Hung Journal of Inequalities and Applications 2013, 2013:40http://www.journalofinequalitiesandapplications.com/content/2013/1/40

RESEARCH Open Access

Existence conditions for symmetricgeneralized quasi-variational inclusionproblemsNguyen Van Hung*

*Correspondence:ngvhungdhdt@yahoo.comDepartment of Mathematics, DongThap University, 783 Pham Huu LauStreet, Ward 6, Cao Lanh City,Vietnam

AbstractIn this paper, we establish an existence theorem by using the Kakutani-Fan-Glicksbergfixed-point theorem for a symmetric generalized quasi-variational inclusion problemin real locally convex Hausdorff topological vector spaces. Moreover, the closednessof the solution set for this problem is obtained. As special cases, we also derive theexistence results for symmetric weak and strong quasi-equilibrium problems. Theresults presented in the paper improve and extend the main results in the literature.MSC: 90B20; 49J40

Keywords: symmetric generalized quasi-variational inclusion problem; symmetricweak quasi-equilibrium problem; symmetric strong quasi-equilibrium problem;Kakutani-Fan-Glicksberg fixed-point theorem; existence; closedness

1 IntroductionLet X and Z be real locally convex Hausdorff spaces, A ⊂ X be a nonempty subset andC ⊂ Z be a closed convex pointed cone. Let F : A × A → Z be a given set-valued map-ping. Ansari et al. [] introduced the following two problems (in short, (VEP) and (SVEP)),respectively:Find x ∈ A such that

F(x, y) �⊂ – intC, ∀y ∈ A,

and find x ∈ A such that

F(x, y)⊂ C, ∀y ∈ A.

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 havebeen studied by many authors; see, for example, [, ] and references.In , Long et al. [] introduced a generalized strong vector quasi-equilibrium prob-

lem (for short, (GSVQEP)). Let X, Y and Z be real locally convex Hausdorff topologicalvector spaces, A ⊂ X and B ⊂ Y be nonempty compact convex subsets and C ⊂ Z be anonempty closed convex cone, and let S : A → A, T : A → B, F : A × B × A → Z beset-valued mappings.

© 2013 Hung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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(GSVQEP): Find x ∈ A and y ∈ T(x) such that x ∈ S(x) and

F(x, y,x)⊂ C, ∀x ∈ S(x),

where x is a strong solution of (GSVQEP).Recently, Plubtieng and Sitthithakerngkietet [] considered the system of generalized

strong vector quasi-equilibrium problems (in short, (SGSVQEPs)). This model is a gen-eral problem which contains (GVEP) and (QSVQEP). Let X, Y , Z be real locally convexHausdorff topological vector spaces, A⊂ X and B⊂ Y be nonempty compact convex sub-sets andC ⊂ Z be a nonempty closed convex cone. Let S,S : A → A, T,T : A→ B andF,F : A×B×A → Z be set-valued mappings. They considered (SGSVQEPs) as follows.Find (x, u) ∈ A×A and z ∈ T(x), v ∈ T(u) such that x ∈ S(x), u ∈ S(u) and

F(x, z, y)⊂ C, ∀y ∈ S(x)

and

F(u, v, y)⊂ C, ∀y ∈ S(u),

where (x, u) is a strong solution of (SGSVQEPs).Very recently, new symmetric strong vector quasi-equilibrium problems (in short,

(SSVQEP)) in Hausdorff locally convex spaces were introduced by Chen et al. []. LetX, Y , Z be real locally convex Hausdorff topological vector spaces, A ⊂ X and B ⊂ Y benonempty compact convex subsets and C ⊂ Z be a nonempty closed convex cone. LetS,S : A × A → A, T,T : A × A → B and F,F : A × B × A → Z be set-valued map-pings. They considered the following (SSVQEP):Find (x, u) ∈ A × A and z ∈ T(x, u), v ∈ T(x, u) such that x ∈ S(x, u), u ∈ S(x, u)

and

F(x, z, y)⊂ C, ∀y ∈ S(x, u)

and

F(u, v, y)⊂ C, ∀y ∈ S(x, u),

where (x, u) is a strong solution for the (SSVQEP).Motivated by the research works mentioned above, in this paper, we introduce symmet-

ric generalized quasi-variational inclusion problems. Let X, Y , Z be real locally convexHausdorff topological vector spaces and A ⊂ X, B⊂ Y be nonempty compact convex sub-sets. LetKi,Pi : A×A→ A,Ti : A×A → B and Fi : A×B×A→ Z , i = , , be set-valuedmappings.Now, we adopt the following notations (see [–]). Letters w, m and s are used for weak,

middle and strong problems, respectively. For subsets U and V under consideration, we

