- Research
- Open Access
On the existence theorems of solutions for generalized vector variational inequalities
- Shih-sen Chang^{1}Email author,
- Salahuddin^{2} and
- Gang Wang^{3}
https://doi.org/10.1186/s13660-015-0856-4
© Chang et al. 2015
- Received: 6 April 2015
- Accepted: 5 October 2015
- Published: 21 November 2015
Abstract
In this paper, we prove some existence theorems of solutions for two classes of generalized vector variational inequalities and Minty generalized vector variational inequalities.
Keywords
- generalized vector variational inequalities
- Minty generalized vector variational inequality
- cones
- quasilinear type operator
- Minty lemma
- KKM mapping
MSC
- 46A50
- 47H04
1 Introduction
The variational inequality theory, which is mainly due to Stampacchia [1], provides very powerful techniques for studying problems arising in mechanics, optimization, transportation, economics, contact problems in elasticity, and other branches of mathematics. The free boundary value problem can be studied effectively in the framework of variational inequalities, the traffic assignment problem is a variational inequality problems. The theory of vector variational inequalities was initiated by Giannessi [2]. The theory has shown to be very useful in studying problems arising in pure and applied sciences, engineering and technology, financial mathematics, transportation, and other problems of practical interest, see [3–10]. In recent years, a considerable number of generalizations of vector variational inequalities have been considered, studied, and applied in various directions. The general variational inequality problems provide us with a unified, simple, innovative, and natural frame work for studying a wide class of problems involving unilateral, moving boundary, obstacle, free boundary and equilibrium problems. The existence problems of solutions for this problems are helpful in the sense that one would like to know if a solution of a general variational inequality exists before one actually devises some plausible algorithms for solving the problems; while existence results of the solution for Stampacchia variational inequalities were abundant in the last years, this is not the case of general variational inequalities of Stampacchia type [1].
Inspired and motivated by the recent work [11–21], the purpose of this paper is first to study a new class of quasilinear type operators and then to establish some existence theorems of solutions for a class of generalized vector variational inequalities and Minty generalized vector variational inequalities.
2 Preliminaries
- (i)
\(\lambda C(x)\subset C(x)\), for \(\lambda>0\);
- (ii)
\(C(x)+C(x)\subset C(x)\);
- (iii)
\(C(x)\cap(-C(x))=\{0\}\).
Theorem 2.1
[22]
Let \(f:I\subseteq R\to R\) be a function. Then f is a quasilinear if and only if f is monotone.
Theorem 2.3
[22]
Definition 2.4
Proposition 2.5
Let X and Y be two real linear spaces and \(A:D\subseteq X\to Y\) be a quasilinear operator. Then \(\lambda A:D\to Y\) is quasilinear for all \(\lambda\in R\).
Proof
Definition 2.6
Let U and V be two real linear spaces and \(A:D\subseteq U\to V\) be an operator. A is said to be c-monotone, if for every \(v\in V\) the set \(A^{-1}(v)\) is convex.
Proposition 2.7
Let X and Y be two real linear spaces, \(D\subseteq X\) be a convex set and \(A:D\to Y\) be a quasilinear mapping. Then A is c-monotone.
Proof
Definition 2.8
Let D be an open subset of a topological space X and \(f:D\to Y\) be an operator where Y is an arbitrary set. Then f is said to be locally injective, if for each \(x\in D\), it admits a neighborhood \(U_{x}\subseteq D\) such that f is injective on \(U_{x}\) i.e., \(f(u)\neq f(v)\) for all \(u,v \in U_{x}\), \(u\neq v\).
Proposition 2.9
Let X be a topological real linear space, Y be a real linear space and \(D\subseteq X\) be a convex and open subset. Let \(A:D\to Y\) be quasilinear and locally injective. Then A is injective.
