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Existence conditions for symmetric generalized quasivariational inclusion problems
Journal of Inequalities and Applications volume 2013, Article number: 40 (2013)
Abstract
In this paper, we establish an existence theorem by using the KakutaniFanGlicksberg fixedpoint theorem for a symmetric generalized quasivariational inclusion problem in real locally convex Hausdorff topological vector spaces. Moreover, the closedness of the solution set for this problem is obtained. As special cases, we also derive the existence results for symmetric weak and strong quasiequilibrium problems. The results presented in the paper improve and extend the main results in the literature.
MSC:90B20, 49J40.
1 Introduction
Let X and Z be real locally convex Hausdorff spaces, A\subset X be a nonempty subset and C\subset Z be a closed convex pointed cone. Let F:A\times A\to {2}^{Z} be a given setvalued mapping. Ansari et al. [1] introduced the following two problems (in short, (VEP) and (SVEP)), respectively:
Find \overline{x}\in A such that
and find \overline{x}\in A such that
The problem (VEP) is called the weak vector equilibrium problem and the problem (SVEP) is called the strong vector equilibrium problem. Later, these two problems have been studied by many authors; see, for example, [2, 3] and references.
In 2008, Long et al. [4] introduced a generalized strong vector quasiequilibrium problem (for short, (GSVQEP)). Let X, Y and Z be real locally convex Hausdorff topological vector spaces, A\subset X and B\subset Y be nonempty compact convex subsets and C\subset Z be a nonempty closed convex cone, and let S:A\to {2}^{A}, T:A\to {2}^{B}, F:A\times B\times A\to {2}^{Z} be setvalued mappings.
(GSVQEP): Find \overline{x}\in A and \overline{y}\in T(\overline{x}) such that \overline{x}\in S(\overline{x}) and
where \overline{x} is a strong solution of (GSVQEP).
Recently, Plubtieng and Sitthithakerngkietet [5] considered the system of generalized strong vector quasiequilibrium problems (in short, (SGSVQEPs)). This model is a general problem which contains (GVEP) and (QSVQEP). Let X, Y, Z be real locally convex Hausdorff topological vector spaces, A\subset X and B\subset Y be nonempty compact convex subsets and C\subset Z be a nonempty closed convex cone. Let {S}_{1},{S}_{2}:A\to {2}^{A}, {T}_{1},{T}_{2}:A\to {2}^{B} and {F}_{1},{F}_{2}:A\times B\times A\to {2}^{Z} be setvalued mappings. They considered (SGSVQEPs) as follows.
Find (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x}), \overline{v}\in {T}_{2}(\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x}), \overline{u}\in {S}_{2}(\overline{u}) and
and
where (\overline{x},\overline{u}) is a strong solution of (SGSVQEPs).
Very recently, new symmetric strong vector quasiequilibrium problems (in short, (SSVQEP)) in Hausdorff locally convex spaces were introduced by Chen et al. [6]. Let X, Y, Z be real locally convex Hausdorff topological vector spaces, A\subset X and B\subset Y be nonempty compact convex subsets and C\subset Z be a nonempty closed convex cone. Let {S}_{1},{S}_{2}:A\times A\to {2}^{A}, {T}_{1},{T}_{2}:A\times A\to {2}^{B} and {F}_{1},{F}_{2}:A\times B\times A\to {2}^{Z} be setvalued mappings. They considered the following (SSVQEP):
Find (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) and
and
where (\overline{x},\overline{u}) is a strong solution for the (SSVQEP).
Motivated by the research works mentioned above, in this paper, we introduce symmetric generalized quasivariational inclusion problems. Let X, Y, Z be real locally convex Hausdorff topological vector spaces and A\subset X, B\subset Y be nonempty compact convex subsets. Let {K}_{i},{P}_{i}:A\times A\to {2}^{A}, {T}_{i}:A\times A\to {2}^{B} and {F}_{i}:A\times B\times A\to {2}^{Z}, i=1,2, be setvalued mappings.
Now, we adopt the following notations (see [7–9]). Letters w, m and s are used for weak, middle and strong problems, respectively. For subsets U and V under consideration, we adopt the following notations:
Let \alpha \in \{\text{w, m, s}\}. We consider the following for symmetric generalized quasivariational inclusion problem (in short, (SQVIP_{ α })).
(SQVIP): Find (\overline{x},\overline{u})\in A\times A such that \overline{x}\in {K}_{1}(\overline{x},\overline{u}), \overline{u}\in {K}_{2}(\overline{x},\overline{u}) and
We denote that {\mathrm{\Xi}}_{\alpha}(F) is the solution set of (SQVIP_{ α }).
The symmetric generalized quasivariational inclusion problems include as special cases symmetric generalized vector quasiequilibrium problems, vector quasiequilibrium problems, symmetric vector quasivariational inequality problems, variational relation problems, etc. In recent years, a lot of results for the existence of solutions for symmetric vector quasiequilibrium problems, vector quasiequilibrium problems, vector quasivariational inequality problems, variational relation problems and optimization problems have been established by many authors in different ways. For example, equilibrium problems [1–6, 10–21], variational inequality problems [22–24], variational relation problems [7, 25], optimization problems [22, 26] and the references therein.
The structure of our paper is as follows. In the first part of this article, we introduce the model symmetric generalized quasivariational inclusion problem. In Section 2, we recall definitions for later use. In Section 3, we establish an existence and closedness theorem by using the KakutaniFanGlicksberg fixedpoint theorem for a symmetric generalized quasivariational inclusion problem. Applications to symmetric weak and strong vector quasiequilibrium problems are presented in Section 4.
2 Preliminaries
In this section, we recall some basic definitions and some of their properties.
Let X, Y be two topological vector spaces, A be a nonempty subset of X and F:A\to {2}^{Z} be a setvalued mapping.