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adopt the following notations:

(u, v) w U ×V means ∀u ∈U ,∃v ∈ V ,

(u, v) m U ×V means ∃v ∈ V ,∀u ∈ U ,

(u, v) s U ×V means ∀u ∈U ,∀v ∈ V .

Let α ∈ {w, m, s}. We consider the following for symmetric generalized quasi-variationalinclusion problem (in short, (SQVIPα)).(SQVIPα): Find (x, u) ∈ A×A such that x ∈ K(x, u), u ∈ K(x, u) and

(y, z)αP(x, u)× T(x, u) satisfying ∈ F(x, z, y),

(y, v)αP(x, u)× T(x, u) satisfying ∈ F(u, v, y).

We denote that �α(F) is the solution set of (SQVIPα).The symmetric generalized quasi-variational inclusion problems include as special cases

symmetric generalized vector quasi-equilibrium problems, vector quasi-equilibriumproblems, symmetric vector quasi-variational inequality problems, variational relationproblems, etc. In recent years, a lot of results for the existence of solutions for symmet-ric vector quasi-equilibrium problems, vector quasi-equilibrium problems, vector quasi-variational inequality problems, variational relation problems and optimization problemshave been established by many authors in different ways. For example, equilibrium prob-lems [–, –], variational inequality problems [–], variational relation problems[, ], optimization problems [, ] 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 quasi-variational inclusion problem. In Section , we recalldefinitions for later use. In Section , we establish an existence and closedness theoremby using the Kakutani-Fan-Glicksberg fixed-point theorem for a symmetric generalizedquasi-variational inclusion problem. Applications to symmetric weak and strong vectorquasi-equilibrium problems are presented in Section .

2 PreliminariesIn this section, we recall some basic definitions and some of their properties.

Definition [, ] Let X, Y be two topological vector spaces, A be a nonempty subsetof X and F : A → Z be a set-valued mapping.

(i) F is said to be lower semicontinuous (lsc) at x ∈ A if F(x)∩U �= ∅ for some openset U ⊆ Y implies the existence of a neighborhood N of x such that F(x)∩U �= ∅,∀x ∈N . F is said to be lower semicontinuous in A if it is lower semicontinuous at allx ∈ A.

(ii) F is said to be upper semicontinuous (usc) at x ∈ A if for each open set U ⊇G(x),there is a neighborhood N of x such that U ⊇ F(x), ∀x ∈N . F is said to be uppersemicontinuous in A if it is upper semicontinuous at all x ∈ 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 ∈ A, y ∈ F(x)} is a closed subset in

A× Y .

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Definition [] Let X, Y be two topological vector spaces, A be a nonempty subset ofX, F : A→ Y be a multifunction and C ⊂ Y be a nonempty closed convex cone.

(i) F is called upper C-continuous at x ∈ A if for any neighborhood U of the origin inY , there is a neighborhood V of x such that

F(x)⊂ F(x) +U +C, ∀x ∈ V .

(ii) F is called lower C-continuous at x ∈ A if for any neighborhood U of the origin inY , there is a neighborhood V of x such that

F(x)⊂ F(x) +U –C, ∀x ∈ V .

Definition [] Let X, Y be two topological vector spaces andA be a nonempty subset ofX and C ⊂ Y be a nonempty closed convex cone. A set-valued mapping F : A→ Y is saidto be type II C-lower semicontinuous at x ∈ A if for each y ∈ F(x) and any neighborhoodU of the origin in Y , there exists a neighborhood U(x) of x such that

F(x)∩ (y +U –C) �= ∅, ∀x ∈U(x)∩A.

Definition [, ] Let X and Y be two topological vector spaces and A be a nonemptyconvex subset of X. A set-valued mapping F : A → Y is said to be C-convex if for anyx, y ∈ A and t ∈ [, ], one has

F(tx + ( – t)y

) ⊂ tF(x) + ( – t)F(y) –C.

