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Symmetric strong vector quasiequilibrium problems in Hausdorff locally convex spaces
Journal of Inequalities and Applications volumeÂ 2011, ArticleÂ number:Â 56 (2011)
Abstract
In this article, a new symmetric strong vector quasiequilibrium problem in real locally convex Hausdorff topological vector spaces is introduced and studied. An existence theorem of solutions for the symmetric strong vector quasiequilibrium problem by using KakutaniFanGlicksberg fixed point theorem is obtained. Moreover, the closedness of the solution set for this problem is derived. The results presented in this article improve and extend some known results according to Long et al. [Math. Comput. Model. 47, 445451 (2008)], Somyot and Kanokwan [Fixed Point Theory Appl. doi:10.1155/2011/475121], and Wang et al. [Bull. Malays. Math. Sci. Soc. http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/200911022_R1.pdf].
2000 MSC: 49J40; 90C29.
1 Introduction
Equilibrium problem was introduced by Blum and Oettli [1] (see, also Noor and Oettli [2]). It provides a unified model of several classes of problems, for example, optimization problems, problems of Nash equilibrium, fixed point problems, variational inequalities, and complementarity problems. In recent years, there has been an increasing interest in the study of vector equilibrium problems. A lot of results for existence of solutions for vector equilibrium problems and vector variational inequalities have been established by many authors in different ways. For details, we refer the reader to various studies (see, e.g., [3â€“15] and the references therein).
Let X and Z be real locally convex Hausdorff spaces, K âŠ‚ X be a nonempty subset, and C âŠ‚ Z be a closed convex pointed cone. Let F : K Ã— K â†’ 2 ^{Z} be a given setvalued mapping. Ansari et al. [16] introduced two types of setvalued vector equilibrium problems as follows. The first type is the weak vector equilibrium problem: finding x âˆˆ K such that
The second type is the strong vector equilibrium problem (SVEP): finding x âˆˆ K such that
If int C â‰ âˆ… and x satisfies (1.1), then we call x a weak efficient solution for the vector equilibrium problem. If x satisfies (1.2), then we call x a strong efficient solution for the vector equilibrium problem.
It is worth mentioning that many existing results of the vector equilibrium problem are obtained under the assumption that the dual C* of the ordering cone C has a weak* compact base. As we know, for a normed space, the dual cone C* has a weak* compact base if and only if int C â‰ âˆ… (see [17]). However, in many cases, the ordering cone has an empty interior. For example, in the classical Banach spaces l^{p} and L^{p} (Î©), where 1 < p < âˆž, the standard ordering cone has an empty interior. Thus, it is interesting for the study of the existence of solutions and the properties of the solution sets for this case.
On the other hand, it is well known that a strong efficient solution of vector equilibrium problem is an ideal solution. It is better than other solutions, such as efficient solution, weak efficient solution, proper efficient solution, and supper efficient solution (see [18]). Hence, it is important to study the existence of strong efficient solution and the properties of the strong efficient solution set. Very recently, Long et al. [19] gave an existence theorem for generalized strong vector quasiequilibrium problem (GSVQEP) and discussed the stability of strong efficient solutions. Somyot and Kanokwan [20] derived an existence theorem for system of generalized strong vector quasiequilibrium problem (SGSVQEP) which extends the main results of Long et al. [19].
Motivated and inspired by the research studies mentioned above, in this article, we consider a type of symmetric strong vector quasiequilibrium problem (SSVQEP) without assuming that the dual of the ordering cone has a weak* compact base. Let X, Y, and Z be real locally convex Hausdorff topological vector spaces, K âŠ‚ X and D âŠ‚ Y be nonempty subsets, and C âŠ‚ Z be a nonempty closed convex cone. Suppose that S_{1}, S_{2} : K Ã— K â†’ 2 ^{K} , T_{1}, T_{2} : K Ã— K â†’ 2 ^{D} and F_{1}, F_{2} : K Ã— D Ã— K â†’ 2 ^{Z} are setvalued mappings. We consider the following SSVQEP: finding \left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right)\xe2\u02c6\u02c6K\xc3\u2014K and \mathrm{\xc8\xb3}\xe2\u02c6\u02c6{T}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \stackrel{\xcc\u201e}{v}\xe2\u02c6\u02c6{T}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) such that \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6{S}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \mathrm{\AA \xab}\xe2\u02c6\u02c6{S}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right),
and
We call this \left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) a strong efficient solution for the SSVQEP.
In this article, we establish an existence theorem of the solutions for the SSVQEP by using KakutaniFanGlicksberg fixed point theorem. We also discuss the closedness of the solution set for this problem. The results presented in this article improve and extend some known results according to Long et al. [19], Somyot and Kanokwan [20], and Wang et al. [21].
2 Preliminary results
Throughout this article, we suppose that X, Y, and Z are real locally convex Hausdorff topological vector spaces, K âŠ‚ X and D âŠ‚ Y are nonempty compact convex subsets, and C âŠ‚ Z is a nonempty closed convex cone. Suppose that S_{1}, S_{2} : K Ã— K â†’ 2 ^{K} , T_{1}, T_{2} : K Ã— K â†’ 2 ^{D} and F_{1}, F_{2} : K Ã— D Ã— K â†’ 2 ^{Z} are setvalued mappings.
For our main results, we need some definitions and lemmas as follows.
Definition 2.1. Let X and Y be two topological vector spaces and T : X â†’ 2 ^{Y} be a setvalued mapping.