(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. F is said to be lower semicontinuous in A if it is lower semicontinuous at all {x}_{0}\in A.

(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. F is said to be upper semicontinuous in A if it is upper semicontinuous at all {x}_{0}\in A.

(iii)
F is said to be continuous in A if it is both lsc and usc 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.
Definition 2 [27]
Let X, Y be two topological vector spaces, A be a nonempty subset of X, F:A\to {2}^{Y} be a multifunction and C\subset Y be a nonempty closed convex cone.

(i)
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
F(x)\subset F({x}_{0})+U+C,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in V. 
(ii)
F is called lower 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
F({x}_{0})\subset F(x)+UC,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in V.
Definition 3 [6]
Let X, Y be two topological vector spaces and A be a nonempty subset of X and C\subset Y be a nonempty closed convex cone. A setvalued mapping F:A\to {2}^{Y} is said to be type II Clower semicontinuous at {x}_{0}\in A if for each y\in F({x}_{0}) and any neighborhood U of the origin in Y, there exists a neighborhood U({x}_{0}) of {x}_{0} such that
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 Cconvex if for any x,y\in A and t\in [0,1], one has
F is said to be Cconcave if F is Cconvex.
Definition 5 [28]
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 t\in [0,1], we have
Lemma 6 [28]
Let X, Y be two topological vector spaces, A be a nonempty convex subset of X and F:A\to {2}^{Y} be a multifunction.

(i)
If F is upper semicontinuous at {x}_{0}\in A with closed values, then F is closed at {x}_{0}\in A;

(ii)
If F is closed at {x}_{0}\in A and Y is compact, then F is upper semicontinuous at {x}_{0}\in A.

(iii)
If F has compact values, then F is usc at {x}_{0}\in A if and only if, for each net \{{x}_{\alpha}\}\subseteq A which converges to {x}_{0}\in A and for each net \{{y}_{\alpha}\}\subseteq F({x}_{\alpha}), there are {y}_{0}\in F({x}_{0}) and a subnet \{{y}_{\beta}\} of \{{y}_{\alpha}\} such that {y}_{\beta}\to {y}_{0}.
Lemma 7 (KakutaniFanGlickcberg (see [29]))
Let A be a nonempty compact convex subset of a locally convex Hausdorff vector topological space X. If F:A\u27f6{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}).
3 Main results
In this section, we discuss the existence and closedness of the solution sets of symmetric generalized quasivariational inclusion problems by using the KakutaniFanGlicksberg fixed point theorem.
Theorem 8 For each \{i=1,2\}, assume for the problem (SQVIP_{ α }) that

(i)
{K}_{i} is usc in A\times A with nonempty convex closed values and {P}_{i} is lsc in A\times A with nonempty closed values;

(ii)
{T}_{i} is usc in A\times A with nonempty convex compact values if \alpha =w (or \alpha =m) and {T}_{i} is lsc in A\times A with nonempty convex values if \alpha =s;