F is said to be C-concave if -F is C-convex.

Definition [] Let X and Y be two topological vector spaces and A be a nonemptyconvex subset ofX. A set-valuedmapping F : A→ Y is said to be properlyC-quasiconvexif for any x, y ∈ A and t ∈ [, ], we have

either F(x)⊂ F(tx + ( – t)y

)+C,

or F(y)⊂ F(tx + ( – t)y

)+C.

Lemma [] Let X, Y be two topological vector spaces, A be a nonempty convex subsetof X and F : A → Y be a multifunction.

(i) If F is upper semicontinuous at x ∈ A with closed values, then F is closed at x ∈ A;(ii) If F is closed at x ∈ A and Y is compact, then F is upper semicontinuous at x ∈ A.(iii) If F has compact values, then F is usc at x ∈ A if and only if, for each net {xα} ⊆ A

which converges to x ∈ A and for each net {yα} ⊆ F(xα), there are y ∈ F(x) and asubnet {yβ} of {yα} such that yβ → y.

Lemma (Kakutani-Fan-Glickcberg (see [])) Let A be a nonempty compact convex sub-set of a locally convex Hausdorff vector topological space X . If F : A –→ A is upper semi-continuous and for any x ∈ A,F(x) is nonempty, convex and closed, then there exists anx* ∈ A such that x* ∈ F(x*).

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3 Main resultsIn this section, we discuss the existence and closedness of the solution sets of symmetricgeneralized quasi-variational inclusion problems by using the Kakutani-Fan-Glicksbergfixed point theorem.

Theorem For each {i = , }, assume for the problem (SQVIPα) that(i) Ki is usc in A×A with nonempty convex closed values and Pi is lsc in A×A with

nonempty closed values;(ii) Ti is usc in A×A with nonempty convex compact values if α = w (or α =m) and Ti

is lsc in A×A with nonempty convex values if α = s;(iii) for all (x, z,u) ∈ A× B×A, ∈ Fi(x, z,Pi(x,u));(iv) for all (x, z,u) ∈ A× B×A, the set {a ∈ Ki(x,u) : ∈ Fi(a, z, y),∀y ∈ Pi(x,u)} is

convex;(v) the set {(x, z, y) ∈ A× B×A : ∈ Fi(x, z, y)} is closed.

Then the (SQVIPα) has a solution, i.e., there exist (x, u) ∈ A × A such that x ∈ K(x, u),u ∈ K(x, u) and

(y, z)αP(x, u)× T(x, u) satisfying ∈ F(x, z, y),

(y, v)αP(x, u)× T(x, u) satisfying ∈ F(u, v, y).

Moreover, the solution set of the (SQVIPα) is closed.

Proof Similar arguments can be applied to three cases. We present only the proof for thecase where α =m.Indeed, for all (x, z,u, v) ∈ A×B×A×B, define mappings:�m,�m : A×B×A→ A by

�m(x, z,u) ={a ∈ K(x,u) : ∈ F(a, z, y),∀y ∈ P(x,u)

},

and

�m(x, v,u) ={b ∈ K(x,u) : ∈ F(b, v, y),∀y ∈ P(x,u)

}.

(a) Show that �m(x, z,u) and �m(x, v,u) are nonempty convex sets.Indeed, for all (x, z,u) ∈ A× B×A and (x, v,u) ∈ A× B×A, for each {i = , },

Ki(x,u), Pi(x,u) are nonempty. Thus, by assumptions (i), (ii) and (iii), we have�m(x, z,u) and �m(x, v,u) are nonempty. On the other hand, by the condition (iv),we also have �m(x, z,u), �m(x, v,u) are convex.

