R E S E A R C H
Open Access
On the existence and essential components
of solution sets for systems of generalized
quasi-variational relation problems
Nguyen Van Hung and Phan Thanh Kieu
**Correspondence:
Department of Mathematics, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Vietnam
Abstract
In this paper, we study the existence of a solution for a system of quasi-variational 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 quasi-variational inclusions and to systems of weak vector quasi-equilibrium 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
Keywords: system quasi-variational relation problems; existence; essential components; system quasi-variational inclusion problems; system vector quasi-equilibrium problems
1 Introduction and preliminaries
Variational relation problems were first introduced and studied by Luc in []. These prob-lems include as special cases variational inclusion probprob-lems, vector equilibrium probprob-lems, 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 sta-bility of solutions in different models; see for example [–] and the references therein.
In , Fort [] 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 [] introduced the notion of essential components of the set of fixed points of a single-valued 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 [, ], vector variational inequality problems [],etc.
Very recently, Yang and Pu [] introduced and studied the system of strong vector quasi-equilibrium 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 sys-tem of generalized quasi-variational 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
variational relation problems. These results are then applied to systems of quasi-variational inclusion problems and systems of weak vector quasi-equilibrium problems.
Now, we pass to our problem setting. LetI={, . . . ,n}be an index set. For eachi∈I, let
XiandYibe two real locally convex Hausdorff topological vector spaces andKia nonempty convex compact subset ofXi. Denote
Kˆi=
j∈I,j=i
Kj, K=
i∈I
Ki=Ki×Kˆi, X=
i∈I
Xi.
For eachx∈K, we can writex= (xi,xˆi). For eachi∈I, letSi,Ti:K→Kibe set-valued mappings, and letRi(xi,xˆi,yi) be a relation linkingxi∈Ki,xˆi∈Kˆi andyi∈Ki. We con-sider the following system of generalized quasi-variational relation problems (in short, (SQVR)).
(SQVR): Find(x¯i,x¯ˆi)∈Ksuch that, for eachi∈I,x¯i∈Si(x¯)and
Ri(x¯i,x¯ˆi,yi)holds, ∀yi∈Ti(x¯),
wherex¯is a solution of (SQVR). We denote by(R) the solution set of (SQVR). Next, we recall some basic definitions and some of their properties.
LetX,Ybe two topological vector spaces; letAbe a nonempty subset ofXandF:A→
Y be a multifunction.
(i) Fis said to belower semicontinuous(lsc)atx∈AifF(x)∩U=∅for some open
setU⊆Yimplies the existence of a neighborhoodNofxsuch thatF(x)∩U=∅,
∀x∈N.
(ii) Fis said to beupper semicontinuous(usc)atx∈Aif, for each open setU⊇G(x),
there is a neighborhoodNofxsuch thatU⊇F(x),∀x∈N.
(iii) Fis said to becontinuous atx∈Aif it is both lsc and usc atx∈A.
(iv) Fis said to beclosedifGraph(F) ={(x,y) :x∈A,y∈F(x)}is a closed subset in A×Y.
LetA,B,Cbe convex sets in topological vector spaces andR(x,y,z) be a relation between elements of the three sets. The relationRis said to beclosedif the set{(x,y,z)∈A×B×C:
R(x,y,z) holds}is closed. The relationRis said to beconvexin the first variable if whenever
R(x,y,z) holds andR(x,y,z) holds, thenR(λx+ ( –λ)x,y,z) holds, for allx,x∈A, λ∈[, ].
Definition .([]) LetX,Ybe two topological vector spaces,Ais a nonempty subset of
X, andF:A→Y be a multifunction; andC⊂Yis a nonempty, closed, and convex cone.
F is calledupper C-continuous at x∈A, if, for any neighborhoodUof the origin inY, there is a neighborhoodVofxsuch that
F(x)⊂F(x) +U+C, ∀x∈V.
Definition .([]) LetXandYbe two topological vector spaces andAbe a nonempty convex subset ofX. A set-valued mappingF:A→Yis said to beproperly C-quasiconvex
if, for anyx,y∈Aandλ∈[, ], we have either F(x)⊂Ftx+ ( –t)y+C
Lemma .([]) Let X,Y be two Hausdorff topological vector spaces and F:X→Y be
a multivalued map.