(i)
T is said to be upper semicontinuous at x âˆˆ X if, for any neighborhood U of T(x), there exists a neighborhood V of x such that T(t) âŠ‚ U, for all t âˆˆ V. T is said to be upper semicontinuous on X if it is upper semicontinuous at each x âˆˆ X.

(ii)
T is said to be lower semicontinuous at x âˆˆ X if, for any open set U with T(x) âˆ© U â‰ âˆ…, there exists a neighborhood V of x such that T(x') âˆ© U â‰ âˆ…, for all x' âˆˆ V. T is said to be lower semicontinuous on X if it is lower semicontinuous at each x âˆˆ X.

(iii)
T is said to be continuous on X if it is both upper semicontinuous and lower semicontinuous on X.

(iv)
T is said to be closed if, Graph(T) = {(x, y): x âˆˆ X, y âˆˆ T(x)} is a closed subset in X Ã— Y.
Definition 2.2. Let W be a topological vector space and D âŠ‚ W be a nonempty set. A setvalued mapping G : D â†’ 2 ^{Z} is said to be type I Clower semicontinuous at x_{0} if, for any neighborhood U of 0 in Z, there exists a neighborhood U(x_{0}) of x_{0} such that
Definition 2.3. Let W be a topological vector space and D âŠ‚ W be a nonempty set. A setvalued mapping G : D â†’ 2 ^{Z} is said to be type II Clower semicontinuous at x_{0} if, for each z âˆˆ G(x_{0}) and any neighborhood U of 0 in Z, there exists a neighborhood U(x_{0}) of x_{0} such that
Definition 2.4. Let W be a topological vector space and D âŠ‚ W be a nonempty convex set. A setvalued mapping G : D â†’ 2 ^{Z} is said to be Cproperly quasiconvex if, for any x, y âˆˆ D, t âˆˆ [0, 1], we have
Definition 2.5. Let W be a topological vector space and D âŠ‚ W be a nonempty set. A setvalued mapping G : D â†’ 2 ^{Z} is said to be Cconvex if, for any x, y âˆˆ D and t âˆˆ [0, 1], one has
G is said to be Cconcave if G is Cconvex.
Definition 2.6. Let X be a Hausdorff topological vector space and K âŠ‚ X be a nonempty set. A setvalued mapping G : K â†’ 2 ^{X} is said to be a KKM mapping if, for any finite set {x_{1}, ..., x_{ n } } âŠ‚ K, the relation
holds, where co{x_{1}, ..., x_{ n } } denotes the convex hull of {x_{1}, ..., x_{ n } }.
Lemma 2.1. ([22]) Let X and Y be two Hausdorff topological spaces and T : X â†’ 2 ^{Y} be a setvalued mapping.

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

(ii)
If T is closed and Y is compact, then T is upper semicontinuous.
Lemma 2.2. (KakutaniFanGlicksberg [23]) Let X be a locally convex Hausdorff topological vector space and K be a nonempty compact convex subset of X. Let T : K â†’ 2 ^{K} be a upper semicontinuous setvalued mapping with nonempty closed convex values. Then, there exists \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6K such that \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6T\left(\stackrel{\xcc\u201e}{x}\right).
Lemma 2.3. ([24]) Let X and Y be two Hausdorff topological vector spaces and T : X â†’ 2 ^{Y} be a setvalued mapping with compact values. Then, T is upper semicontinuous on x âˆˆ X if and only if for any set {x_{ Î± } } with x_{ Î± } â†’ x, and y_{ Î± } âˆˆ T(x_{ Î± } ), there exists y âˆˆ T(x), and a subset {y_{ Î² } } of {y_{ Î± } }, such that y_{ Î² } â†’ y.
Lemma 2.4. ([25]) Let D be a nonempty convex compact subset of Hausdorff topological vector space X and E be a subset of D Ã— D such that

(i)
for each x âˆˆ D, (x, x) âˆ‰ E;

(ii)
for each x âˆˆ D, {y âˆˆ D : (x, y) âˆˆ E} is convex;