(iii)
for all (x,z,u)\in A\times B\times A, 0\in {F}_{i}(x,z,{P}_{i}(x,u));

(iv)
for all (x,z,u)\in A\times B\times A, the set \{a\in {K}_{i}(x,u):0\in {F}_{i}(a,z,y),\mathrm{\forall}y\in {P}_{i}(x,u)\} is convex;

(v)
the set \{(x,z,y)\in A\times B\times A:0\in {F}_{i}(x,z,y)\} is closed.
Then the (SQVIP_{ α }) has a solution, i.e., there exist (\overline{x},\overline{u})\in A\times A such that \overline{x}\in {K}_{1}(\overline{x},\overline{u}), \overline{u}\in {K}_{2}(\overline{x},\overline{u}) and
Moreover, the solution set of the (SQVIP_{ α }) is closed.
Proof Similar arguments can be applied to three cases. We present only the proof for the case where \alpha =m.
Indeed, for all (x,z,u,v)\in A\times B\times A\times B, define mappings: {\mathrm{\Phi}}_{m},{\mathrm{\Pi}}_{m}:A\times B\times A\to {2}^{A} by
and

(a)
Show that {\mathrm{\Phi}}_{m}(x,z,u) and {\mathrm{\Pi}}_{m}(x,v,u) are nonempty convex sets.
Indeed, for all (x,z,u)\in A\times B\times A and (x,v,u)\in A\times B\times A, for each \{i=1,2\}, {K}_{i}(x,u), {P}_{i}(x,u) are nonempty. Thus, by assumptions (i), (ii) and (iii), we have {\mathrm{\Phi}}_{m}(x,z,u) and {\mathrm{\Pi}}_{m}(x,v,u) are nonempty. On the other hand, by the condition (iv), we also have {\mathrm{\Phi}}_{m}(x,z,u), {\mathrm{\Pi}}_{m}(x,v,u) are convex.

(b)
We will prove {\mathrm{\Phi}}_{m} and {\mathrm{\Pi}}_{m} are upper semicontinuous in A\times B\times A with nonempty closed values.
First, we show that {\mathrm{\Phi}}_{m} is upper semicontinuous in A\times B\times A with nonempty closed values. Since A is a compact set, by Lemma 6(ii), we need only to show that {\mathrm{\Phi}}_{m} is a closed mapping. Indeed, let a net \{({x}_{n},{z}_{n},{u}_{n}):n\in I\}\subset A\times B\times A such that ({x}_{n},{z}_{n},{u}_{n})\to (x,z,u)\in A\times B\times A, and let {a}_{n}\in {\mathrm{\Phi}}_{m}({x}_{n},{z}_{n},{u}_{n}) such that {a}_{n}\to {a}_{0}. Now we need to show that {a}_{0}\in {\mathrm{\Phi}}_{m}(x,z,u). Since {a}_{n}\in {K}_{1}({x}_{n},{u}_{n}) and {K}_{1} is upper semicontinuous with nonempty closed values by Lemma 6(i), hence {K}_{1} is closed, thus we have {a}_{0}\in {K}_{1}(x,u). Suppose the contrary {a}_{0}\notin {\mathrm{\Phi}}_{m}(x,z,u). Then \mathrm{\exists}{y}_{0}\in {P}_{1}(x,u) such that
By the lower semicontinuity of {P}_{1}, there is a net \{{y}_{n}\} such that {y}_{n}\in {P}_{1}({x}_{n},{u}_{n}), {y}_{n}\to {y}_{0}. Since {a}_{n}\in {\mathrm{\Phi}}_{m}({x}_{n},{z}_{n},{u}_{n}), we have
By the condition (v) and (2), we have
This is a contradiction between (3) and (1). Thus, {a}_{0}\in {\mathrm{\Phi}}_{m}(x,z,u). Hence, {\mathrm{\Phi}}_{m} is upper semicontinuous in A\times B\times A with nonempty closed values. Similarly, we also have {\mathrm{\Pi}}_{m}(x,v,u) is upper semicontinuous in A\times B\times A with nonempty closed values.