Proof
Now we prove that for each \(u\in A(D)\), \(A^{-1}(u)\) contains only one point. Assume that there exist \(x,y\in A^{-1}(u)\), \(x\neq y\). Since A is locally injective, there exists a neighborhood \(U_{x}\) of x contained in D such that \(A(z)\neq A(x)\) for \(z\neq x\), \(z\in U_{x}\). Since A is c-monotone and \([x,y]\subseteq A^{-1}(u)\), \(z\in U_{x}\cap(x,y]\). Hence \(A(z)=u\) and \(A(z)\neq A(x)=u\), a contradiction. □
Definition 2.10
Lemma 2.11
[23]
Remark 2.12
Proposition 2.13
Let X be a real linear space, Y be a real linear metric space. Let \(A: D\subseteq X\to Y\) be an injective convex and quasilinear mapping and it is continuous on line segments. Then the operator \(A^{-1}:A(D)\to D\) is quasilinear.
Proof
Theorem 2.14
Let X be a real linear space, Y be a real linear space and also metric space. Let \(A:D\subseteq X\to Y\) be continuous on line segments, quasilinear with its domain D and convex. Then \(A(D)\) is convex.
Proof
Definition 2.15
Proposition 2.16
Let X, Y, Z be real linear spaces, and let \(D\subseteq X\) and \(A:D\to Y\), \(B:A(D)\to Z\) be two quasilinear operators. Then \(B \circ A:D\to Z\) is quasilinear.
Proof
Definition 2.17
Theorem 2.18
Theorem 2.19
- (i)
\(x-y\in-\operatorname{int}C\) and \(x\notin-\operatorname{int}C\Rightarrow y\notin-\operatorname{int}C\);
- (ii)
\(x+y\in-C\) and \(x+z\notin-\operatorname{int}C\Rightarrow z-y\notin-\operatorname{int}C\);
- (iii)
\(x+z-y\notin-\operatorname{int}C\) and \(-y\in-C\Rightarrow x+z\notin-\operatorname{int}C\);
- (iv)
\(x+y\notin-\operatorname{int}C\) and \(y-z\in-C\Rightarrow x+z\notin-\operatorname{int}C\).
3 Existence of solutions for generalized vector variational inequalities
In this section we present some existence results of solutions for vector variational inequalities (1) and (3).
Definition 3.1
(Knaster-Kuratowski-Mazurkiewicz)
Lemma 3.2
[25]
An operator \(T:X\to X^{*}\) is called weak to \(\|\cdot\|\)-sequentially continuous at \(x\in X\) if, for every sequence \(\{x_{n}\}\) which converges weakly to x, we have \(\{T(x_{n})\}\to T(x)\) in the topology of the norm of \(X^{*}\). An operator \(T:X\to X\) is called weak to weak sequentially continuous at \(x\in X\) if, for every sequence \(\{x_{n}\}\) which converges weakly to x, we find that \(\{T(x_{n})\}\) converges to \(T(x)\).
Lemma 3.3
If \(P\subset Q\subset X\) where Q is weakly compact and P is weakly sequentially closed, then P is weakly compact.
Proof
From the Eberlein-Smulian theorem [26], Q is weakly sequentially compact. Let \(\{x_{k}\}\subseteq P\), hence \(\{x_{k}\}\subseteq Q\), which is weakly sequentially compact. Hence there exists \(\{ x_{k_{n}}\}\subseteq\{x_{k}\}\), weakly converges to a point \(x\in Q\). But \(\{x_{k_{n}}\}\subseteq P\), which is weakly sequentially closed, hence \(x \in P\). Thus P is weakly sequentially compact. Therefore from the Eberlein-Smulian theorem P is weakly compact. □
Lemma 3.4
- (a)
\(\{x_{i}\}_{i\in I}\) converges to x in the weak topology of X and \(\{x_{i}^{*}\}_{i\in I}\) converges to \(x^{*}\) in the topology of norm of \(X^{*}\).
- (b)
\(\{x_{i}\}_{i\in I}\) converges to x in the topology of norm of X and \(\{x_{i}^{*}\}_{i\in I}\) converges to \(x^{*}\) in the weak ^{∗} topology of \(X^{*}\).
Now we are ready to state our first main result.