(iii)
for each y âˆˆ D, {x âˆˆ D : (x, y) âˆˆ E} is open in D.
Then, there exists \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6D such that \left(\stackrel{\xcc\u201e}{x},y\right)\xe2\u02c6\u2030E for all y âˆˆ D.
Lemma 2.5. ([26]) Let K be a nonempty subset of a topological vector space X and F : K â†’ 2 ^{X} be a KKM mapping with closed values. Assume that there exists a nonempty compact convex subset B of K such that D={\xe2\u2039\u201a}_{x\xe2\u02c6\u02c6B}F\left(x\right) is compact. Then, {\xe2\u2039\u201a}_{x\xe2\u02c6\u02c6K}F\left(x\right)\xe2\u2030\mathrm{\xe2\u02c6\dots}.
3 Main results
In this section, we apply KakutaniFanGlicksberg fixed point theorem to prove an existence theorem of solutions for the SSVQEP. Moreover, we prove the closedness of the solution set for this problem.
Lemma 3.1. Let W be a topological vector space and D âŠ‚ W be a nonempty subset. Let G : D â†’ 2 ^{Z} be a setvalued mapping.

(i)
If G is lower semicontinuous, then G is type II Clower semicontinuous.

(ii)
If G is type I (C)lower semicontinuous, then G is type II Clower semicontinuous.

(iii)
If G is singlevalued mapping, then G is type I (C)lower semicontinuous â‡” G is type II Clower semicontinuous.
Proof. (i) It is easy to verify that the assertion (i) holds and so we omit the proof.

(ii)
Suppose that G is type I (C)lower semicontinuous at x _{0} âˆˆ D. Then, for any z âˆˆ G(x _{0}), and for any neighborhood U of 0 in Z, there exists a balanced neighborhood V of 0 in Z such that V âŠ‚ U. Since G is type I (C)lower semicontinuous at x _{0} âˆˆ D, it follows that there exists a neighborhood U(x _{0}) of x _{0} such that
G\left({x}_{0}\right)\xe2\u0160\u201aG\left({x}^{\mathrm{\xe2\u20ac\xb2}}\right)+V+C,\phantom{\rule{1em}{0ex}}\xe2\u02c6\u20ac{x}^{\mathrm{\xe2\u20ac\xb2}}\xe2\u02c6\u02c6U\left({x}_{0}\right)\xe2\u02c6\copyright D.
For z âˆˆ G(x_{0}), there exist y âˆˆ G(x'), v âˆˆ V, c âˆˆ C such that z = y + v + c, and so
Thus,
It follows that G is type II Clower semicontinuous.

(iii)
Suppose that G is type II Clower semicontinuous at x _{0} âˆˆ D. Then, for any neighborhood U of 0 in Z, there exists a balanced neighborhood V of 0 in Z such that V âŠ‚ U. Since G is type II Clower semicontinuous at x _{0} âˆˆ D, it follows that there exists a neighborhood U(x _{0}) of x _{0} such that
G\left({x}^{\mathrm{\xe2\u20ac\xb2}}\right)\xe2\u0160\u201aG\left({x}_{0}\right)+VC,\phantom{\rule{1em}{0ex}}\xe2\u02c6\u20ac{x}^{\mathrm{\xe2\u20ac\xb2}}\xe2\u02c6\u02c6U\left({x}_{0}\right)\xe2\u02c6\copyright D.
Therefore,
This shows that G is type I (C)lower semicontinuous. This completes the proof.
The following example shows that the converse of (i) of Lemma 3.1 is not true.
Example 3.1. Let the setvalued mapping F from R into its subsets be defined by F(0) = [0, 1] and F(x) = {0} for all x â‰ 0. Let C = R_{+} = [0, +âˆž). Then, it is easy to see that F is not lower semicontinuous at 0. In fact, we can find a point y_{0} = 1 âˆˆ F(0) = [0, 1] and a neighborhood U\left({y}_{0}\right)=\left(\frac{1}{2},\frac{3}{2}\right) of y_{0} such that, for any neighborhood U(0) of 0, there exist some x_{0} â‰ 0 âˆˆ U(0) satisfying
This shows that F is not lower semicontinuous at 0. However, we can show that F is type II Clower semicontinuous at 0. In fact, for each y âˆˆ F(0) = [0, 1] and any neighborhood U of 0 in R, there exists Îµ_{0} > 0 such that (Îµ_{0}, Îµ_{0}) âŠ‚ U. Hence, y + (Îµ_{0}, Îµ_{0})  C = (âˆž, y + Îµ_{0}). It is easy to see that, for any neighborhood U(0) of 0,
Thus,
which shows that F is type II Clower semicontinuous at 0.
Theorem 3.1. For each i âˆˆ {1, 2}, let S_{ i } : K Ã— K â†’ 2 ^{K} be continuous setvalued mappings with nonempty compact convex values and T_{ i } : K Ã— K â†’ 2 ^{D} be upper semicontinuous setvalued mappings with nonempty compact convex values. Let F_{ i } : K Ã— D Ã— K â†’ 2 ^{Z} be setvalued mappings which satisfy the following conditions:

(i)
for all (x, y) âˆˆ K Ã— D, F_{ i } (x, y, x) âŠ‚ C;

(ii)
for all (y, z) âˆˆ D Ã— K, F_{ i } (Â·, y, z) are Cconcave;

(iii)
for all (x, y) âˆˆ K Ã— D, F_{ i } (x, y, Â·) are Cproperly quasiconvex;

(iv)
F_{ i } (Â·, Â·, Â·) are type II Clower semicontinuous.
Then, the SSVQEP has a solution. Moreover, the solution set of the SSVQEP is closed.
Proof. For any (x, y, u, v) âˆˆ K Ã— D Ã— K Ã— D, define mappings A, B : K Ã— D Ã— K â†’ 2 ^{K} by

(I)
For any (x, y, u) âˆˆ K Ã— D Ã— K, A(x, y, u) is nonempty.
Indeed, by the assumption, S_{1}(x, u) is nonempty compact convex set for each (x, u) âˆˆ K Ã— K. Set
If E is empty, then it is clear that A(x, y, u) is nonempty. Thus, we consider that E is not empty. For any z âˆˆ S_{1}(x, u), if (a, z) âˆˆ E, then there exists d âˆˆ F_{1}(a, y, z) such that d âˆ‰ C. Hence, there exists an open neighborhood U of 0 in Z such that (d + U) âˆ© C = âˆ… and so
By the type II Clower semicontinuity of F_{1}, there exists an open neighborhood U_{1} of a in S_{1}(x, u) such that
From (3.2), we know that there exists d' âˆˆ F_{1}(a', y, z) such that d' âˆˆ d + U  C. By (3.1), d' âˆ‰ C. Hence,
Thus, for any z âˆˆ S_{1}(x, z), {a âˆˆ S_{1}(x, u): (a, z) âˆˆ E} is open in S_{1}(x, u). For any a âˆˆ S_{1}(x, u), (a, z_{1}) âˆˆ E, (a, z_{2}) âˆˆ E, t âˆˆ [0, 1], it follows that
By condition (iii), we have
We claim that F_{1}(a, y, tz_{1} + (1  t)z_{2}) âŠ„ C. If not, then by (3.5), we have
which contradicts (3.4). Hence, F_{1}(a, y, tz_{1} + (1  t)z_{2}) âŠ„ C. Thus, for any a âˆˆ S_{1}(x, u), {z âˆˆ S_{1}(x, u): (a, z) âˆˆ E} is convex in S_{1}(x, u). The condition (i) implies that for any a âˆˆ S_{1}(x, u), (a, a) âˆ‰ E. By Lemma 2.4, there exists a âˆˆ S_{1}(x, u) such that (a, z) âˆ‰ E for all z âˆˆ S_{1}(x, u) i.e., F_{1}(a, y, z) âŠ‚ C for all z âˆˆ S_{1}(x, u). Hence, A(x, y, u) â‰ âˆ….

(II)
For any (x, y, u) âˆˆ K Ã— D Ã— K, A(x, y, u) is convex.
In fact, let a_{1}, a_{2} âˆˆ A(x, y, u), t âˆˆ [0, 1]. Then, a_{1}, a_{2} âˆˆ S_{1}(x, u),
and
By the convexity of S_{1}(x, u), we have ta_{1} + (1  t)a_{2} âˆˆ S_{1}(x, y). It follows from (3.6) and (3.7) that
Thus, ta_{1} + (1  t)a_{2} âˆˆ A(x, y, u). Therefore, A(x, y, u) is convex.

