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Scalarized system of nonsmooth vector quasivariational inequalities with applications to Debreu type vector equilibrium problems
Journal of Inequalities and Applicationsvolume 2015, Article number: 130 (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.
Introduction
A system of vector quasiequilibrium problems (SVQEP, for short) is one where we find $\bar{x}\in K$ such that, for each $i\in I$,
where I is an index set, $K=\prod_{i\in I}K_{i}$ with $K_{i}$ nonempty convex subsets of a Hausdorff locally convex topological vector space $X_{i}$, and for each $i\in I$, $A_{i}:K\rightarrow2^{K_{i}}$ is a setvalued map with nonempty values ($2^{K_{i}}$ represents the set of all subsets of $K_{i}$), $\varphi_{i}:K\times K_{i}\rightarrow Y_{i}$, with $Y_{i}$ another Hausdorff locally convex topological vector space, is a bifunction and $C_{i}:K\rightarrow 2^{Y}_{i}$ is a setvalued map such that, for each $x\in K$, $C_{i}(x)$ is a closed, convex, and pointed ($C_{i}(x)\cap( C_{i}(x))=\{\mathbf{0}\}$) cone in $Y_{i}$ with a nonempty topological interior, $\operatorname{int}(C_{i}(x))$.
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].
This paper is concerned with the study of existence of solutions to an important class of (SVQEP); namely, the Debreu type equilibrium problem for vectorvalued functions (DVEP, for short) which seeks to find an $\bar{x}\in K$ such that, for each $i\in I$, $\bar{x} _{i}\in A_{i}(\bar{x})$, and
where $f_{i}: K=K^{i}\times K_{i}\rightarrow Y_{i}$ is a vectorvalued function and
Here, we write $x=(x^{i},x_{i})$ for each $x\in K=K^{i}\times K_{i}$ and $\bar{x}$ is said to be a solution of (DVEP).
The following system of problems, called a system of nonsmooth vector quasivariational inequality problems (SNVQVI, for short), is essential as a tool to study (DVEP). In this system one is interested in finding $\bar{x}\in K$ such that, for each $i\in I$,
here $h_{i}:K\times K_{i}\rightarrow Y_{i}$ are bifunctions.
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.
Definitions and preliminaries
In this section, we lay out the basic definitions and necessary background required in what follows.
Throughout this section, unless stated otherwise, X and Y are two normed space and $Y^{*}$ is the topological dual of Y. Suppose that K is a nonempty subset of X and that $C:K\rightarrow2^{Y}$ is a setvalued map such that, for each $x\in K$, $C(x)$ is a closed, convex, and pointed cone in Y with nonempty interior, i.e. $\operatorname {int}(C(x))\neq\emptyset$, the positive dual of C is a setvalued map $C^{*}:K\rightarrow 2^{Y^{*}}$ defined as
Although the following result is generally accepted, we include the proof for completeness.
Lemma 2.1
Suppose that $C:K\rightarrow2^{Y}$ is a setvalued map such that, for each $x\in K$, $C(x)$ is a closed, convex, and pointed cone in Y with $\operatorname{int}(C(x))\neq\emptyset$, then
Proof
