 Research
 Open access
 Published:
On the existence and essential components of solution sets for systems of generalized quasivariational relation problems
Journal of Inequalities and Applications volume 2014, Article number: 250 (2014)
Abstract
In this paper, we study the existence of a solution for a system of quasivariational relation problems (in short, (SQVR)). Moreover, we discuss the existence of essentially connected components of the solution set for (SQVR). Then the obtained results are applied to systems of quasivariational inclusions and to systems of weak vector quasiequilibrium problems. The results presented in the paper improve and extend many results from the literature. Some examples are given to illustrate our results.
MSC:47J20, 49J40.
1 Introduction and preliminaries
Variational relation problems were first introduced and studied by Luc in [1]. These problems include as special cases variational inclusion problems, vector equilibrium problems, vector variational inequality problems and vector optimization problems, etc. Later, the results of many authors had been extended and studied as regards the existence and stability of solutions in different models; see for example [2–11] and the references therein.
In 1950, Fort [12] first introduced the notion of essential fixed points of a continuous mapping from a compact metric space into itself and proved that any mapping can be approximately closed by a mapping whose fixed points are all essential. Later, Kinoshita [13] introduced the notion of essential components of the set of fixed points of a singlevalued map and proved that there exists at least one essential component of the set of its fixed points. Recently, the essential components of the solution set have been studied for vector equilibrium problems [14, 15], vector variational inequality problems [16], etc. Very recently, Yang and Pu [17] introduced and studied the system of strong vector quasiequilibrium problems (in short, (SSVQEP)) and also obtained the existence of essential components for these problems.
Motivated by the research works mentioned above, in this paper, we introduce the system of generalized quasivariational relation problems. Then we establish some existence theorems of solution sets for this problem. Moreover, we also obtain an existence theorem for essentially connected components of the set of solutions for a system of generalized quasivariational relation problems. These results are then applied to systems of quasivariational inclusion problems and systems of weak vector quasiequilibrium problems.
Now, we pass to our problem setting. Let I=\{1,\dots ,n\} be an index set. For each i\in I, let {X}_{i} and {Y}_{i} be two real locally convex Hausdorff topological vector spaces and {K}_{i} a nonempty convex compact subset of {X}_{i}. Denote
For each x\in K, we can write x=({x}_{i},{x}_{\stackrel{\u02c6}{i}}). For each i\in I, let {S}_{i},{T}_{i}:K\to {2}^{{K}_{i}} be setvalued mappings, and let {R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i}) be a relation linking {x}_{i}\in {K}_{i}, {x}_{\stackrel{\u02c6}{i}}\in {K}_{\stackrel{\u02c6}{i}} and {y}_{i}\in {K}_{i}. We consider the following system of generalized quasivariational relation problems (in short, (SQVR)).
(SQVR): Find ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
where \overline{x} is a solution of (SQVR). We denote by \mathrm{\Sigma}(R) the solution set of (SQVR).
Next, we recall some basic definitions and some of their properties.
Let X, Y be two topological vector spaces; let A be a nonempty subset of X and F:A\to {2}^{Y} be a multifunction.

(i)
F is said to be lower semicontinuous (lsc) at {x}_{0}\in A if F({x}_{0})\cap U\ne \mathrm{\varnothing} for some open set U\subseteq Y implies the existence of a neighborhood N of {x}_{0} such that F(x)\cap U\ne \mathrm{\varnothing}, \mathrm{\forall}x\in N.

(ii)
F is said to be upper semicontinuous (usc) at {x}_{0}\in A if, for each open set U\supseteq G({x}_{0}), there is a neighborhood N of {x}_{0} such that U\supseteq F(x), \mathrm{\forall}x\in N.

(iii)
F is said to be continuous at {x}_{0}\in A if it is both lsc and usc at {x}_{0}\in A.

(iv)
F is said to be closed if Graph(F)=\{(x,y):x\in A,y\in F(x)\} is a closed subset in A\times Y.
Let A, B, C be convex sets in topological vector spaces and R(x,y,z) be a relation between elements of the three sets. The relation R is said to be closed if the set \{(x,y,z)\in A\times B\times C:R(x,y,z)\text{holds}\} is closed. The relation R is said to be convex in the first variable if whenever R({x}_{1},y,z) holds and R({x}_{2},y,z) holds, then R(\lambda {x}_{1}+(1\lambda ){x}_{2},y,z) holds, for all {x}_{1},{x}_{2}\in A, \lambda \in [0,1].
Definition 1.1 ([18])
Let X, Y be two topological vector spaces, A is a nonempty subset of X, and F:A\to {2}^{Y} be a multifunction; and C\subset Y is a nonempty, closed, and convex cone. F is called upper Ccontinuous at {x}_{0}\in A, if, for any neighborhood U of the origin in Y, there is a neighborhood V of {x}_{0} such that
Definition 1.2 ([18])
Let X and Y be two topological vector spaces and A be a nonempty convex subset of X. A setvalued mapping F:A\to {2}^{Y} is said to be properly Cquasiconvex if, for any x,y\in A and \lambda \in [0,1], we have
Lemma 1.1 ([18])
Let X, Y be two Hausdorff topological vector spaces and F:X\to {2}^{Y} be a multivalued map.