(III)
A is upper semicontinuous on K Ã— D Ã— K.
Since K is compact, we only need to show that A is a closed mapping. Let {(x_{ Î± } , y_{ Î± } , u_{ Î± } ): Î± âˆˆ I} âŠ‚ K Ã— D Ã— K be a set with (x_{ Î± } , y_{ Î± } , u_{ Î± } ) â†’ (x, y, u) âˆˆ K Ã— D Ã— K. Let v_{ Î± } âˆˆ A(x_{ Î± } , y_{ Î± } , u_{ Î± } ) with v_{ Î± } â†’ v. We will show that v âˆˆ A(x, y, u). Since S_{1} is upper semicontinuous mapping with nonempty closed values, it follows that S_{1} is a closed mapping. It follows from v_{ Î± } âˆˆ S_{1}(x_{ Î± } , u_{ Î± } ) and (x_{ Î± } , u_{ Î± } , v_{ Î± } ) â†’ (x, u, v) that we have v âˆˆ S_{1}(x, u). Now we claim that v âˆˆ A(x, y, u). If not, then there exists z_{1} âˆˆ S_{1}(x, u) such that
Hence, there exists d âˆˆ F_{1}(v, y, z_{1}) such that d â‰ C, and so there exists an open neighborhood U of 0 in Z such that (d + U) âˆ© C + âˆ…. Therefore,
Since F_{1} is type II Clower semicontinuous, for d âˆˆ F_{1}(v, y, z_{1}) and U, there exists a neighborhood U(v, y, z_{1}) of (v, y, z_{1}) such that, for all \left({v}^{\mathrm{\xe2\u20ac\xb2}},{y}^{\mathrm{\xe2\u20ac\xb2}},{z}_{1}^{\mathrm{\xe2\u20ac\xb2}}\right)\xe2\u02c6\u02c6U\left(v,y,{z}_{1}\right)\xe2\u02c6\copyright \left(K\xc3\u2014D\xc3\u2014K\right),
Since (x_{ Î± } , u_{ Î± } ) â†’ (x, u) and S is lower semicontinuous, for z_{1} âˆˆ S_{1}(x, u), there exists z_{ Î± } âˆˆ S_{1}(x_{ Î± } , u_{ Î± } ) such that z_{ Î± } â†’ z_{1}. Thus, (v_{ Î± } , y_{ Î± } , z_{ Î± } ) â†’ (v, y, z_{1}). It follows from (3.10) that there exists Î±_{0} âˆˆ I such that, for Î± â‰¥ Î±_{0},
From v_{ Î± } âˆˆ A(v_{ Î± } , y_{ Î± } , z_{ Î± } ), we have
Because z_{ Î± } âˆˆ S_{1}(x_{ Î± } , u_{ Î± } ), we get F_{1}(v_{ Î± } , y_{ Î± } , z_{ Î± } ) âŠ‚ C. It follows from (3.9) that
which contradicts (3.11). Hence, A is a closed mapping.
Similarly, we know for any (x, v, u) âˆˆ K Ã— D Ã— K, B is upper semicontinuous on K Ã— D Ã— K with nonempty closed convex values.

