banach space, let K be a compact convex subset of X, and let A, B : K → X * be two continuous mappings. Then the system of variational inequalities has a solution (x, y) ∈ K × K and the set of solutions of (.) is closed.
We give an example to show the mathematical and physical significance of the main results of this paper.
This element x  must be a solution of the following system of multivariate variational inequalities: (.) In fact, we have for all i = , , . . . , N . Hence x  must satisfy (.). In addition, the system of multivariate variational inequalities (.) is equivalent to

Conclusion
In this article, we use an ingenious mathematical method to prove the existence theorem of solutions for a kind of system of multivariate variational inequalities:
Here K is a nonempty closed convex and bounded subset of a reflexive Banach space X and A  , A  , . . . , A N are N -variables monotone demi-continuous mappings from K N into X * . This system of multivariate variational inequalities has a solution. The set of solutions of this system of multivariate variational inequalities is closed convex in K N . If A  , A  , . . . , A N are also strictly monotone, this system of multivariate variational inequalities has a unique solution.