(i)
If F is upper semicontinuous with closed values, then F is closed.

(ii)
If F is closed and Y is compact, then F is upper semicontinuous.
Lemma 1.2 ([19])
Let X, Y be two Hausdorff topological vector spaces and F:X\to {2}^{Y} be a setvalued mapping with compact values. Then F is upper semicontinuous {x}_{0}\in X if and only if, for each net \{{x}_{\alpha}\}\subseteq X which converges to {x}_{0}\in X and for each net \{{y}_{\alpha}\}\subseteq F({x}_{\alpha}), there exist {y}_{0}\in F({x}_{0}) and a subnet \{{y}_{\beta}\} of \{{y}_{\alpha}\} such that {y}_{\beta}\to {y}_{0}.
Lemma 1.3 ([20])
Let A be a nonempty compact convex subset of a Hausdorff topological vector space X. Suppose that M:A\to {2}^{A}\cup \{\mathrm{\varnothing}\} be a setvalued map with the following conditions:

(i)
for each at x\in A, M(x) is convex;

(ii)
for each at x\in A, x\notin M(x);

(iii)
for each at y\in A, {M}^{1}(y)=\{x\in A:y\in M(x)\} is open in A.
Then there exists {x}_{0}\in A such that M({x}_{0})=\mathrm{\varnothing}.
Lemma 1.4 (KakutaniFanGlicksberg [21])
Let A be a nonempty compact convex subset of a locally convex Hausdorff vector topological space X. If F:A\to {2}^{A} is upper semicontinuous and for any x\in A, F(x) is nonempty, convex, and closed, then there exists an {x}^{\ast}\in A such that {x}^{\ast}\in F({x}^{\ast}).
2 Existence of solutions
In this section, we establish an existence theorem of solutions for system of generalized quasivariational relation problems.
Theorem 2.1 For each i\in I, assume that

(i)
{S}_{i} is upper semicontinuous in K with nonempty compact convex values;

(ii)
{T}_{i} is lower semicontinuous in K with nonempty convex values;

(iii)
for all ({x}_{i},{x}_{\stackrel{\u02c6}{i}})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}, {R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{x}_{i}) holds;

(iv)
the set \{({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:{R}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i})\mathit{\text{does not hold}}\} is convex in {K}_{i};

(v)
the relation {R}_{i} is convex in the first variable and closed.
Then the (SQVR) has a solution, i.e., there exists ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
Moreover, the solution set of the (SQVR) is closed.
Proof We define a setvalued map: \mathrm{\Phi}:K\to {2}^{K} by \mathrm{\Phi}(x)={\prod}_{i\in I}{\mathrm{\Phi}}_{i}(x), where
(The map Φ is called the bestreply map; see [14, 17].)
For each i\in I:

(I)
For any x\in K, we show that {\mathrm{\Phi}}_{i}(x)\ne \mathrm{\varnothing} is nonempty.
Indeed, for all x\in K, we define a setvalued map {M}_{i}:{S}_{i}(x)\to {2}^{{S}_{i}(x)}\cup \{\mathrm{\varnothing}\} by

(a)
For each {x}_{i}\in {S}_{i}(x), by condition (iv), {M}_{i}(x) is a convex set.

(b)
For each {x}_{i}\in {S}_{i}(x), by condition (iii), {x}_{i}\notin {M}_{i}({x}_{i}).

(c)
For each {y}_{i}\in {T}_{i}(x), by condition (v), the set {M}_{i}^{1}({y}_{i})=\{{x}_{i}\in {S}_{i}(x):{y}_{i}\in {M}_{i}(x)\}=\{{x}_{i}\in {S}_{i}(x):{R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\text{does not hold}\} is open in {S}_{i}(x).
By Lemma 1.3, there exists {\overline{x}}_{i}\in {S}_{i}(x) such that {M}_{i}({\overline{x}}_{i})=\mathrm{\varnothing}, i.e., {\mathrm{\Phi}}_{i}(x)\ne \mathrm{\varnothing}.

