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Scalarized system of nonsmooth vector quasivariational inequalities with applications to Debreu type vector equilibrium problems
 Mohammed M Alshahrani^{1}Email author
https://doi.org/10.1186/s136600150656x
© Alshahrani; licensee Springer. 2015
 Received: 16 February 2015
 Accepted: 1 April 2015
 Published: 11 April 2015
Abstract
In this work, we utilize a scalarization method to introduce a system of nonsmooth vector quasivariational inequalities. We also study their relationship to Debreu type vector equilibrium problems. Then we establish some existence results for solutions of these systems by using maximal element theorems for a family of setvalued maps.
Keywords
 scalarization
 system of nonsmooth vector quasivariational inequalities
 system of Debreu type equilibrium problem for vectorvalued functions
 system of vector quasiequilibrium problems
 Clarke generalized directional derivative
 maximal element theorem
 Φcondensing maps
1 Introduction
This concept of a system of vector quasiequilibrium problems was first introduced and studied by Ansari et al. [1]. It generalizes the vector equilibrium problem (VEP, for short) which has received a lot of attention by many researchers in recent years [2–7]. This problem is important because it serves as a unified framework for many problems in optimization, such as vector variational inequalities, vector variationallike inequalities, vector complementarity problems, and vector optimization problems. Particularly, vector equilibrium problems are successful in expressing optimality conditions for constrained extremum problems and equilibrium conditions for network flow and economic problems; see [3, 8] and the references therein. For more on this topic and its applications, we refer the reader to the review paper [9].
In this paper, we introduce a bilinear form that is suitable for the data in (DVEP) and we use it to write (SNVQVI) in a scalarized form. Considering the data of (DVEP), the scalarization method we used is one where we choose a continuous linear functional from a set in the dual space of \(Y_{i}\) that is closely related to the ordering cone in \(Y_{i}\), then we pair this functional with the range of \(f_{i}\). The resulting function, which is nonsmooth, is used to define a bilinear form; namely Clarke’s generalized directional derivative. This bilinear form helps finally write a scalarized version of (SNVQVI). Utilizing this method, we are able to establish some existence results for the scalarized version of (SNVQVI) that unify and improve many results in the literature [10–13].
The rest of the paper is organized as follows. Section 2 presents the necessary background needed. In Section 3, we describe the scalarization method and introduce a scalarized system of nonsmooth quasivariational inequality problems. We also introduce the scalarized Debreu type equilibrium problem for a vectorvalued function and investigate its relations to (DVEP) and to the scalarized system of nonsmooth quasivariational inequality problems. In Section 4, we establish the main results concerning the existence of solutions of the scalarized system of nonsmooth quasivariational inequality problems.
2 Definitions and preliminaries
In this section, we lay out the basic definitions and necessary background required in what follows.
Although the following result is generally accepted, we include the proof for completeness.
Lemma 2.1
Proof
Definition 2.1
(see [15])
The next lemma summarizes the basic properties of the Clarke generalized directional derivative.
Lemma 2.2
(see [15])
 (i)
The function \(\phi^{\circ}(x;\cdot)\) is finite, positively homogeneous, and subadditive on X.
 (ii)
The function \(\phi^{\circ}(\cdot;\cdot)\) is upper semicontinuous.
Definition 2.2
Remark 2.1
It is clear from the previous definition that the Clarke generalized partial directional derivative inherits all properties of the Clarke generalized directional derivative listed in Lemma 2.2.
Definition 2.3
We shall recall the definition of a strictly compactly Lipschitz function. This concept is very essential for scalarizing (SNVQVI).
Definition 2.4
(see [16])
 1.
a setvalued map \(\mathfrak{R}: K\rightarrow\operatorname{Comp}(Y)\), where \(\operatorname{Comp}(Y)\) is the set of all compact subset of Y,
 2.
a function \(r:(0,1]\times K\times K \rightarrow Y\),
 3.
a neighborhood U of \(\bar{x}\), and
 4.
a neighborhood \(\mathcal{O}\) of 0 in X,
 (i)\(\lim_{t\rightarrow0^{+}, x\rightarrow\bar{x}} {r(t,x;v)}=0\) for each \(v\in \mathcal{O}\) and$$\lim_{\substack{t\rightarrow0^{+}\\ x\rightarrow\bar{x}\\ v\rightarrow0 }} {r(t,x;v)}=0; $$
 (ii)for all \(x\in U\), \(v\in\mathcal{O}\) and \(t\in(0,1]\), we have$$\frac{\psi(x+tv)f(x)}{t} \in\mathfrak{R}(v) + r(t,x;v); $$
 (iii)
\(\mathfrak{R}(0)=\{0\}\) and the setvalued map ℜ is upper semicontinuous at the origin (that is, for each neighborhood of \(\mathfrak{R}(0)\) in Y there exists a neighborhood U of 0 in X such that, for each \(x\in U\), \(\mathfrak{R}(x)\in W\)).
Remark 2.2
 (i)
If Y is finite dimensional, then ψ is strictly compactly Lipschitz at \(\bar{x}\) if and only if it is locally Lipschitz at \(\bar{x}\).