(c)
Now we need to prove the solution set {\mathrm{\Xi}}_{m}(F)\ne \mathrm{\varnothing}.
Define the setvalued mappings {\mathrm{\Theta}}_{m},{\mathrm{\Omega}}_{m}:A\times B\times A:\to {2}^{A\times B} by
and
Then {\mathrm{\Theta}}_{m}, {\mathrm{\Omega}}_{m} are upper semicontinuous and \mathrm{\forall}(x,z,u)\in A\times B\times A, \mathrm{\forall}(x,v,u)\in A\times B\times A, {\mathrm{\Theta}}_{m}(x,z,u) and {\mathrm{\Theta}}_{m}(x,v,u) are nonempty closed convex subsets of A\times B\times A.
Define the setvalued mapping H:(A\times B)\times (A\times B)\to {2}^{(A\times B)\times (A\times B)} by
Then H is also upper semicontinuous and \mathrm{\forall}((x,z),(u,v))\in (A\times B)\times (A\times B), H((x,z),(u,v)) is a nonempty closed convex subset of (A\times B)\times (A\times B).
By Lemma 7, there exists a point (({x}^{\ast},{z}^{\ast}),({v}^{\ast},{u}^{\ast}))\in (A\times B)\times (A\times B) such that (({x}^{\ast},{z}^{\ast}),({u}^{\ast},{v}^{\ast}))\in H(({x}^{\ast},{z}^{\ast}),({u}^{\ast},{v}^{\ast})), that is,
which implies that {x}^{\ast}\in {\mathrm{\Phi}}_{m}({x}^{\ast},{z}^{\ast},{u}^{\ast}), {z}^{\ast}\in {T}_{1}({x}^{\ast},{u}^{\ast}), {u}^{\ast}\in {\mathrm{\Pi}}_{m}({x}^{\ast},{v}^{\ast},{u}^{\ast}) and {v}^{\ast}\in {T}_{2}({x}^{\ast},{u}^{\ast}). Hence, there exists ({x}^{\ast},{u}^{\ast})\in A\times A, {z}^{\ast}\in {T}_{1}({x}^{\ast},{u}^{\ast}), {v}^{\ast}\in {T}_{2}({x}^{\ast},{u}^{\ast}) such that {x}^{\ast}\in {K}_{1}({x}^{\ast},{u}^{\ast}), {u}^{\ast}\in {K}_{2}({x}^{\ast},{u}^{\ast}), satisfying
and
i.e., (SQVIP_{ α }) has a solution.

(d)
Now we prove that {\mathrm{\Xi}}_{m}(F) is closed. Indeed, let a net \{({x}_{n},{u}_{n}),n\in I\}\in {\mathrm{\Xi}}_{m}(F): ({x}_{n},{u}_{n})\to ({x}_{0},{u}_{0}). We need to prove that ({x}_{0},{u}_{0})\in {\mathrm{\Xi}}_{m}(F). Indeed, by the lower semicontinuity of {P}_{i},i=1,2, for any {y}_{0}\in {P}_{i}({x}_{0},{u}_{0}), there exists {y}_{n}\in {P}_{i}({x}_{n},{u}_{n}) such that {y}_{n}\to {y}_{0}. Since ({x}_{n},{u}_{n})\in {\mathrm{\Xi}}_{m}(F), there exists {z}_{n}\in {T}_{1}({x}_{n},{u}_{n}), {v}_{n}\in {T}_{2}({x}_{n},{u}_{n}), {x}_{n}\in {K}_{1}({x}_{n},{u}_{n}), {u}_{n}\in {K}_{2}({x}_{n},{u}_{n}) such that
0\in {F}_{1}({x}_{n},{z}_{n},{y}_{n}),
and
Since {K}_{1}, {K}_{2} are upper semicontinuous in A\times A with nonempty closed values, by Lemma 6(i), we have {K}_{1}, {K}_{2} are closed. Thus, {x}_{0}\in {K}_{1}({x}_{0},{u}_{0}), {u}_{0}\in {K}_{2}({x}_{0},{u}_{0}). Since {T}_{1}, {T}_{2} are upper semicontinuous in A\times A with nonempty compact values, there exists {z}_{0}\in {T}_{1}({x}_{0},{u}_{0}) and {v}_{0}\in {T}_{2}({x}_{0},{u}_{0}) such that {z}_{n}\to {z}_{0}, {v}_{n}\to {v}_{0} (taking subnets if necessary). By the condition (v) and ({x}_{n},{z}_{n},{u}_{n},{v}_{n})\to ({x}_{0},{z}_{0},{u}_{0},{v}_{0}), we have
and
This means that ({x}_{0},{u}_{0})\in {\mathrm{\Xi}}_{m}(F). Thus, {\mathrm{\Xi}}_{m}(F) is a closed set.
□
If {K}_{1}(x,u)={P}_{1}(x,u)={S}_{1}(x,u), {K}_{2}(x,u)={P}_{2}(x,u)={S}_{2}(x,u), \alpha =m, and {F}_{1}(x,z,y)={G}_{1}(x,z,y)C, {F}_{2}(u,z,y)={G}_{2}(u,z,y)C, with {S}_{1},{S}_{2}:A\times A\to {2}^{A}, {G}_{1},{G}_{2}:A\times B\times A\to {2}^{Z} are setvalued mappings, and C\subset Z is a nonempty closed convex cone. Then (SQVIP_{ α }) becomes (SSVQEP) studied in [6].
In this special case, we have the following corollary.
Corollary 9 For each \{i=1,2\}, assume for the problem (SSVQEP) that