(II)
We show that {\mathrm{\Phi}}_{i}(x) is convex.
Let {a}_{i}^{1},{a}_{i}^{2}\in {\mathrm{\Phi}}_{i}(x) and \lambda \in [0,1] and put {a}_{i}=\lambda {a}_{i}^{1}+(1\lambda ){a}_{i}^{2}. Since {a}_{i}^{1},{a}_{i}^{2}\in {S}_{i}(x) and {S}_{i}(x) is convex, we have {a}_{i}\in {S}_{i}(x). Thus, for {a}_{i}^{1},{a}_{i}^{2}\in {\mathrm{\Phi}}_{i}(x), it follows that
By (v), {R}_{i} is convex in the first variable, we have
i.e., {a}_{i}\in {\mathrm{\Phi}}_{i}(x). Therefore, {\mathrm{\Phi}}_{i}(x) is convex.

(III)
We will prove that {\mathrm{\Phi}}_{i} is upper semicontinuous in K with nonempty compact values.
Since K is a compact set, it suffices to show that {\mathrm{\Phi}}_{i} is a closed mapping. Indeed, let {\{({a}_{i}^{\alpha},{x}^{\alpha})\}}_{\alpha \in \mathrm{\Lambda}} be any a net in Graph({\mathrm{\Phi}}_{i}) such that ({a}_{i}^{\alpha},{x}^{\alpha})\to ({a}_{i}^{0},{x}^{0}). Now, we need only prove that {a}_{i}^{0}\in {\mathrm{\Phi}}_{i}({x}^{0}). Since {a}_{i}^{\alpha}\in {S}_{i}({x}^{\alpha}) and {S}_{i} is upper semicontinuous at {x}^{0}\in K with nonempty compact values, we have {S}_{i} is closed at {x}^{0}\in K, thus, {a}_{i}^{0}\in {S}_{i}({x}^{0}). Suppose to the contrary {a}_{i}^{0}\notin {\mathrm{\Phi}}_{i}({x}^{0}). Then \mathrm{\exists}{y}_{i}^{0}\in {T}_{i}({x}^{0}) such that
By the lower semicontinuity of {T}_{i}, there is a net \{{y}_{i}^{\alpha}\} with {y}_{i}^{\alpha}\in {T}_{i}({x}^{\alpha}) such that {y}_{i}^{\alpha}\to {y}_{i}^{0}. Since {a}_{i}^{\alpha}\in {\mathrm{\Phi}}_{i}({x}^{\alpha}),
By condition (v) and (2.2),
This is a contradiction between (2.1) and (2.3). Thus, {a}_{i}^{0}\in {\mathrm{\Phi}}_{i}({x}^{0}). Hence, {\mathrm{\Phi}}_{i} is upper semicontinuous in K with nonempty compact values.
By the definition of the mapping Φ is upper semicontinuous with nonempty compact values. By Lemma 1.4, there exists ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and

(IV)
Now we prove that \mathrm{\Sigma}(R) is closed.
Let a net \{{x}_{\alpha},\alpha \in I\}\in \mathrm{\Sigma}(R): {x}_{\alpha}\to {x}_{0}. We need to prove that {x}_{0}\in \mathrm{\Sigma}(R). Indeed, by the lower semicontinuity of {T}_{i}, for any {y}_{i}^{0}\in {T}_{i}({x}^{0}), there exists {y}_{i}^{\alpha}\in {T}_{i}({x}^{\alpha}) such that {y}_{i}^{\alpha}\to {y}_{i}^{0}. As {x}^{\alpha}\in \mathrm{\Sigma}(R),
Since {S}_{i} is upper semicontinuous with nonempty and closed values, by Lemma 1.1(i), we find that {S}_{i} is closed. Thus, {x}^{0}\in {S}_{i}({x}^{0}). By condition (v),
This means that {x}^{0}\in \mathrm{\Sigma}(R). Thus \mathrm{\Sigma}(R) is a closed set. □
If we let {S}_{i}(x)={T}_{i}(x)={K}_{i} for each i\in I and x\in X, then (SQVR) becomes the following system of variational relation problems (in short, (SVR)). So, we obtain following result.
Corollary 2.1 For each i\in I, assume that

(i)
for all ({x}_{i},{x}_{\stackrel{\u02c6}{i}})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}, {R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{x}_{i}) holds;

(ii)
the set \{({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:{R}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i})\mathit{\text{does not hold}}\} is convex in {K}_{i};

(iii)
the relation {R}_{i} is convex in the first variable and closed.
Then the (SVR) has a solution, i.e., there exists ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I,
Moreover, the solution set of the (SVR) is closed.
If I is a singleton, {S}_{i}(x)=S(x), {T}_{i}(x)=T(x), then (SQVR) reduces to the following quasivariational relation problem (in short, (QVR)). So, we also obtain the following result.
Corollary 2.2 Assume that