(b) We will prove �m and �m are upper semicontinuous in A× B×A with nonemptyclosed values.First, we show that �m is upper semicontinuous in A× B×A with nonempty

closed values. Since A is a compact set, by Lemma (ii), we need only to show that�m is a closed mapping. Indeed, let a net {(xn, zn,un) : n ∈ I} ⊂ A× B×A such that(xn, zn,un)→ (x, z,u) ∈ A× B×A, and let an ∈ �m(xn, zn,un) such that an → a.Now we need to show that a ∈ �m(x, z,u). Since an ∈ K(xn,un) and K is uppersemicontinuous with nonempty closed values by Lemma (i), hence K is closed,

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thus we have a ∈ K(x,u). Suppose the contrary a /∈ �m(x, z,u). Then∃y ∈ P(x,u) such that

/∈ F(a, z, y). ()

By the lower semicontinuity of P, there is a net {yn} such that yn ∈ P(xn,un),yn → y. Since an ∈ �m(xn, zn,un), we have

∈ F(an, zn, yn). ()

By the condition (v) and (), we have

∈ F(a, z, y). ()

This is a contradiction between () and (). Thus, a ∈ �m(x, z,u). Hence, �m isupper semicontinuous in A× B×A with nonempty closed values. Similarly, we alsohave �m(x, v,u) is upper semicontinuous in A×B×A with nonempty closed values.

(c) Now we need to prove the solution set �m(F) �= ∅.Define the set-valued mappings �m,�m : A× B×A :→ A×B by

�m(x, z,u) =(�m(x, z,u),T(x,u)

), ∀(x, z,u) ∈ A× B×A

and

�m(x, v,u) =(�m(x, v,u),T(x,u)

), ∀(x, v,u) ∈ A× B×A.

Then �m, �m are upper semicontinuous and ∀(x, z,u) ∈ A× B×A,∀(x, v,u) ∈ A× B×A, �m(x, z,u) and �m(x, v,u) are nonempty closed convexsubsets of A× B×A.Define the set-valued mapping H : (A× B)× (A× B)→ (A×B)×(A×B) by

H((x, z), (u, v)

)=

(�m(x, z,u),�m(x, v,u)

), ∀(

(x, z), (u, v)) ∈ (A×B)× (A×B).

Then H is also upper semicontinuous and ∀((x, z), (u, v)) ∈ (A× B)× (A× B),H((x, z), (u, v)) is a nonempty closed convex subset of (A× B)× (A× B).By Lemma , there exists a point ((x*, z*), (v*,u*)) ∈ (A× B)× (A× B) such that

((x*, z*), (u*, v*)) ∈H((x*, z*), (u*, v*)), that is,

(x*, z*

) ∈ �m(x*, z*,u*

),

(u*, v*

) ∈ �m(x*, v*,u*

),

which implies that x* ∈ �m(x*, z*,u*), z* ∈ T(x*,u*), u* ∈ �m(x*, v*,u*) andv* ∈ T(x*,u*). Hence, there exists (x*,u*) ∈ A×A, z* ∈ T(x*,u*), v* ∈ T(x*,u*) suchthat x* ∈ K(x*,u*), u* ∈ K(x*,u*), satisfying

∈ F(x*, z*, y

), ∀y ∈ P

(x*,u*

),

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and

∈ F(u*, v*, y

), ∀y ∈ P

(x*,u*

),

i.e., (SQVIPα) has a solution.(d) Now we prove that �m(F) is closed. Indeed, let a net {(xn,un),n ∈ I} ∈ �m(F):

(xn,un)→ (x,u). We need to prove that (x,u) ∈ �m(F). Indeed, by the lowersemicontinuity of Pi, i = , , for any y ∈ Pi(x,u), there exists yn ∈ Pi(xn,un) suchthat yn → y. Since (xn,un) ∈ �m(F), there exists zn ∈ T(xn,un), vn ∈ T(xn,un),xn ∈ K(xn,un), un ∈ K(xn,un) such that

∈ F(xn, zn, yn),

and

∈ F(un, vn, yn).

Since K, K are upper semicontinuous in A×A with nonempty closed values, byLemma (i), we have K, K are closed. Thus, x ∈ K(x,u), u ∈ K(x,u). SinceT, T are upper semicontinuous in A×A with nonempty compact values, thereexists z ∈ T(x,u) and v ∈ T(x,u) such that zn → z, vn → v (taking subnetsif necessary). By the condition (v) and (xn, zn,un, vn)→ (x, z,u, v), we have

∈ F(x, z, y),

and

∈ F(u, v, y).