(i)
{S}_{i} is continuous in A\times A with nonempty convex closed values;

(ii)
{T}_{i} is usc in A\times A with nonempty convex compact values;

(iii)
for all (x,z,u)\in A\times B\times A, {G}_{i}(x,z,{S}_{i}(x,u))\subset C;

(iv)
for all (x,z,u)\in A\times B\times A, the set \{a\in {S}_{i}(x,u):{G}_{i}(a,z,y)\subset C,\mathrm{\forall}y\in {S}_{i}(x,u)\} is convex;

(v)
the set \{(x,z,y)\in A\times B\times A:{G}_{i}(x,z,y)\subset C\} is closed.
Then the (SSVQEP) has a solution, i.e., there exist (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) and
and
Moreover, the solution set of the (SSVQEP) is closed.
Remark 10 Chen et al. [6] obtained an existence result of (SSVQEP). However, the assumptions in Theorem 3.1 in [6] are different from the assumptions in Corollary 9. The following example shows that all assumptions of Corollary 9 are satisfied. But Theorem 3.1 in [6] is not fulfilled.
Example 11 Let X=Y=Z=\mathbb{R}, A=B=[0,1], C=[0,+\mathrm{\infty}) and let {S}_{1}(x)={S}_{2}(x)=[0,1], {G}_{1},{G}_{2},F:[0,1]\times [0,1]\times [0,1]\to {2}^{\mathbb{R}} and
and
We show that assumptions of Corollary 9 are easily seen to be fulfilled. Hence, by Corollary 9, (SSVQEP) has a solution. But F is neither type II Clower semicontinuous nor Cconcave at {x}_{0}=\frac{1}{2}. Thus, Theorem 3.1 in [6] does not work.
If {K}_{1}(x,u)={P}_{1}(x,u)={S}_{1}(x), {K}_{2}(x,u)={P}_{2}(x,u)={S}_{2}(u), {T}_{1}(x,u)={T}_{1}(x), {T}_{2}(x,u)={T}_{2}(u), \alpha =m and {F}_{1}(x,z,y)={G}_{1}(x,z,y)C, {F}_{2}(u,z,y)={G}_{2}(u,z,y)C, with {S}_{1},{S}_{2}:A\to {2}^{A}, {G}_{1},{G}_{2}:A\times B\times A\to {2}^{Z} are setvalued mappings, and C\subset Z is a nonempty closed convex cone. Then (SQVIP_{ α }) becomes (SGSVQEP) studied in [5].
In this special case, we have the following corollary.
Corollary 12 For each \{i=1,2\}, assume for the problem (SGSVQEP) that

(i)
{S}_{i} is continuous in A with nonempty convex closed values;

(ii)
{T}_{i} is usc in A with nonempty convex compact values;

(iii)
for all (x,z)\in A\times B, {G}_{i}(x,z,{S}_{i}(x))\subset C;

(iv)
for all (x,z)\in A\times B, the set \{a\in {S}_{i}(x):{G}_{i}(a,z,y)\subset C,\mathrm{\forall}y\in {S}_{i}(x)\} is convex;