Assume that $x\in\operatorname{int}(C(\bar{x}))$ then there exists $\epsilon>0$ such that $x+\epsilon e \in C(\bar{x})$ for all $e \in\mathbb{B}$ where $\mathbb{B}$ is the unit ball. Now assume, to the contrary, that there exists $w_{0}^{*}\in C^{*}(\bar{x})\setminus\{\mathbf{0}\}$ such that $\langle w_{0}^{*}, x\rangle= 0$, then
and, in particular, we have
which implies that $w^{*}_{0} = 0$; a contradiction.
Conversely, assume that $\langle w^{*}, x\rangle> 0$ for all $w^{*} \in C^{*}(\bar{x})\setminus\{\mathbf{0}\}$ and that $x\notin \operatorname{int}(C(\bar{x}))$. Since $\operatorname{int}(C(\bar {x} ))$ is a nonempty convex set, then, by the strong separation theorem, there exists a nonzero $w_{0}^{*}\in Y^{*}$ such that
It is now clear that $\langle w^{*}_{0}, y \rangle\geq0 $ for all $y\in \operatorname{int}(C(\bar{x}))$. Because, otherwise, if $\langle w^{*}_{0}, y_{0} \rangle< 0 $ for some $y_{0} \in\operatorname{int}(C(\bar{x}))$ and since $C(\bar{x})$ is a cone, we can find large enough $\lambda>0$, such that $\langle w^{*}_{0}, x \rangle> \langle w^{*}_{0}, \lambda y_{0} \rangle$. Since $C(\bar{x})$ is closed and by the continuity of $w_{0}^{*}$, we have $\langle w^{*}_{0}, y \rangle\geq0 $ for all $y\in C(\bar{x})$; which shows that $w^{*}_{0} \in C^{*}(\bar{x})\setminus\{\mathbf{0}\}$. Now if $\langle w^{*}_{0}, x \rangle> 0$ then we can find $z_{0}\in\operatorname{int}(C(\bar{x}))$ and small enough $\mu>0$ such that $\langle w^{*}_{0}, x \rangle> \langle w^{*}_{0}, \mu z_{0} \rangle$. Therefore $\langle w^{*}_{0}, x \rangle\leq0$. This implies that $\langle w^{*}_{0}, x\rangle =0$ because $C(\bar{x})$ is pointed. This is a contradiction. □
We recall that a setvalued map $\psi: X \rightarrow Y$ is said to be locally Lipschitz at $x \in X$, see [14], if there exist a positive constant l and a neighborhood $U \subset\operatorname {Dom}(\psi):=\{y\in X: \psi(y)\neq\emptyset\}$ of x such that
where $\mathbb{B}_{Y}$ is the unit ball in Y. If ψ is locally Lipschitz at every $x\in K$ with K being a nonempty subset of X, we say that ψ is locally Lipschitz on K. We, also, recall that a realvalued function $\phi:X\rightarrow\mathbb{R}$ from a normed space to the real number line ℝ is said to be upper semicontinuous on a subset E of X if
It is called positively homogeneous on X if $\phi(t x)=t\phi (x)$ for all $x\in X$ and any positive scalar t. It is, also, called subadditive on X if for every $x,y \in X$, $\phi(x+y)\leq\phi (x)+\phi(y)$. A positively homogeneous function is clearly subadditive if and only if it is convex.
Definition 2.1
(see [15])
Let $\phi:X\rightarrow\mathbb{R}$ be locally Lipschitz at $x\in X$. The Clarke generalized directional derivative of ϕ at $x\in X$ in the direction of another vector $d\in X$, denoted by $\phi^{\circ }(x;d)$, is defined as
where $y\in X$ and t is a positive number.
The next lemma summarizes the basic properties of the Clarke generalized directional derivative.
Lemma 2.2
(see [15])
Let $\phi :X\rightarrow \mathbb{R}$ be locally Lipschitz at $x\in X$. The following assertions hold true.