 (ii)If ψ is strictly compactly Lipschitz on K, then the function \(w^{*}\circ\psi: K\rightarrow\mathbb{R}\) defined asis locally Lipschitz for all \(w^{*}\in Y^{*}\).$$\bigl(w^{*}\circ\psi\bigr) (x)=\bigl\langle w^{*}, \psi(x)\bigr\rangle \quad \text{for all } x\in K $$
Definition 2.5
Let K be a nonempty subset of X, \(\psi:K\rightarrow Y\) be a strictly compactly Lipschitz function on K, and \(w^{*}\in Y^{*}\). We say that ψ is \(w^{*}\)Clarkepseudoconvex if the realvalued function \(w^{*}\circ\psi: K\rightarrow\mathbb{R}\) is Clarkepseudoconvex on K, where \(\langle\cdot,\cdot\rangle\) is the pairing of \(Y^{*}\) and Y.
We end this section by giving the mayor tools to establish existence of solutions for (DVEP). Theorem 2.1, due to Deguire et al. [17], is a particular form of a maximal element theorem for a family of setvalued maps.
Definition 2.6
(see [18])
 (i)
\(\Phi(A)=0\) if and only if A is relatively compact.
 (ii)
\(\Phi(\overline{\operatorname{conv}}(A))= \Phi(A)\), where \(\overline{\operatorname{conv}}(A)\) stands for the convex closure of A.
 (iii)
\(\Phi(A\cup B)=\max{\{\Phi(A), \Phi(B)\}}\).
Definition 2.7
(see [18])
Remark 2.3
 (i)
(iii) in Definition 2.6 implies that if \(A\subseteq B\), then \(\Phi(A)\leq\Phi(B)\);
 (ii)
any setvalued map defined on a compact set is Φcondensing;
 (iii)
if Z is, also, locally convex, then any compact setvalued map is Φcondensing for every measure of noncompactness Φ;
 (iv)
if \(T:D\rightarrow2^{Z}\) is Φcondensing and \(T':D\rightarrow2^{Z}\) is a setvalued map such that \(T'(x)\subseteq T(x)\) for all \(x\in D\), then \(T'\) is also Φcondensing.
Theorem 2.1
 (i)
\(G_{i}(x)\) is convex, for all \(i\in I\) and all \(x\in K\);
 (ii)
\(x_{i}\notin G_{i}(x)\) for all \(i\in I\) and all \(x\in K\), where \(x_{i}\) is the ith component of x;
 (iii)
\(G^{1}(y_{i})\) is open for all \(i\in I\) and for all \(y_{i}\in K_{i}\), and
 (iv)
there exist a nonempty and compact subset N of K and a nonempty, compact, and convex subset \(B_{i}\) of \(K_{i}\) for each \(i\in I\) such that, for all \(x\in K\setminus N\), there exists \(i_{0}\in I\) satisfying \(G_{i_{0}}(x)\cap B_{i_{0}} \neq \emptyset\).
Remark 2.4
 (iv)′:

the setvalued map \(G:K\rightarrow2^{K}\), defined as \(G(x):=\prod_{i\in I}G_{i}(x)\) for all \(x\in K\), is Φcondensing;
3 Scalarization
Proposition 3.1
Every solution of (\(w^{*}\)DVEP) is a solution of (DVEP).
Proof
Proposition 3.2
Let \(f_{i}:K\rightarrow Y_{i}\) be strictly compactly Lipschitz at \(\bar{x}\in K\) and the function \(f(\bar {x}^{i},\cdot)\) is \(w^{*}\)Clarkepseudoconvex for all \(w^{*} \in C_{i}^{*}(\bar {x})\setminus\{\mathbf{0}\}\). If \(\bar{x}\) is a solution to (SSNQVI), then it is a solution to (\(w^{*}\)DVEP).
Proof
The proof is immediate from the definition of \(w^{*}\)Clarkepseudoconvexity of f in the second argument. □
4 Main results
From this point onward, we assume that I is any index set, countable or uncountable, and for each \(i\in I\), \(K_{i}\) is a nonempty convex subset of a Hausdorff topological vector space \(X_{i}\), \(Y_{i}\) is a real locally convex topological vector space, \(K=\prod_{i\in I}K_{i}\), \(C_{i}: K\rightarrow2^{Y_{i}}\) is a setvalued map such that, for all \(x\in K\), \(C_{i}(x)\) is a closed, convex, and pointed cone with nonempty interior and \(C_{i}^{*}(x)\) is its positive dual, as defined in (1). Furthermore, we assume that, for each \(i\in I\), \(A_{i}:K\rightarrow2^{K_{i}}\) is a setvalued map such that, for all \(x\in K\), \(A_{i}(x)\) is nonempty and convex, \(A^{1}_{i}(y_{i})\) is open in K for all \(y_{i}\in K_{i}\) and the set \(\mathfrak{F}_{i}:=\{x\in K: x_{i} \in A_{i}(x)\}\) is closed in K, where \(x_{i}\) is the ith component of x.
Theorem 4.1
Proof
Corollary 4.1
The following result establishes the existence of solutions to (SSNQVI) in the presence of Φcondensing maps.
Theorem 4.2
Proof
Declarations
Acknowledgements
The author is thankful to King Fahd University of Petroleum and Minerals (KFUPM), Dhahran Saudi Arabia for providing excellent research facilities to carry out this research. This work is supported by KFUPM through internally university funded research grant number SP121035 (IN121035).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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