(v)
the set \{(x,z,y)\in A\times B\times A:{G}_{i}(x,z,y)\subset C\} is closed.
Then the (SGSVQEP) has a solution, i.e., there exist (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x}), \overline{v}\in {T}_{2}(\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x}), \overline{u}\in {S}_{2}(\overline{u}) and
and
Moreover, the solution set of the (SGSVQEP) is closed.
Remark 13 In [5], PlubtiengSitthithakerngkiet also obtained an existence result of (SGSVQEP). However, the assumptions in Theorem 3.1 in [5] are different from the assumptions in Corollary 12. The following example shows that in this special case, all assumptions of Corollary 12 are satisfied. But Theorem 3.1 in [5] is not fulfilled.
Example 14 Let X=Y=Z=\mathbb{R}, A=B=[0,1], C=[0,+\mathrm{\infty}) and let {S}_{1}(x)={S}_{2}(x)=[0,1], F:[0,1]\times [0,1]\times [0,1]\to {2}^{\mathbb{R}} and
and
We show that all assumptions of Corollary 12 are satisfied. So, by this corollary, the considered problem has solutions. However, F is not lower (C)continuous at {x}_{0}=\frac{1}{2}. Also, Theorem 3.1 in [5] does not work.
If {K}_{1}(\overline{x},\overline{u})={P}_{1}(\overline{x},\overline{u})={K}_{2}(\overline{x},\overline{u})={P}_{2}(\overline{x},\overline{u})=S(\overline{x}), {T}_{1}(\overline{x},\overline{u})={T}_{2}(\overline{x},\overline{u})=\{z\} and {F}_{1}(x,z,y)={F}_{2}(u,z,y)=G(x,y)C, for each \overline{x},\overline{u}\in A and S:A\to {2}^{A}, G:A\times A\to {2}^{Z} are setvalued mappings, and C\subset Z is a nonempty closed convex cone. Then (SQVIP_{ α }) becomes (SVQEP) studied in [21].
In this special case, we also have the following corollary.
Corollary 15 Assume for the problem (SVQEP) that

(i)
S is continuous in A with nonempty convex closed values;

(ii)
for all x\in A, G(x,S(x))\subset C;

(iii)
for all x\in A, the set \{a\in S(x):G(a,y)\subset C,\mathrm{\forall}y\in S(x)\} is convex;

(iv)
the set \{(x,y)\in A\times A:G(x,y)\subset C\} is closed.
Then the (SVQEP) has a solution, i.e., there exists \overline{x}\in S(\overline{x}) such that
Moreover, the solution set of the (SVQEP) is closed.
The following example shows that in this special case, all assumptions of Corollary 15 are satisfied. But Theorem 3.3 in [21] is not fulfilled.
Example 16 Let X, Y, Z, A, B, C as in Example 14, and let S(x)=[0,1], G:[0,1]\times [0,1]\to {2}^{\mathbb{R}} and
We show that all assumptions of Corollary 15 are satisfied. So, (SVQEP) has a solution. However, G is not upper Ccontinuous at {x}_{0}=\frac{1}{2}. Also, Theorem 3.3 in [21] does not work.
The following example shows that all assumptions of Corollary 9, Corollary 12 and Corollary 15 are satisfied. However, Theorem 3.1 in [6], Theorem 3.1 in [5] and Theorem 3.3 in [21] are not fulfilled. The reason is that G is not properly Cquasiconvex.
Example 17 Let A, B, X, Y, Z, C as in Example 14, and let S:[0,1]\to {2}^{\mathbb{R}}, G:[0,1]\times [0,1]\to {2}^{\mathbb{R}}, {S}_{1}(x,u)={S}_{2}(x,u)=S(x)=[0,1], {T}_{1}(x,u)={T}_{2}(x,u)=T(x,u)=\{z\} and
We show that all assumptions of Corollary 9, Corollary 12 and Corollary 15 are satisfied. However, G is not properly Cquasiconvex at {x}_{0}=\frac{1}{2}. Thus, it gives the case where Corollary 9, Corollary 12 and Corollary 15 can be applied but Theorem 3.1 in [6], Theorem 3.1 in [5] and Theorem 3.3 in [21] do not work.
4 Applications
Since our symmetric vector quasiequilibrium problems include many rather general problems as particular cases mentioned in Section 1, from the results of Section 2 we can derive consequences for such special cases. In this section, we discuss only some corollaries for symmetric weak and strong quasiequilibrium problems as examples.
Let X, Y, Z, A, B be as in Section 1, and C\subset Z be a nonempty closed convex cone. Let {S}_{i},{P}_{i}:A\times A\to {2}^{A}, {T}_{i}:A\times A\to {2}^{B} be setvalued mappings and {f}_{i}:A\times B\times A\to Z, i=1,2 be vectorvalued functions. We consider the two following symmetric weak and strong vector quasiequilibrium problems (in short, (SWQVEP) and (SSQVEP)), respectively.
(SWQVEP): Find (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) satisfying
(SSQVEP): Find (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) satisfying
Corollary 18 For each \{i=1,2\}, assume for the problem (SWQVEP) that