This means that (x,u) ∈ �m(F). Thus, �m(F) is a closed set. �

If K(x,u) = P(x,u) = S(x,u), K(x,u) = P(x,u) = S(x,u), α = m, and F(x, z, y) =G(x, z, y) –C, F(u, z, y) =G(u, z, y) –C, with S,S : A×A → A,G,G : A×B×A → Z

are set-valued mappings, and C ⊂ Z is a nonempty closed convex cone. Then (SQVIPα)becomes (SSVQEP) studied in [].In this special case, we have the following corollary.

Corollary For each {i = , }, assume for the problem (SSVQEP) that(i) Si is continuous in A×A with nonempty convex closed values;(ii) Ti is usc in A×A with nonempty convex compact values;(iii) for all (x, z,u) ∈ A× B×A, Gi(x, z,Si(x,u))⊂ C;(iv) for all (x, z,u) ∈ A× B×A, the set {a ∈ Si(x,u) :Gi(a, z, y)⊂ C,∀y ∈ Si(x,u)} is

convex;(v) the set {(x, z, y) ∈ A× B×A :Gi(x, z, y)⊂ C} is closed.

Then the (SSVQEP) has a solution, i.e., there exist (x, u) ∈ A × A and z ∈ T(x, u), v ∈T(x, u) such that x ∈ S(x, u), u ∈ S(x, u) and

G(x, z, y)⊂ C, ∀y ∈ S(x, u),

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and

G(u, v, y)⊂ C, ∀y ∈ S(x, u).

Moreover, the solution set of the (SSVQEP) is closed.

Remark Chen et al. [] obtained an existence result of (SSVQEP). However, the as-sumptions in Theorem . in [] are different from the assumptions in Corollary . Thefollowing example shows that all assumptions of Corollary are satisfied. But Theorem .in [] is not fulfilled.

Example Let X = Y = Z = R, A = B = [, ], C = [,+∞) and let S(x) = S(x) = [, ],G,G,F : [, ]× [, ]× [, ]→ R and

T(x,u) = T(x,u) =

⎧⎨⎩[, ] if x = u =

,[, ] otherwise.

and

G(x, z, y) =G(u, z, y) = F(x, z, y) =

⎧⎨⎩[ , ] if x = z = y =

,[, ] otherwise.

We show that assumptions of Corollary are easily seen to be fulfilled. Hence, by Corol-lary , (SSVQEP) has a solution. But F is neither type II C-lower semicontinuous norC-concave at x =

. Thus, Theorem . in [] does not work.

If K(x,u) = P(x,u) = S(x), K(x,u) = P(x,u) = S(u), T(x,u) = T(x), T(x,u) = T(u),α =m and F(x, z, y) =G(x, z, y)–C, F(u, z, y) =G(u, z, y)–C, with S,S : A → A,G,G :A× B× A → Z are set-valued mappings, and C ⊂ Z is a nonempty closed convex cone.Then (SQVIPα) becomes (SGSVQEP) studied in [].In this special case, we have the following corollary.

Corollary For each {i = , }, assume for the problem (SGSVQEP) that(i) Si is continuous in A with nonempty convex closed values;(ii) Ti is usc in A with nonempty convex compact values;(iii) for all (x, z) ∈ A× B, Gi(x, z,Si(x))⊂ C;(iv) for all (x, z) ∈ A× B, the set {a ∈ Si(x) :Gi(a, z, y)⊂ C,∀y ∈ Si(x)} is convex;(v) the set {(x, z, y) ∈ A× B×A :Gi(x, z, y)⊂ C} is closed.

Then the (SGSVQEP) has a solution, i.e., there exist (x, u) ∈ A×A and z ∈ T(x), v ∈ T(u)such that x ∈ S(x), u ∈ S(u) and

G(x, z, y)⊂ C, ∀y ∈ S(x)

and

G(u, v, y)⊂ C, ∀y ∈ S(u).

Moreover, the solution set of the (SGSVQEP) is closed.

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Remark In [], Plubtieng-Sitthithakerngkiet also obtained an existence result of(SGSVQEP). However, the assumptions in Theorem . in [] are different from the as-sumptions in Corollary . The following example shows that in this special case, all as-sumptions of Corollary are satisfied. But Theorem . in [] is not fulfilled.