(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
For an index set I, let $X=\prod_{i\in I}X_{i}$ with each $X_{i}$ a normed space and let $\phi_{i}: X\rightarrow\mathbb{R}$ be locally Lipschitz at $x \in X$. The Clarke generalized partial directional derivative of ϕ at x in the direction of $d_{i}\in X_{i}$, denoted by $\phi ^{\circ}_{i}(x;d_{i})$, is defined to be a Clarke generalized directional derivative of the function $g: X_{i}\rightarrow\mathbb{R}$ defined as
at $x_{i}$ in the direction of $d_{i}$, where $x_{i}$ is the ith component of x and
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
Let K be a nonempty subset of X and let $\phi:K\rightarrow\mathbb {R}$ be a locally Lipschitz function at $x\in K$. We say that ϕ is Clarkepseudoconvex at x if
The function ϕ is said to be Clarkepseudoconvex on K if it is Clarkepseudoconvex at every point in K.
We shall recall the definition of a strictly compactly Lipschitz function. This concept is very essential for scalarizing (SNVQVI).
Definition 2.4
(see [16])
Let K be a nonempty subset of X, a function $\psi:K\rightarrow Y$ is said to be strictly compactly Lipschitz at $\bar{x}\in K$ if there exist

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,
satisfying the following:

(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 $x\in U$, $\mathfrak{R}(x)\in W$).
Moreover, we say that ψ is strictly compactly Lipschitz on K if it is strictly compactly Lipschitz on all $\bar{x}\in K$.
Remark 2.2
In the previous definition, we have the following:

(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 as
$$\bigl(w^{*}\circ\psi\bigr) (x)=\bigl\langle w^{*}, \psi(x)\bigr\rangle \quad \text{for all } x\in K $$is locally Lipschitz for all $w^{*}\in Y^{*}$.
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])
If E is a lattice with a minimal element, denoted by 0, then a mapping $\Phi:2^{Z}\rightarrow E$, with Z a Hausdorff topological vector space, is called a measure of noncompactness of Z if the following conditions hold for any $A, B \in2^{Z}$:

(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])
Let L and Z be as in Definition 2.6. Suppose $\Phi :2^{Z}\rightarrow L$ is a measure of noncompactness of Z and $D\subset Z$. An upper semicontinuous (see [14] for a definition) setvalued map $T:D\rightarrow2^{Z}$ is said to be Φcondensing if the following implication holds:
Remark 2.3
It should be noted that

(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
Let I be an index set. For each $i\in I$, let $K_{i}$ be a nonempty convex subset of a Hausdorff topological vector space $X_{i}$, and let $G_{i}:K=\prod_{i\in I}K_{i}\rightarrow2^{K_{i}}$ be a setvalued map. Assume that the following conditions hold:

(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 $G_{i_{0}}(x)\cap B_{i_{0}} \neq \emptyset$.
Then there exists $\bar{x}\in K$ such that $G_{i}(\bar{x})=\emptyset$ for all $i\in I$.
Remark 2.4
If we replace condition (iv) in Theorem 2.1 by the following condition:
 (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;
Scalarization
In this section, we utilize the scalarization method introduced by Alshahrani et al. [12] to present scalarized problems of (DVEP) and (SNVQVI) and establish some relations among them. To this end, let I be an index set, $K=\prod_{i\in I}K_{i}$ with $K_{i}$ nonempty convex subsets of a normed space $X_{i}$. Furthermore, we consider, for each $i\in I$, a setvalued map $A_{i}:K\rightarrow2^{K_{i}}$ with nonempty values, a bifunction $f_{i}:K^{i}\times K_{i}\rightarrow Y_{i}$, with $Y_{i}$ another normed space, and $C_{i}:K\rightarrow2^{K_{i}}$ is a setvalued map such that $C_{i}(x)$ is a closed, convex, and pointed cone in $Y_{i}$ with nonempty interior for each $x\in K$. We write $x=(x^{i},x_{i})\in K=K^{i}\times K_{i}$, with
The scalarized Debreu type equilibrium problem for vectorvalued functions (denoted by wDVEP) is to find $\bar{x}\in K$ such that, for each $i\in I$, $\bar{x}_{i}\in A_{i}(\bar{x})$ and
Proposition 3.1
Every solution of ($w^{*}$DVEP) is a solution of (DVEP).
Proof
Suppose that $\bar{x}$ is a solution of ($w^{*}$DVEP) and assume, to the contrary, that it is not a solution of (DVEP); which means that there exists a $i_{0}\in I$ such that
By Lemma 2.1, we have
which is a contradiction. □
The scalarized system of nonsmooth quasivariational inequality problems (SSNQVI, for short) is to find $\bar{x}\in K$ such that, for each $i\in I$, $\bar{x}_{i} \in A_{i}(\bar{x})$, and
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. □
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
For each $i\in I$, let $f_{i}:K\rightarrow Y_{i}$ be strictly compactly Lipschitz on K. Assume that there exist a nonempty and compact subset N of K and a nonempty, compact, and convex subset $B_{i}$ of $K_{i}$ for all $i\in I$ such that, for all $x\in K\setminus N$, there exist $i_{0}\in I$ and $y_{i_{0}}\in B_{i_{0}}$, satisfying $y_{i_{0}}\in A_{i_{0}}(x)$ and
Then (SSNQVI) has a solution.
Proof
For every $x\in K$ and every $i\in I$, define two setvalued maps $W_{i}:K\rightarrow2^{K_{i}}$ and $G_{i}:K\rightarrow2^{K_{i}}$ as follows:
Because $(w_{i}^{*} \circ f_{i})_{x_{i}}^{\circ}(x;\cdot)$ is positively homogeneous and subadditive, it follows that $(w_{i}^{*} \circ f_{i})_{x_{i}}^{\circ}(x;\cdot)$ is convex and, therefore, $W_{i}(x)$ is a convex set, which in turn shows that $G_{i}(x) $ is also convex for all $x\in K$ and all $i\in I$. Furthermore, $x_{i} \notin W_{i}(x)$ for all $x\in K$ and all $i\in I$, with $x_{i}$ the ith component of x, because the Clarke generalized partial directional derivative vanishes in the direction of the zero vector. This also shows that $x_{i}\notin G_{i}(x)$ for all $x\in K$ and all $i\in I$.
For each $i\in I$ and all $y_{i}\in K_{i}$, the inverse of $G_{i}(x)$ can be written as
Obviously, $A^{1}_{i}(y_{i})$ and $K\setminus\mathfrak{F}_{i}$ are open. To see that $W_{i}^{1}(y_{i})$ is open, note that the complement of the inverse map of $W_{i}$ in K,
is closed due to the upper semicontinuity of the Clarke generalized partial directional derivative in both arguments. Thus $G^{1}_{i}(y_{i})$ is open for all $i\in I$ and all $y_{i}\in K_{i}$. Therefore by Theorem 2.1, there exists $\bar{x}\in K$ such that $G_{i}(\bar {x})=\emptyset$ for all $i\in I$. Since for each $i\in I$ and all $x\in K$, $A_{i}(x)$ is assumed to be nonempty, we must have $G_{i}(\bar{x})=A_{i}(\bar{x})\cap W_{i}(\bar{x} )=\emptyset$. In other words, $\bar{x}$ satisfies, for all $i\in I$, the following:
Thus $\bar{x}$ is a solution of (SSNQVI). □
Corollary 4.1
For each $i\in I$, let $f_{i}:K\rightarrow Y_{i}$ be strictly compactly Lipschitz on K and the function $f_{i}(\bar{x}^{i},\cdot)$ is $w_{i}^{*}$Clarkepseudoconvex for all $w_{i}^{*} \in C_{i}^{*}(x)\setminus\{ \mathbf{0}\}$ and all $x\in K$. Assume that there exist a nonempty and compact subset N of K and a nonempty, compact, and convex subset $B_{i}$ of $K_{i}$ for all $i\in I$ such that, for all $x\in K\setminus N$, there exist $i_{0}\in I$ and $y_{i_{0}}\in B_{i_{0}}$, satisfying $y_{i_{0}}\in A_{i_{0}}(x)$ and
Then (DVEP) has a solution.
Proof
The proof follows from Theorem 4.1 and Propositions 3.1 and 3.2. □
The following result establishes the existence of solutions to (SSNQVI) in the presence of Φcondensing maps.
Theorem 4.2
For each $i\in I$, let $f_{i}:K\rightarrow Y_{i}$ be strictly compactly Lipschitz on K and let the setvalued map $A:K\rightarrow2^{K}$, defined as
be Φcondensing. Then (SSNQVI) has a solution.
Proof
In light of Remark 2.4, we only need to show that the setvalued map $G:K\rightarrow2^{K}$, defined by
where the $G_{i}$ are as defined in the proof of Theorem 4.1, is Φcondensing. From the definition of $G_{i}$, we have $G_{i}(x)\subseteq A_{i}(x)$ for all $i\in I$ and all $x\in K$ and therefore $G(x)\subseteq A(x)$ for all $x\in K$. Because A is Φcondensing, it follows from (iv) and Remark 2.3 that G is Φcondensing as well. □
Remark 4.1
Theorem 4.1 and Theorem 4.2 improve and generalize many existence results in the literature; see for example Theorem 9 in [10], Theorem 4.1 in [11], Theorem 2 in [12] and Theorem 1 in [13].
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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).
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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