(IV)
Define the setvalued mappings H, G : K Ã— D Ã— K â†’ 2^{KÃ—D}by
H\left(x,y,u\right)=\left(A\left(x,y,u\right),{T}_{1}\left(x,u\right)\right),\phantom{\rule{1em}{0ex}}\xe2\u02c6\u20ac\left(x,y,u\right)\xe2\u02c6\u02c6K\xc3\u2014D\xc3\u2014K
and
Then, it is easy to see that H and G are upper semicontinuous mappings with nonempty closed convex values.
Define the setvalued mapping M : (K Ã— D) Ã— (K Ã— D) â†’ 2^{(KÃ—D)Ã—(KÃ—D)}by
Then, we know that M is upper semicontinuous mapping with nonempty closed convex values. By Lemma 2.2, there exists a point \left(\left(\stackrel{\xcc\u201e}{x},\mathrm{\xc8\xb3}\right),\left(\mathrm{\AA \xab},\stackrel{\xcc\u201e}{v}\right)\right)\xe2\u02c6\u02c6\left(K\xc3\u2014D\right)\xc3\u2014\left(K\xc3\u2014D\right) such that \left(\left(\stackrel{\xcc\u201e}{x},\mathrm{\xc8\xb3}\right),\left(\mathrm{\AA \xab},\stackrel{\xcc\u201e}{v}\right)\right)\xe2\u02c6\u02c6M\left(\left(\stackrel{\xcc\u201e}{x},\mathrm{\xc8\xb3}\right),\left(\mathrm{\AA \xab},\stackrel{\xcc\u201e}{v}\right)\right), that is
This implies that \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6A\left(\stackrel{\xcc\u201e}{x},\mathrm{\xc8\xb3},\mathrm{\AA \xab}\right), \mathrm{\xc8\xb3}\xe2\u02c6\u02c6{T}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \mathrm{\AA \xab}\xe2\u02c6\u02c6B\left(\stackrel{\xcc\u201e}{x},\stackrel{\xcc\u201e}{v},\mathrm{\AA \xab}\right), \stackrel{\xcc\u201e}{v}\xe2\u02c6\u02c6{T}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right). Hence, \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6{S}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \mathrm{\xc8\xb3}\xe2\u02c6\u02c6{T}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \mathrm{\AA \xab}\xe2\u02c6\u02c6{S}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), \stackrel{\xcc\u201e}{v}\xe2\u02c6\u02c6{T}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right),
and
Next, we show that the solution set of the SSVQEP is closed. Let {(x_{ Î± } , u_{ Î± } ): Î± âˆˆ I} be a set in the set of solutions of SSVQEP with \left({x}_{\mathrm{\xce\pm}},{u}_{\mathrm{\xce\pm}}\right)\xe2\u2020\u2019\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right), and so there exist y_{ Î± } âˆˆ T_{1}(x_{ Î± } , u_{ Î± } ), v_{ Î± } âˆˆ T_{2}(x_{ Î± } , u_{ Î± } ), x_{ Î± } âˆˆ S_{1}(x_{ Î± } , u_{ Î± } ), u_{ Î± } âˆˆ S_{2}(x_{ Î± } , u_{ Î± } ), such that
and
Since S_{1} and S_{2} are upper semicontinuous setvalued mappings with nonempty closed values, it follows from Lemma 2.1 that S_{1} and S_{2} are closed mappings. Thus, \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6{S}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) and \mathrm{\AA \xab}\xe2\u02c6\u02c6{S}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right). Since T_{1} is upper semicontinuous setvalued mapping with nonempty compact values, by Lemma 2.3, there exist \mathrm{\xc8\xb3}\xe2\u02c6\u02c6{T}_{1}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) and a subset {y_{ Î² } } of {y_{ Î± } } such that {y}_{\mathrm{\xce\xb2}}\xe2\u2020\u2019\mathrm{\xc8\xb3}. Similarly, there exist \stackrel{\xcc\u201e}{v}\xe2\u02c6\u02c6{T}_{2}\left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) and a subset {v_{ Î³ } } of {v_{ Î± } } such that {v}_{\mathrm{\xce\xb3}}\xe2\u2020\u2019\stackrel{\xcc\u201e}{v}. By the condition (iv), similar to the proof of the part (III), we have
and
Hence, \left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right) belongs to the set of solutions of SSVQEP. Thus, the set of solutions of SSVQEP is closed set. This completes the proof.
Now we give an example to show Theorem 3.1 is applicable.
Example 3.2. Let X = Y = Z = R, C = [0, +âˆž) and K = D = [0, 1]. For each x âˆˆ K, u âˆˆ K, let {S}_{1}\left(x,u\right)=\left[\frac{x}{2},1\right], {S}_{2}\left(x,u\right)=\left[0,\frac{u+1}{2}\right], {T}_{1}\left(x,u\right)=\left[0,\frac{1x}{3}+\frac{1u}{2}\right] and {T}_{2}\left(x,u\right)=\left[0,\frac{1x}{2}\right]. Define the setvalued mappings F_{1} and F_{2} as follows:
and
It is easy to verify that all the conditions in Theorem 3.1 are satisfied. By Theorem 3.1, we know that SSVQEP has a solution. Let M be the solution set of SSVQEP. Then,
It is easy to see that M is a closed subset of K Ã— K.
For each i = {1, 2}, if we suppose that S_{ i } is a setvalued mapping from K to K and T_{ i } is a setvalued mapping from K to D, then similar to the proof of Theorem 3.1, we have the following corollary.
Corollary 3.1. Let S_{1}, S_{2} : K â†’ 2 ^{K} be continuous setvalued mappings with nonempty compact convex values and T_{1}, T_{2} : K â†’ 2 ^{D} be upper semicontinuous setvalued mappings with nonempty compact convex values. Let F_{1}, F_{2} : K Ã— D Ã— K â†’ 2 ^{Z} be setvalued mappings satisfy the conditions (i)(iv) of Theorem 3.1. Then, the SGSVQEP has a solution, i.e., there exist \left(\stackrel{\xcc\u201e}{x},\mathrm{\AA \xab}\right)\xe2\u02c6\u02c6K\xc3\u2014K and \mathrm{\xc8\xb3}\xe2\u02c6\u02c6{T}_{1}\left(\mathrm{\AA \xab}\right), \stackrel{\xcc\u201e}{v}\xe2\u02c6\u02c6{T}_{2}\left(\stackrel{\xcc\u201e}{x}\right) such that \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6{S}_{1}\left(\stackrel{\xcc\u201e}{x}\right), \mathrm{\AA \xab}\xe2\u02c6\u02c6{S}_{2}\left(\mathrm{\AA \xab}\right),
and
Moreover, the solution set of the SGSVQEP is closed.
Remark 3.1. In [20], Somyot and Kanokwan also obtained an existence result for SGSVQEP. However, the assumptions of Theorem 3.1 in [20] are quite different from the ones in Corollary 3.1. The following example shows the case, where Corollary 3.1 is applicable, but the hypotheses of the corresponding theorem in [20] cannot be satisfied.
Example 3.3. Let X = Y = Z = R, C = [0, +âˆž) and K = D = [0, 1]. For each x âˆˆ K, u âˆˆ K, let S_{1}(x) = [x, 1], {S}_{2}\left(x\right)=\left[\frac{1x}{2},\frac{1}{2}\right], T_{1}(x) = [1  x, 1] and {T}_{2}\left(x\right)=\left[0,\frac{1+x}{2}\right]. Define the setvalued mappings F_{1} and F_{2} as follows:
and
It is easy to verify that all the conditions in Corollary 3.1 are satisfied. Hence, by Corollary 3.1, SGSVQEP has a solution. Let N be the solution set of SGSVQEP. Then,
It is easy to see that N is a closed subset of K Ã— K. However, the hypothesis (i) of Theorem 3.1 in [20] is not satisfied. Thus, Theorem 3.1 in [20] is not applicable.
If we take S = S_{1} = S_{2}, F = F_{1} = F_{2}, and T = T_{1} = T_{2}; then, from Corollary 3.1, we have the following corollary.
Corollary 3.2. Let S : K â†’ 2 ^{K} be continuous setvalued mappings with nonempty compact convex values and T : K â†’ 2 ^{D} be upper semicontinuous setvalued mappings with nonempty compact convex values. Let F : K Ã— D Ã— K â†’ 2 ^{Z} be setvalued mapping which satisfies the conditions (i)(iv) of Theorem 3.1. Then, the GSVQEP has a solution, i.e., there exist \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6K and \mathrm{\xc8\xb3}\xe2\u02c6\u02c6T\left(\stackrel{\xcc\u201e}{x}\right) such that \stackrel{\xcc\u201e}{x}\xe2\u02c6\u02c6S\left(\stackrel{\xcc\u201e}{x}\right) and
Moreover, the solution set of the GSVQEP is closed.
Remark 3.2. In [19], Long et al. also obtained an existence result for GSVQEP. However, the assumptions in Corollary 3.2 are quite different from the ones in Theorem 3.1 in [19]. The following example shows the case, where Corollary 3.2 is applicable, but the hypotheses of the corresponding theorem in [19] cannot be satisfied.
Example 3.4. Let X = Y = Z = R, C = [0, +âˆž) and K = D = [0, 1]. For each x âˆˆ K, let S\left(x\right)=\left[\frac{x}{3},1\right], T\left(x\right)=\left[0,\frac{x}{2}\right]. Define the setvalued mapping F as follows:
It is easy to verify that all the conditions in Corollary 3.2 are satisfied. Hence, by Corollary 3.2, GSVQEP has a solution. Let O be the solution set of GSVQEP. Then,
It is easy to see that O is a closed subset of K. However, the hypothesis (i) of Theorem 3.1 in [19] is not satisfied. Thus, Theorem 3.1 in [19] is not applicable.
If for any x âˆˆ K, S(x) = T(x) â‰¡ K and F(x, y, z) := F(x, y), then GSVQEP collapses to SVEP. Next we give an existence theorem of SVEP on a noncompact set.
Theorem 3.2. Let X and Z be two real Hausdorff topological vector spaces, K âŠ‚ X a nonempty closed convex subset and C âŠ‚ Z a closed convex cone. Let F : K Ã— K â†’ 2 ^{Z} be a setvalued mapping. Suppose that