Example Let X = Y = Z = R, A = B = [, ], C = [,+∞) and let S(x) = S(x) = [, ],F : [, ]× [, ]× [, ]→ R and

T(x) = T(x) =

⎧⎨⎩[, ] if x =

,[, ] otherwise.

and

G(x, z, y) =G(u, z, y) = F(x, z, y) =

⎧⎨⎩[ , ] if x = z = y =

,[, ] otherwise.

We show that all assumptions of Corollary are satisfied. So, by this corollary, the con-sidered problem has solutions. However, F is not lower (–C)-continuous at x =

. Also,Theorem . in [] does not work.

If K(x, u) = P(x, u) = K(x, u) = P(x, u) = S(x), T(x, u) = T(x, u) = {z} and F(x, z, y) =F(u, z, y) = G(x, y) – C, for each x, u ∈ A and S : A → A, G : A × A → Z are set-valued mappings, and C ⊂ Z is a nonempty closed convex cone. Then (SQVIPα) becomes(SVQEP) studied in [].In this special case, we also have the following corollary.

Corollary Assume for the problem (SVQEP) that(i) S is continuous in A with nonempty convex closed values;(ii) for all x ∈ A, G(x,S(x))⊂ C;(iii) for all x ∈ A, the set {a ∈ S(x) :G(a, y)⊂ C,∀y ∈ S(x)} is convex;(iv) the set {(x, y) ∈ A×A :G(x, y)⊂ C} is closed.

Then the (SVQEP) has a solution, i.e., there exists x ∈ S(x) such that

G(x, y)⊂ C, ∀y ∈ S(x).

Moreover, the solution set of the (SVQEP) is closed.

The following example shows that in this special case, all assumptions of Corollary are satisfied. But Theorem . in [] is not fulfilled.

Example Let X, Y , Z, A, B, C as in Example , and let S(x) = [, ],G : [, ]× [, ]→R and

G(x, y) =

⎧⎨⎩[, ] if x = z = y =

,[ ,

] otherwise.

We show that all assumptions of Corollary are satisfied. So, (SVQEP) has a solution.However,G is not upper C-continuous at x =

. Also, Theorem . in [] does not work.

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The following example shows that all assumptions of Corollary , Corollary andCorollary are satisfied. However, Theorem . in [], Theorem . in [] and Theo-rem . in [] are not fulfilled. The reason is that G is not properly C-quasiconvex.

Example Let A, B, X, Y , Z, C as in Example , and let S : [, ] → R, G : [, ] ×[, ]→ R, S(x,u) = S(x,u) = S(x) = [, ], T(x,u) = T(x,u) = T(x,u) = {z} and

G(x, z, y) =G(u, z, y) =G(x, y) =

⎧⎨⎩[, ] if x = y =

,[ , ] otherwise.

We show that all assumptions of Corollary , Corollary and Corollary are satisfied.However, G is not properly C-quasiconvex at x =

. Thus, it gives the case where Corol-lary , Corollary and Corollary can be applied but Theorem . in [], Theorem .in [] and Theorem . in [] do not work.

4 ApplicationsSince our symmetric vector quasi-equilibrium problems include many rather generalproblems as particular cases mentioned in Section , from the results of Section we canderive consequences for such special cases. In this section, we discuss only some corollar-ies for symmetric weak and strong quasi-equilibrium problems as examples.Let X, Y , Z, A, B be as in Section , and C ⊂ Z be a nonempty closed convex cone.

Let Si,Pi : A × A → A, Ti : A × A → B be set-valued mappings and fi : A × B × A → Z,i = , be vector-valued functions. We consider the two following symmetric weak andstrong vector quasi-equilibrium problems (in short, (SWQVEP) and (SSQVEP)), respec-tively.(SWQVEP): Find (x, u) ∈ A × A and z ∈ T(x, u), v ∈ T(x, u) such that x ∈ S(x, u), u ∈

S(x, u) satisfying

f(x, z, y) /∈ – intC, ∀y ∈ S(x, u),

f(u, v, y) /∈ – intC, ∀y ∈ S(x, u).

(SSQVEP): Find (x, u) ∈ A × A and z ∈ T(x, u), v ∈ T(x, u) such that x ∈ S(x, u), u ∈S(x, u) satisfying

f(x, z, y) ∈ C, ∀y ∈ S(x, u),

f(u, v, y) ∈ C, ∀y ∈ S(x, u).