(i)
S is upper semicontinuous in K with nonempty compact convex values;

(ii)
T is lower semicontinuous in K with nonempty convex values;

(iii)
for all x\in K, R(x,x) holds;

(iv)
the set \{y\in K:R(\cdot ,y)\mathit{\text{does not hold}}\} is convex in K;

(v)
the relation R is convex in the first variable and closed.
Then the (QVR) has a solution, i.e., there exists \overline{x}\in K such that \overline{x}\in S(\overline{x}) and
Moreover, the solution set of the (QVR) is closed.
Remark 2.1

(i)
For each i\in I, let {X}_{i}, {Y}_{i} and {K}_{i}, {K}_{\stackrel{\u02c6}{i}}, K, {T}_{i}={S}_{i} be as in (SQVR). Let {F}_{i}:{K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}\to {2}^{{Y}_{i}} be a setvalued mapping, {C}_{i}\subset {Y}_{i} be a nonempty, closed, and convex cone, and the relation {R}_{i} be defined as follows:
{R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\text{holds}\phantom{\rule{1em}{0ex}}\text{iff}\phantom{\rule{1em}{0ex}}{F}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\subset {C}_{i}.Then (SQVR) becomes the system of strong vector quasiequilibrium problem (in short, (SSVQEP)) studied in [17].

(ii)
Let {S}_{i}(x)={T}_{i}(x)={K}_{i} for each i\in I and x\in X and let {F}_{i}, {C}_{i} as in Remark 2.1(i). Then (SVR) becomes the system of strong vector equilibrium problems (in short, (SSVEP)) studied in [17].

(iii)
If I is a singleton, {S}_{i}(x)=S(x), {T}_{i}(x)=T(x), and let F:K\times K\to {2}^{Y} be a setvalued mapping, C\subset Y be a nonempty, closed, and convex cone, and the relation R be defined as follows:
R(x,y)\text{holds}\phantom{\rule{1em}{0ex}}\text{iff}\phantom{\rule{1em}{0ex}}F(x,y)\subset C.Then (QVR) becomes the strong vector quasiequilibrium problem (in short, (SQVEP)) studied in [17].

(iv)
Yang and Pu [17] obtained some existence results for the system of strong vector quasiequilibrium problems. However, the assumptions of theorems in [17] are different from the assumptions of Theorem 2.1, Corollary 2.1, and Corollary 2.2.

