A kind of system of multivariate variational inequalities and the existence theorem of solutions
- Yanxia Tang^{1}Email author,
- Jinyu Guan^{1},
- Yongchun Xu^{1} and
- Yongfu Su^{2}
https://doi.org/10.1186/s13660-017-1486-9
© The Author(s) 2017
Received: 24 May 2017
Accepted: 25 August 2017
Published: 7 September 2017
Abstract
Keywords
1 Introduction
In 1966, Hartman and Stampacchia [2] proved the following result.
Theorem 1.1
[2]
Let K be a nonempty closed convex and bounded subset of \(R^{n}\). Let \(A: K\rightarrow R^{n}\) be a continuous mapping. Then the variational inequality (1.1) has a solution \(x^{*} \in K\).
In 1967, Browder proved the following more general result (see [15]).
Theorem 1.2
[15]
Let K be a nonempty compact convex subset of a locally convex topological vector space X. Let \(A: K\rightarrow X^{*}\) be a continuous mapping. Then the variational inequality (1.1) has a solution \(x^{*} \in K\).
The variational inequality (1.1) 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.
Definition 1.3
[15]
Let X be a normed space, \(A: X\rightarrow X^{*}\) a mapping, \(x_{0} \in X\). A is said to be demi-continuous at \(x_{0}\), if for any given \(y\in X\), \(A(x_{0}+t_{n}y)\) weak^{∗} converges to \(A(x_{0})\) wherever \(t_{n}\rightarrow0\), \(t_{n}\geq0\).
In 1991, Chang [15] proved the following result in reflexive Banach spaces.
Theorem 1.4
[15]
In 2010, Plubtieng and Sombut [16] proved the following result.
Theorem 1.5
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 2016, Su et al. [17] presented the concept of multivariate fixed point and proved a multivariate fixed point theorem for N-variables contraction mappings which further generalizes Banach contraction mapping principle. In 2016, Luo et al. [18] presented the concept of multivariate best proximity point and proved the multivariate best proximity point theorems in metric spaces for N-variables contraction mappings. In 2017, Xu et al. [19] presented the concept of multivariate contraction mapping in a locally convex topological vector spaces and proved the multivariate contraction mapping principle in such spaces. In 2017, Guan et al. [20] studied a kind of system of N-variables pseudocontractive operator equations and proved the existence theorem of solutions.
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 [15, 16] in reflexive Banach spaces. In order to get the expected results, an ingenious mathematical method is used in this paper.
2 Preliminaries
Let us introduce some conclusions which will be useful for our main results.
Lemma 2.1
[20]
Lemma 2.3
[20]
3 Main results
Definition 3.1
The following is the main result of this paper.
Theorem 3.2
Proof
Corollary 3.3
- (1)the multivariate variational inequalitieshas a solution \((x_{1}^{*},x_{2}^{*}, \ldots,x_{N}^{*}) \in K^{N}\);$$ \bigl\langle A(x_{1},x_{2}, \ldots,x_{N}), y-x_{i}\bigr\rangle \geq0, \quad\forall y \in K, \forall i=1,2,\ldots,N $$(3.8)
- (2)
the set of solutions of (3.8) is closed convex in \(K^{N}\);
- (3)
if A is strictly monotone, then (3.8) has a unique solution.
Proof
Let \(A_{i}=A\) for all \(i=1,2,\ldots,N\) in Theorem 3.2, we can get the conclusion. □
Corollary 3.4
- (1)the multivariate variational inequalitieshas a solution \((x_{1}^{*},x_{2}^{*}, \ldots,x_{N}^{*}) \in K^{N}\);$$ \Biggl\langle A(x_{1},x_{2}, \ldots,x_{N}), y-\frac{1}{N}\sum_{i=1}^{N}x_{i} \Biggr\rangle \geq0,\quad \forall y \in K, $$(3.9)
- (2)
the set of solutions of (3.9) is closed convex in \(K^{N}\);
- (3)
if A is strictly monotone, then (3.9) has a unique solution.
Proof
Next, we prove an existence theorem of solutions for the system of variational inequalities (1.2) in normed spaces.
Theorem 3.5
Let X be a normed space, let K be a compact convex subset of X, and let \(A,B : K\rightarrow X^{*}\) be two continuous mappings. Then the system of variational inequalities (1.2) has a solution \((x,y) \in K\times K\) and the set of solutions of (1.2) is closed.
Proof
It is obvious that Theorem 1.5 is a special form of Theorem 3.5 in reflexive Banach spaces.
Corollary 3.6
Theorem 1.5
We give an example to show the mathematical and physical significance of the main results of this paper.
Example 3.7
4 Conclusion
Declarations
Acknowledgements
This project is supported by the major project of Hebei North University under grant No. ZD201304.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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