Corollary For each {i = , }, assume for the problem (SWQVEP) that(i) Si is continuous in A×A with nonempty convex closed values;(ii) Ti is usc in A×A with nonempty convex compact values;(iii) for all (x, z,u) ∈ A× B×A, fi(x, z,Si(x,u)) /∈ – intC;(iv) for all (x, z,u) ∈ A× B×A, the set {a ∈ Si(x,u) : fi(a, z, y) /∈ – intC,∀y ∈ Si(x,u)} is

convex;(v) the set {(x, z, y) ∈ A× B×A : fi(x, z, y) /∈ – intC} is closed.

Hung Journal of Inequalities and Applications 2013, 2013:40 Page 11 of 12http://www.journalofinequalitiesandapplications.com/content/2013/1/40

Then the (SWQVEP) has a solution, i.e., there exist (x, u) ∈ A × A and z ∈ T(x, u), v ∈T(x, u) such that x ∈ S(x, u), u ∈ S(x, u) satisfying

f(x, z, y) /∈ – intC, ∀y ∈ S(x, u),

f(u, v, y) /∈ – intC, ∀y ∈ S(x, u).

Moreover, the solution set of the (SWQVEP) is closed.

Proof Setting α =m, F(x, z, y) = Z \ (f(x, z, y)+ intC) and F(u, z, y) = Z \ (f(u, z, y)+ intC),problem (SWQVEP) becomes a particular case of (SQVIPα) and Corollary is a directconsequence of Theorem . �

Corollary Assume for the problem (SWQVEP) assumptions (i), (ii), (iii) and (iv) as inCorollary and replace (v) by (v′)(v′) for each i = {, }, fi is continuous in A× B×A.Then the (SWQVEP) has a solution.Moreover, the solution set of the (SWQVEP) is closed.

Proof Weomit the proof since the technique is similar to that forCorollary with suitablemodifications. �

Corollary For each {i = , }, assume for the problem (SSQVEP) that(i) Si is continuous in A×A with nonempty convex closed values;(ii) Ti is usc in A×A with nonempty convex compact values;(iii) for all (x, z,u) ∈ A× B×A, fi(x, z,Si(x,u)) ∈ C;(iv) for all (x, z,u) ∈ A×B×A, the set {a ∈ Si(x,u) : fi(a, z, y) ∈ C,∀y ∈ Si(x,u)} is convex;(v) the set {(x, z, y) ∈ A× B×A : fi(x, z, y) ∈ C} is closed.

Then the (SSQVEP) has a solution, i.e., there exist (x, u) ∈ A × A and z ∈ T(x, u), v ∈T(x, u) such that x ∈ S(x, u), u ∈ S(x, u) satisfying

f(x, z, y) ∈ C, ∀y ∈ S(x, u),

f(u, v, y) ∈ C, ∀y ∈ S(x, u).

Moreover, the solution set of the (SSQVEP) is closed.

Proof Setting α = m, F(x, z, y) = f(x, z, y) – C and F(u, z, y) = f(u, z, y) – C, problem(SSQVEP) becomes a particular case of (SQVIPα) and the Corollary is a direct con-sequence of Theorem . �

Corollary Assume for the problem (SSQVEP) assumptions (i), (ii), (iii) and (iv) as inCorollary and replace (v) by (v′)(v′) for each i = {, }, fi is continuous in A× B×A.Then the (SSQVEP) has a solution.Moreover, the solution set of the (SSQVEP) is closed.

Competing interestsThe author declares that he has no competing interests.

Hung Journal of Inequalities and Applications 2013, 2013:40 Page 12 of 12http://www.journalofinequalitiesandapplications.com/content/2013/1/40

AcknowledgementsThe author thanks the two anonymous referees for their valuable remarks and suggestions, which helped them toimprove the article considerably.

Received: 5 July 2012 Accepted: 9 January 2013 Published: 6 February 2013

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doi:10.1186/1029-242X-2013-40Cite this article as: Hung: Existence conditions for symmetric generalized quasi-variational inclusion problems.Journal of Inequalities and Applications 2013 2013:40.