Theorem 3.5
Let X be a real Banach space and \(X^{*}\) be its topological dual. Let K be a weakly compact convex subset of X, \(A:K\to X^{*}\), and \(a:K \to X\) be two given operators. Assume that \(C:K\to2^{X}\) is a mapping with closed convex solid cone values and for each \(x\in K\), \(\operatorname{int}C(x) \neq \emptyset\). If A is weak to \(\|\cdot\|\)-sequentially continuous, a is quasilinear and weak to weak sequentially continuous. Then the generalized vector variational inequality (1) admits a solution.
Proof
Corollary 3.6
Assume that K is a weakly compact convex subset of X and \(X^{*}\) its topological dual. Let \(A:K\to X^{*}\) be a single-valued operator. Assume that \(C:K\to2^{X}\) is a mapping with closed convex solid cone values and for each \(x\in K\), \(\operatorname{int}C(x) \neq \emptyset\). If A is weak to \(\|\cdot\|\)-sequentially continuous, then (2) admits a solution.
Theorem 3.7
Proof
Theorem 3.8
- (a)
a is quasilinear,
- (b)if \(\{x_{n}\}\subseteq K\) converges weakly to \(x\in K\) then$$\lim\inf_{n\to\infty}\bigl\langle A(x_{n}),y\bigr\rangle \leq\bigl\langle A(x), y\bigr\rangle \notin-\operatorname{int}C(x),\quad \forall y \in K, $$
- (c)
the function \(x\mapsto\langle A(x), a(x)\rangle: K \to X\) is sequentially weakly lower semi-continuous.
Proof
Theorem 3.9
- (a)
a is quasilinear,
- (b)if \(\{x_{n}\}\subseteq K\) converges weakly to \(x\in K\) then$$\lim\inf_{n\to\infty}\bigl\langle A(x_{n}),y\bigr\rangle \leq\bigl\langle A(x), y\bigr\rangle \notin-\operatorname{int}C(x), \quad \forall y \in K, $$
- (c)
the function \(x\mapsto\langle A(x), a(x)\rangle: K \to X\) is sequentially weakly lower semi-continuous,
- (d)there exists \(y_{0}\in K\) such that$$\lim\inf_{\|x\|\to\infty, x\in K}\bigl\langle A(x), a(x)-a(y_{0}) \bigr\rangle \in-\operatorname{int}C(x). $$
Proof
In a similar way, we can prove that \(G(y)\) (see Theorem 3.8) is weakly sequentially closed for all \(y\in K\). From Theorem 3.7, \(G(y)\) is closed for all \(y\in K\) and \(G(y_{0})\) is weakly compact. The rest of the proof is similar to the proof of Theorem 3.5. □
Definition 3.10
Remark 3.11
If operator \(A:K\to X^{*}\) is hemi-continuous and monotone in the Minty-Browder sense then the solutions of the Minty generalized vector variational inequality (3) and the Minty general vector variational inequality (4) coincide.
Theorem 3.12
- (i)
If \(A: K \to K\) is hemi-continuous and K is convex, then every solution \(x\in K\) of (4) is also a solutions of general vector variational inequality (2).
- (ii)
If A is monotone on the convex set K, then every solution \(x\in K\) of the Minty generalized vector variational inequality (2) is also a solution of the Minty general vector variational inequality (4).
Definition 3.13
Remark 3.14
If \(a\equiv \mathrm{id}_{D}\) we obtain the definition of Minty-Browder monotonicity and pseudomonotonicity, respectively. It is well known [27, 28] that monotonicity implies pseudomonotonicity but the converse is not true.
Theorem 3.15
- (i)
If \(A: K \to K \) is monotone with respect to a, then every solution \(x\in K\) of generalized vector variational inequality (1) is also a solution of the Minty generalized vector variational inequality (3).
- (ii)
If A is hemi-continuous and a is strictly quasilinear then every solution \(x\in K\) of the Minty generalized vector variational inequality (3) is also a solution of generalized vector variational inequality (1).
Proof
Declarations
Acknowledgements
The authors would like to express their thanks to the editors and the referees for their kind and helpful comments and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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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