(i)
for any x âˆˆ K, F(x, x) âŠ‚ C;

(ii)
for any x âˆˆ K, the set {y âˆˆ K : F(x, y) âŠ„ C} is empty or convex;

(iii)
for any y âˆˆ K, the set {x âˆˆ K : F(x, y) âŠ‚ C} is closed;

(iv)
there exist a nonempty compact subset E of K and a nonempty convex compact subset D of K such that, for each x âˆˆ K\E, there exists y âˆˆ D such that F(x, y) âŠ„ C.
Then, the SVEP has a solution. Moreover, the solution set of the SVEP is compact.
Proof. We define G : K â†’ 2 ^{K} as follows:
It follows from condition (i) that for any y âˆˆ K, we have y âˆˆ G(y) and so G(y) â‰ âˆ…. We claim that G is a KKM mapping. Suppose to the contrary that there exists a finite subset {y_{1}, ..., y_{ n } } of K, and there exists \mathrm{\xc8\xb3}\xe2\u02c6\u02c6\mathsf{\text{co}}\left\{{y}_{1},\xe2\u20ac\xa6,{y}_{n}\right\} such that \mathrm{\xc8\xb3}\xe2\u02c6\u2030{\xe2\u2039\u0192}_{i=1}^{n}G\left({y}_{i}\right). Hence, \mathrm{\xc8\xb3}={\xe2\u02c6\u2018}_{i=1}^{n}{t}_{i}{y}_{i} for some t_{ i } â‰¥ 0, 1 â‰¤ i â‰¤ n, with {\xe2\u02c6\u2018}_{i=1}^{n}{t}_{i}=1, and \mathrm{\xc8\xb3}\xe2\u02c6\u2030G\left({y}_{i}\right) for all i = 1, ..., n. Therefore,
Equation 3.12 implies that {y}_{i}\xe2\u02c6\u02c6\left\{z\xe2\u02c6\u02c6K,F\left(\mathrm{\xc8\xb3},z\right)\xe2\u0160\u201eC\right\}, for all i = 1, ..., n. By condition (ii), we have
which contradicts condition (i). Hence, G is a KKM mapping. Applying conditions (iii) and (iv), we deduce that {\xe2\u2039\u201a}_{y\xe2\u02c6\u02c6D}G\left(y\right) is a closed subset of E. Now, G satisfies all the assumptions of Lemma 2.5 and hence {\xe2\u2039\u201a}_{y\xe2\u02c6\u02c6K}G\left(y\right)\xe2\u2030\mathrm{\xe2\u02c6\dots}. This means that SVEP has a solution. By condition (iii), the solution set of SVEP is closed and by condition (iv), it is subset of the compact set E. Thus, the solution set of SVEP is compact. This completes the proof.
Remark 3.3. Theorem 3.2 is different from Theorem 3.1 of Wang et al. [21] in the following two aspects.