(v)
Corollary 2.2 is a particular case of Theorems 3.1 and 3.3 in [2]. However, the assumptions and proof methods in Theorems 3.1 and 3.3 in [2] are different from the assumptions and proof methods of Corollary 2.2.
In the next example all assumptions of Corollary 2.2 are satisfied, but Theorem 3.3 in [17] is not applicable. The reason is that F is neither upper Ccontinuous nor properly Cquasiconvex.
Example 2.1 Let I=\{1\}, {X}_{i}={Y}_{i}=\mathbb{R}, {K}_{i}=[0,1], C={\mathbb{R}}_{+}, and let {S}_{i},{T}_{i}:{K}_{i}\times {K}_{i}\to {2}^{{K}_{i}} and F:{K}_{i}\times {K}_{i}\to {2}^{{Y}_{i}} be defined by
We let the relation {R}_{i} be defined by {R}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i}) holding iff {F}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})=F(x,y)\subseteq {\mathbb{R}}_{+}. We show that all assumptions of Corollary 2.2 are satisfied. However, F is neither upper Ccontinuous nor properly Cquasiconvex at {x}_{0}=\frac{1}{2}.
Firstly, we prove that F is not upper Ccontinuous at {x}_{0}=\frac{1}{2}. Indeed, we let a neighborhood U=[\frac{1}{6},\frac{1}{6}] of the origin in Z, then for any neighborhood V=[\frac{1}{2}\epsilon ,\frac{1}{2}+\epsilon ] of {x}_{0}=\frac{1}{2}, where \epsilon >0, we choose \frac{1}{2}\ne {x}^{\ast}\in V and y=\frac{1}{2}. Then
Next, we show that F is not properly Cquasiconvex at {x}_{0}=\frac{1}{2}. Indeed, we let y=\frac{1}{2}, \lambda =\frac{1}{2}, and {x}_{1}=0, {x}_{2}=1. Then
Thus, it gives also the case where Corollary 2.2 can be applied, but Theorem 3.3 in [17] does not work.
3 Essential components
In this section, we discuss the existence of essential components for (SQVR).
First, we recall some notions; see [17, 22, 23]. Let A be a nonempty and compact subset of a linear normed X. Denote by M the set of all upper semicontinuous maps R:A\to {2}^{A} with nonempty convex compact values. For any R,P\in M, we define \xi (R,P)={sup}_{x\in A}H(R(x),P(x)), where H is the Hausdorff metric defined in A. It is easy to verify that (M,\xi ) is a metric space. For each R\in M, we denote by \mathrm{\Xi}(R) the set of all fixed points of R. By KakutaniFanGlicksberg’s fixed point theorem, \mathrm{\Xi}(R) is a nonempty compact set.
For each R\in M, the connected component of a point x\in \mathrm{\Xi}(R) is the union of all the connected subsets of \mathrm{\Xi}(R) containing x. Note that the connected components are connected closed subsets of \mathrm{\Xi}(R), and, since A is compact, thus, all the connected components are connected compact. It is easy to see that the connected components of two distinct points of \mathrm{\Xi}(R) either coincide or are disjoint, so that all connected components constitute a decomposition of \mathrm{\Xi}(R) into connected pairwise disjoint compact subsets, i.e.,
where Λ is an index set. For each \alpha \in \mathrm{\Lambda}, {\mathrm{\Xi}}_{\alpha}(R) is a nonempty connected compact subset of \mathrm{\Xi}(R), and for any \alpha ,\beta \in \mathrm{\Lambda} (\alpha \ne \beta), {\mathrm{\Xi}}_{\alpha}(R)\cap {\mathrm{\Xi}}_{\beta}(R)=\mathrm{\varnothing}.
Definition 3.1 ([22])
Let R\in M and E be a nonempty and closed subset of \mathrm{\Xi}(R). E is said to be an essential set of \mathrm{\Xi}(R) if, for each open set O\supset E, there exists an open neighborhood U of R in M such that for any {R}^{\prime}\in U with \mathrm{\Xi}({R}^{\prime})\cap O\ne \mathrm{\varnothing}. If a connected component {\mathrm{\Xi}}_{\alpha}(R) of \mathrm{\Xi}(R) is an essential set with respect to M, then {\mathrm{\Xi}}_{\alpha}(R) is said be an essential component of \mathrm{\Xi}(R) with respect to M.
Lemma 3.1 ([23])
For any R\in M, there is at least one essential component of \mathrm{\Xi}(R) with respect to M.
Next, we discuss the existence of essential components for (SQVR).
Let Ω be the collection of all (SQVR) satisfying the conditions of Theorem 2.1. For each \omega \in \mathrm{\Omega}, denote by \mathrm{\Psi}(\omega ) the solution set of ω. It is easy to see that \mathrm{\Psi}(\omega )=\mathrm{\Xi}(\mathrm{\Phi}), where Φ is the bestreply map of ω. By the proof of Theorem 2.1, we know \mathrm{\Phi}\in M.
Definition 3.2 Let \omega \in \mathrm{\Omega} and {\mathrm{\Xi}}_{\alpha} a connected component of \mathrm{\Psi}(\omega ). {\mathrm{\Xi}}_{\alpha} is said to be essential if it, as a connected component of \mathrm{\Xi}(\mathrm{\Phi}), is an essential component of \mathrm{\Xi}(\mathrm{\Phi}) with respect to M, where Φ is the bestreply map of ω.
By Lemma 3.1, we obtain the following result.
Theorem 3.1 For any \omega \in \mathrm{\Omega}, there is at least one essential component of \mathrm{\Psi}(\omega ).
Remark 3.1

(i)
In the special case of Remark 2.1, Theorem 3.1 improves and extends Theorems 4.1 and 4.2 in [17].

(ii)
In 2011, Khanh and Quan [24] studied the existence of essential components for generalized KKM points. Moreover, Khanh and Quan also applied these results to optimizationrelated problems. However, the assumptions and proof methods are very different from the assumptions and proof methods in Theorem 3.1.

(iii)
If I is a singleton, then (SQVR) becomes the variational relation problem (in short, (VR)) studied by Khanh and Luc in [3]. However, Khanh and Luc discussed some kinds of semicontinuous sets as outercontinuous, innercontinuous, inneropen, and outeropen solution sets for (VR), while our Theorem 3.1 discusses the existence of essential components for (SQVR) by using KakutaniFanGlicksberg’s fixed point theorem.
4 Applications (I): system of quasivariational inclusion problems
For each i\in I, let {X}_{i}, {Y}_{i} and {K}_{i}, {K}_{\stackrel{\u02c6}{i}}, K, {T}_{i}={S}_{i} be as in (SQVR). Let {F}_{i}:{K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}\to {2}^{{Y}_{i}} be a setvalued mapping. We consider the following system of quasivariational inclusion problems (in short, (SQIP)): Find ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
where \overline{x} is a solution of (SQIP).
Definition 4.1 Let X, Y, Z be topological vector spaces. Suppose F:X\times Y\times X\to {2}^{Z} is a multifunction. F is said to be generalized \{0\}quasiconvex (in the first variable) in a convex set A\subset X, if, whenever 0\in F({x}_{1},y,z) and 0\in F({x}_{2},y,z), then 0\in F(\lambda {x}_{1}+(1\lambda ){x}_{2},y,z), \mathrm{\forall}{x}_{1},{x}_{2}\in A, \mathrm{\forall}\lambda \in [0,1].
Theorem 4.1 For each i\in I, assume that