(i)
{S}_{i} is continuous in A\times A with nonempty convex closed values;

(ii)
{T}_{i} is usc in A\times A with nonempty convex compact values;

(iii)
for all (x,z,u)\in A\times B\times A, {f}_{i}(x,z,{S}_{i}(x,u))\notin int\mathrm{C};

(iv)
for all (x,z,u)\in A\times B\times A, the set \{a\in {S}_{i}(x,u):{f}_{i}(a,z,y)\notin int\mathrm{C},\mathrm{\forall}y\in {S}_{i}(x,u)\} is convex;

(v)
the set \{(x,z,y)\in A\times B\times A:{f}_{i}(x,z,y)\notin int\mathrm{C}\} is closed.
Then the (SWQVEP) has a solution, i.e., there exist (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) satisfying
Moreover, the solution set of the (SWQVEP) is closed.
Proof Setting \alpha =m, {F}_{1}(x,z,y)=Z\setminus ({f}_{1}(x,z,y)+int\mathrm{C}) and {F}_{2}(u,z,y)=Z\setminus ({f}_{2}(u,z,y)+int\mathrm{C}), problem (SWQVEP) becomes a particular case of (SQVIP_{ α }) and Corollary 18 is a direct consequence of Theorem 8. □
Corollary 19 Assume for the problem (SWQVEP) assumptions (i), (ii), (iii) and (iv) as in Corollary 18 and replace (v) by (v′)
(v′) for each i=\{1,2\}, {f}_{i} is continuous in A\times B\times A.
Then the (SWQVEP) has a solution. Moreover, the solution set of the (SWQVEP) is closed.
Proof We omit the proof since the technique is similar to that for Corollary 18 with suitable modifications. □
Corollary 20 For each \{i=1,2\}, assume for the problem (SSQVEP) that

(i)
{S}_{i} is continuous in A\times A with nonempty convex closed values;

(ii)
{T}_{i} is usc in A\times A with nonempty convex compact values;

(iii)
for all (x,z,u)\in A\times B\times A, {f}_{i}(x,z,{S}_{i}(x,u))\in C;

(iv)
for all (x,z,u)\in A\times B\times A, the set \{a\in {S}_{i}(x,u):{f}_{i}(a,z,y)\in C,\mathrm{\forall}y\in {S}_{i}(x,u)\} is convex;

(v)
the set \{(x,z,y)\in A\times B\times A:{f}_{i}(x,z,y)\in C\} is closed.
Then the (SSQVEP) has a solution, i.e., there exist (\overline{x},\overline{u})\in A\times A and \overline{z}\in {T}_{1}(\overline{x},\overline{u}), \overline{v}\in {T}_{2}(\overline{x},\overline{u}) such that \overline{x}\in {S}_{1}(\overline{x},\overline{u}), \overline{u}\in {S}_{2}(\overline{x},\overline{u}) satisfying
Moreover, the solution set of the (SSQVEP) is closed.
Proof Setting \alpha =m, {F}_{1}(x,z,y)={f}_{1}(x,z,y)C and {F}_{2}(u,z,y)={f}_{2}(u,z,y)C, problem (SSQVEP) becomes a particular case of (SQVIP_{ α }) and the Corollary 20 is a direct consequence of Theorem 8. □
Corollary 21 Assume for the problem (SSQVEP) assumptions (i), (ii), (iii) and (iv) as in Corollary 20 and replace (v) by (v′)
(v′) for each i=\{1,2\}, {f}_{i} is continuous in A\times B\times A.
Then the (SSQVEP) has a solution. Moreover, the solution set of the (SSQVEP) is closed.
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Hung, N.V. Existence conditions for symmetric generalized quasivariational inclusion problems. J Inequal Appl 2013, 40 (2013). https://doi.org/10.1186/1029242X201340
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DOI: https://doi.org/10.1186/1029242X201340
Keywords
 symmetric generalized quasivariational inclusion problem
 symmetric weak quasiequilibrium problem
 symmetric strong quasiequilibrium problem
 KakutaniFanGlicksberg fixedpoint theorem
 existence
 closedness