(a)
The condition (iv) in Theorem 3.2 is weaker than the condition (iv) in Theorem 3.1 of Wang et al. [21]; hence, Theorem 3.2 generalizes Theorem 3.1 of Wang et al. [21];

(b)
Theorem 3.2 is proved using FanKKM lemma, while Theorem 3.1 of Wang et al. [21] was proved using Brouwer fixed point theorem.
Example 3.5. Let X = Z = R, K = C = [0, +âˆž). Define the setvalued mapping F as follows:
If we take E = [0, 1] âˆª{2}, D = [0, 2], it is easy to verify that all the conditions in Theorem 3.2 are satisfied. Hence, by Theorem 3.2, SVEP has a solution. Let P be the solution set of SVEP. Then, P = [0, 1]. It is obvious that P is a compact subset of K.
Corollary 3.3. Let X, Z, K, C and F be as in Theorem 3.2. Assume that the conditions (i), (ii), and (iv) of Theorem 3.2 and the following condition holds:
(iii') for any y âˆˆ K, F(Â·, y) is type II Clower semicontinuous.
Then, the SVEP has a solution. Moreover, the solution set of the SVEP is compact.
Proof. By Theorem 3.2, we only need to show that for any y âˆˆ K, the set
is closed.
Indeed, let {x_{ Î± } } âŠ‚ G(y) be an arbitrary set such that x_{ Î± } â†’ x_{0}. We need to show that x_{0} âˆˆ G(y). Since x_{ Î± } âˆˆ K and K is closed, we have x_{0} âˆˆ K. In addition, for each Î±,
We claim that F(x_{0}, y) âŠ‚ C. If not, there exists z âˆˆ F(x_{0}, y) such that z âˆ‰ C. Hence, there exists a neighborhood U of 0 in Z such that (z + U) âˆ© C = âˆ…, which implies
By condition (iii'), we know that there exists Î±_{0} such that, for Î± â‰¥ Î±_{0},
Hence,
which contradicts (3.13). Hence, x_{0} âˆˆ G(y) and so G(y) is closed. This completes the proof.
Remark 3.4. It follows from Lemma 3.1 that Corollary 3.3 generalizes Theorem 3.2 of Wang et al. [21].
Corollary 3.4. Let X, Z, K, C, and F be the same as in Theorem 3.2. Assume that the conditions (i), (iii), and (iv) of Theorem 3.2 and the following condition holds:
(ii') for any x âˆˆ K, F(x, Â·) is Cproperly quasiconvex.
Then, the SVEP has a solution. Moreover, the solution set of the SVEP is closed.
Proof. By Theorem 3.2, we only need to show that, for any x âˆˆ K, the set
is convex. Indeed, let y_{1}, y_{2} âˆˆ G(x), t âˆˆ [0, 1],
By condition (iii), we have
We claim that F(x, ty_{1} + (1  t)y_{2}) âŠ„ C. If not, then by (3.15), we have
which contradicts (3.14). Hence, F(x, ty_{1} + (1  t)y_{2}) âŠ„ C. Thus, ty_{1} + (1  t)y_{2} âˆˆ G(x), and so G(x) is convex. This completes the proof.
Abbreviations
 GSVQEP:

generalized strong vector quasiequilibrium problem
 SGSVQEP:

system of generalized strong vector quasiequilibrium problem
 SSVQEP:

symmetric strong vector quasiequilibrium problem
 SVEP:

strong vector equilibrium problem.
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Acknowledgements
The authors are grateful to the editor and referees for their valuable comments and suggestions. This study was supported by the Key Program of NSFC (Grant No. 70831005), the National Natural Science Foundation of China (11171237), and the Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00050).
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BC carried out the study of the existence theorem of solutions and the closedness of the solution set for the symmetric strong vector quasiequilibrium problems and drafted the manuscript. NJH participated in the design of the study and gave some examples to show the main results. YJC conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Chen, B., Huang, Nj. & Cho, Y.J. Symmetric strong vector quasiequilibrium problems in Hausdorff locally convex spaces. J Inequal Appl 2011, 56 (2011). https://doi.org/10.1186/1029242X201156
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DOI: https://doi.org/10.1186/1029242X201156