(i)
{S}_{i} is continuous in K with nonempty compact convex values;

(ii)
for all ({x}_{i},{x}_{\stackrel{\u02c6}{i}})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}, 0\in {F}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{x}_{i});

(iii)
the set \{({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:0\notin {F}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i})\} is convex in {K}_{i};

(iv)
for all ({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}, {F}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i}) is generalized \{0\}quasiconvex in {K}_{i};

(v)
the set \{({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:0\in {F}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\} is closed.
Then the (SQIP) has a solution, i.e., there exists ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
Moreover, the solution set of the (SQIP) is closed.
Proof Let the relation {R}_{i} be defined as follows:
Then the problem (SQIP) becomes a particular case of (SQVR) and Theorem 4.1 is a direct consequence of Theorem 2.1. □
Next, we discuss the existence of essential components for (SQIP).
Let Θ be the collection of all (SQIP) satisfying the conditions of Theorem 4.1. For each \theta \in \mathrm{\Theta}, denote by \mathrm{\Delta}(\theta ) the solution set of θ. It is easy to see that \mathrm{\Delta}(\theta )=\mathrm{\Xi}(\mathrm{\Phi}), where Φ is the bestreply map of θ. By the proof of Theorem 4.1, we know \mathrm{\Phi}\in M.
Definition 4.2 Let \theta \in \mathrm{\Theta} and {\mathrm{\Xi}}_{\alpha} be a connected component of \mathrm{\Delta}(\theta ). {\mathrm{\Xi}}_{\alpha} is said to be essential if it, as a connected component of \mathrm{\Xi}(\mathrm{\Phi}), is an essential component of \mathrm{\Xi}(\mathrm{\Phi}) with respect to M, where Φ is the bestreply map of θ.
By Lemma 3.1, we also obtain the following result.
Theorem 4.2 For any \theta \in \mathrm{\Theta}, there is at least one essential component of \mathrm{\Delta}(\theta ).
5 Applications (II): system of weak vector quasiequilibrium problems
For each i\in I, let {X}_{i}, {Y}_{i} and {K}_{i}, {K}_{\stackrel{\u02c6}{i}}, K, {T}_{i}={S}_{i} be as in (SQVR). Let {f}_{i}:{K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}\to {Y}_{i} be a vector function and {C}_{i}\subset {Y}_{i} be a nonempty, closed, and convex cone. We consider the following weak system of quasiequilibrium problems (in short, (WSQVEP)).
(WSQVEP): Find ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
Definition 5.1 Let X, Y, Z be topological vector spaces and C\subset Z be a nonempty, closed, and convex cone. Suppose f:X\times Y\times X\to Z is a vector function. f is said to be weakly Cquasiconvex (in the first variable) in a convex set A\subset X, if whenever f({x}_{1},y,z)\notin intC and f({x}_{2},y,z)\notin intC, then f(\lambda {x}_{1}+(1\lambda ){x}_{2},y,z)\notin intC, \mathrm{\forall}{x}_{1},{x}_{2}\in A, \mathrm{\forall}\lambda \in [0,1].
Theorem 5.1 For each i\in I, assume that

(i)
{S}_{i} is continuous in K with nonempty compact convex values;

(ii)
for all ({x}_{i},{x}_{\stackrel{\u02c6}{i}})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}, {f}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{x}_{i})\notin int{C}_{i};

(iii)
the set \{({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:{f}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i})\in int{C}_{i}\} is convex in {K}_{i};

(iv)
for all ({x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}, {f}_{i}(\cdot ,{x}_{\stackrel{\u02c6}{i}},{y}_{i}) is weakly {C}_{i}quasiconvex in {K}_{i};

(v)
the set \{({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}:{f}_{i}({x}_{i},{x}_{\stackrel{\u02c6}{i}},{y}_{i})\notin int{C}_{i}\} is closed.
Then the (WSQVEP) has a solution, i.e., there exists ({\overline{x}}_{i},{\overline{x}}_{\stackrel{\u02c6}{i}})\in K such that, for each i\in I, {\overline{x}}_{i}\in {S}_{i}(\overline{x}) and
Moreover, the solution set of the (WSQVEP) is closed.
Proof Let the relation {R}_{i} be defined as follows:
Then the problem (WSQVEP) becomes a particular case of (SQVR) and Theorem 5.1 is a direct consequence of Theorem 2.1. □
Definition 5.2 ([14])
Let X and Z be two topological vector spaces and A\subseteq X be nonempty convex set, and C\subset Z be a nonempty, closed, and convex cone. Suppose f:A\to Z is a vector function. f is called Ccontinuous at {x}_{0}\in A if, for any open neighborhood V of the zero element θ in Z, there exists an open neighborhood U of {x}_{0} in A such that
and Ccontinuous in A if it is Ccontinuous at every point of A.
Remark 5.1 Lin et al. [14] obtained some existence results of (WSVQEP). However, the assumptions of Theorems 3.1, 3.3, and 3.4 in [14] are different from the assumptions in Theorem 5.1. Example 5.1 shows that all the assumptions of Theorem 5.1 are satisfied. But Theorem 3.1 in [14] does not work. The reason is that {f}_{i} is not{C}_{i}continuous.
Example 5.1 Let I=\{1\}, {X}_{i}={Y}_{i}={Z}_{i}=\mathbb{R}, {K}_{i}=[0,1], {C}_{i}={\mathbb{R}}_{+}, and let {S}_{i}:{K}_{i}\to {2}^{{K}_{i}} and f:{K}_{i}\to \mathbb{R} be defined by
We show that all assumptions of Theorem 5.1 are satisfied. However, f is notCcontinuous at {x}_{0}=\frac{1}{6}. Thus, it gives the case where Theorem 5.1 can be applied but Theorems 3.1, 3.3, and 3.4 in [14] do not work.
Remark 5.2 If we let {S}_{i}(x)={K}_{i} for each i\in I. Then (WSQVEP) becomes the system of vector equilibrium problem (in short, (SVEP)). If we let I=\{1\}, then (WSQVEP) becomes the strong vector equilibrium problem (in short, (VEP)). The problems (SVEP) and (VEP) are studied in [14].
Definition 5.3 ([14])
Let X and Z be two topological vector spaces and A\subseteq X be a nonempty convex set, and C\subset Z be a nonempty, closed, and convex cone. Suppose f:A\to Z is a vector function. f is called Cquasiconvexlike if, for any {x}_{i},{x}_{2}\in A and each \lambda \in [0,1], we have
Example 5.2 shows that in the special cases of Remarks 5.1 and 5.2, all the assumptions of Theorem 5.1 are satisfied. But Theorems 3.1, 3.3, and 3.4 in [14] do not work. The reason is that {f}_{i} is not{C}_{i}quasiconvexlike.
Example 5.2 Let {X}_{i}, {Y}_{i}, {Z}_{i}, {K}_{i}, {C}_{i} be as in Example 5.1, and let {S}_{i}:[0,1]\to {2}^{{K}_{i}} and f:[0,1]\to \mathbb{R} be defined by
We show that all assumptions of Theorem 5.1 are satisfied. However, f is notCquasiconvexlike at {x}_{0}=\frac{1}{6}. Indeed, let \lambda =\frac{1}{2} and {x}_{1}=0, {x}_{2}=\frac{1}{3}. Then
Thus, Theorem 5.1 can be applied but Theorems 3.1, 3.3, and 3.4 in [14] do not work.
Theorem 5.2 Assume for the problem (WSQVEP) assumptions (i), (ii), (iii), and (iv) are as in Theorem 5.1 and replace (v) by (v′)
(v′) {f}_{i} is continuous in {K}_{i}\times {K}_{\stackrel{\u02c6}{i}}\times {K}_{i}.
Then the (WSQVEP) has a solution. Moreover, the solution set of the (WSQVEP) is closed.
Proof We omit the proof since the technique is similar to that for Theorem 5.1 with suitable modifications. □
Next, we discuss the existence of essential components for (WSQVEP).
Let ϒ be the collection of all (WSQVEP) satisfying the conditions of Theorem 5.1. For each \gamma \in \mathrm{\Upsilon}, denote by \mathrm{\Pi}(\gamma ) the solution set of γ. It is easy to see that \mathrm{\Pi}(\gamma )=\mathrm{\Xi}(\mathrm{\Phi}), where Φ is the bestreply map of γ. By the proof of Theorem 5.1, we know that \mathrm{\Phi}\in M.
Definition 5.4 Let \gamma \in \mathrm{\Upsilon} and {\mathrm{\Xi}}_{\alpha} a connected component of \mathrm{\Pi}(\gamma ). {\mathrm{\Xi}}_{\alpha} is said to be essential if it, as a connected component of \mathrm{\Xi}(\mathrm{\Phi}), is an essential component of \mathrm{\Xi}(\mathrm{\Phi}) with respect to M, where Φ is the bestreply map of γ.
By Lemma 3.1, we also obtain the following result.
Theorem 5.3 For any \gamma \in \mathrm{\Upsilon}, there is at least one essential component of \mathrm{\Pi}(\gamma ).
Remark 5.3 In the special case of Remarks 5.1 and 5.2, Theorem 5.3 improves and extends Theorems 4.1 and 4.4 in [14].
References
Luc DT: An abstract problem in variational analysis. J. Optim. Theory Appl. 2008, 138: 65–76. 10.1007/s1095700893719
Agarwal RP, Balaj M, O’Regan D: A unifying approach to variational relation problems. J. Optim. Theory Appl. 2012, 155: 417–429. 10.1007/s109570120090x
Khanh PQ, Luc DT: Stability of solutions in parametric variational relation problems. SetValued Anal. 2008, 16: 1015–1035. 10.1007/s1122800801010
Balaj M, Lin LJ: Existence criteria for the solutions of two types of variational relation problems. J. Optim. Theory Appl. 2013, 156: 232–246. 10.1007/s1095701201360
Latif A, Luc DT: Variational relation problems: existence of solutions and fixed points of contraction mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 315
Balaj M, Luc DT: On mixed variational relation problems. Comput. Math. Appl. 2010, 60: 2712–2722. 10.1016/j.camwa.2010.09.026
Yang Z: On existence and essential stability of solutions of symmetric variational relation problems. J. Inequal. Appl. 2014., 2014: Article ID 5
Hung NV: Continuity of solutions for parametric generalized quasivariational relation problems. Fixed Point Theory Appl. 2012., 2012: Article ID 102
Hung NV: Sensitivity analysis for generalized quasivariational relation problems in locally G convex spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 158
Lin LJ, Ansari QH: System of quasivariational relations with applications. Nonlinear Anal. 2010, 72: 1210–1220. 10.1016/j.na.2009.08.005
Pu YJ, Yang Z: Stability of solutions for variational relation problem with applications. Nonlinear Anal. 2012, 75: 1758–1767. 10.1016/j.na.2011.09.007
Fort MK: Essential and nonessential fixed points. Am. J. Math. 1950, 72: 315–322. 10.2307/2372035
Kinoshita S: On essential components of the set of fixed points. Osaka J. Math. 1952, 4: 19–22.
Lin Z, Yang H, Yu J: On existence and essential components of the solution set for the system of vector quasiequilibrium problems. Nonlinear Anal. 2005, 63: 2445–2452. 10.1016/j.na.2005.03.049
Lin Z, Yu J: The existence of solutions for the system of generalized vector quasiequilibrium problems. Appl. Math. Lett. 2005, 18: 415–422. 10.1016/j.aml.2004.07.023
Hou SH, Gong XH, Yang XM: Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. J. Optim. Theory Appl. 2010, 146: 387–398. 10.1007/s1095701096567
Yang Z, Pu YJ: On the existence and essential components for solution set for system of strong vector quasiequilibrium problems. J. Glob. Optim. 2013, 55: 253–259. 10.1007/s108980119830y
Luc DT Lecture Notes in Economics and Mathematical Systems. In Theory of Vector Optimization. Springer, Berlin; 1989.
Ferro F: Optimization and stability results through cone lower semicontinuity. SetValued Anal. 1997, 5: 365–375. 10.1023/A:1008653120360
Yannelis N, Prabhakar ND: Existence of maximal elements and equilibria in linear topological spaces. J. Math. Econ. 1983, 12: 233–245. 10.1016/03044068(83)900411
Holmes RB: Geometric Functional Analysis and Its Application. Springer, New York; 1975.
Hillas J: On the definition of the strategic stability of equilibria. Econometrica 1990, 58: 1365–1390. 10.2307/2938320
Jiang JH: Essential component of the set of fixed points of the multivalued mappings and its application to the theory of games. Sci. Sin. 1963, 12: 951–964.
Khanh PQ, Quan NH: Generic stability and essential components of generalized KKM points and applications. J. Optim. Theory Appl. 2011, 148: 488–504. 10.1007/s1095701097644
Acknowledgements
The authors thank the editor and the two anonymous referees for their valuable remarks and suggestions, which helped them to considerably improve the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.
About this article
Cite this article
Hung, N.V., Kieu, P.T. On the existence and essential components of solution sets for systems of generalized quasivariational relation problems. J Inequal Appl 2014, 250 (2014). https://doi.org/10.1186/1029242X2014250
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014250