(i) IfFis upper semicontinuous with closed values,thenFis closed. (ii) IfFis closed andYis compact,thenFis upper semicontinuous.
Lemma .([]) Let X,Y be two Hausdorff topological vector spaces and F:X→Y be
a set-valued mapping with compact values.Then F is upper semicontinuous x∈X if and
only if,for each net{xα} ⊆X which converges to x∈X and for each net{yα} ⊆F(xα),there
exist y∈F(x)and a subnet{yβ}of{yα}such that yβ→y.
Lemma .([]) Let A be a nonempty compact convex subset of a Hausdorff topological vector space X.Suppose that M:A→A∪ {∅}be a set-valued map with the following
conditions:
(i) for each atx∈A,M(x)is convex; (ii) for each atx∈A,x∈/M(x);
(iii) for each aty∈A,M–(y) ={x∈A:y∈M(x)}is open inA. Then there exists x∈A such that M(x) =∅.
Lemma .(Kakutani-Fan-Glicksberg []) Let A be a nonempty compact convex subset of a locally convex Hausdorff vector topological space X.If F:A→Ais upper
semicontin-uous and for any x∈A,F(x)is nonempty,convex,and closed,then there exists an x∗∈A such that x∗∈F(x∗).
2 Existence of solutions
In this section, we establish an existence theorem of solutions for system of generalized quasi-variational relation problems.
Theorem . For each i∈I,assume that
(i) Siis upper semicontinuous inKwith nonempty compact convex values;
(ii) Tiis lower semicontinuous inKwith nonempty convex values;
(iii) for all(xi,xˆi)∈Ki×Kˆi,Ri(xi,xˆi,xi)holds;
(iv) the set{(xˆi,yi)∈Kˆi×Ki:Ri(·,xˆi,yi)does not hold}is convex inKi;
(v) the relationRiis convex in the first variable and closed.
Then the(SQVR)has a solution,i.e.,there exists(x¯i,x¯ˆi)∈K such that,for each i∈I,x¯i∈
Si(x¯)and
Ri(x¯i,x¯ˆi,yi)holds, ∀yi∈Ti(x¯).
Moreover,the solution set of the(SQVR)is closed.
Proof We define a set-valued map::K→Kby(x) =
i∈Ii(x), where
i(x) =
ai∈Si(x) :Ri(ai,xˆi,yi) holds,∀yi∈Ti(x)
, ∀x∈K. (The mapis called the best-reply map; see [, ].)
For eachi∈I:
Indeed, for allx∈K, we define a set-valued mapMi:Si(x)→Si(x)∪ {∅}by
Mi(x) =yi∈Ti(x) :Ri(xi,xˆi,yi) does not hold
, for eachxi∈Si(x).
(a) For eachxi∈Si(x), by condition (iv),Mi(x)is a convex set.
(b) For eachxi∈Si(x), by condition (iii),xi∈/Mi(xi).
(c) For eachyi∈Ti(x), by condition (v), the set
M–
i (yi) ={xi∈Si(x) :yi∈Mi(x)}={xi∈Si(x) :Ri(xi,xˆi,yi)does not hold}is open in
Si(x).
By Lemma ., there existsx¯i∈Si(x) such thatMi(x¯i) =∅,i.e.,i(x)=∅. (II) We show thati(x) is convex.
Letai,ai ∈i(x) andλ∈[, ] and putai=λai+ ( –λ)ai. Sinceai,ai ∈Si(x) andSi(x) is convex, we haveai∈Si(x). Thus, forai,ai ∈i(x), it follows that
Ri(ai,xˆi,yi) holds, ∀yi∈Ti(x).
By (v),Riis convex in the first variable, we have
Ri
λai+ ( –λ)ai,xˆi,yi
holds, ∀λ∈[, ],∀yi∈Ti(x),
i.e.,ai∈i(x). Therefore,i(x) is convex.
(III) We will prove thatiis upper semicontinuous inKwith nonempty compact values. SinceK is a compact set, it suffices to show thati is a closed mapping. Indeed, let
{(aα i,x
α)}
α∈be any a net inGraph(i) such that (aαi,x
α)→(a
i,x). Now, we need only prove thata
i ∈i(x). Sinceaαi ∈Si(xα) andSiis upper semicontinuous atx∈K with nonempty compact values, we haveSiis closed atx∈K, thus,ai ∈Si(x). Suppose to the contraryai ∈/i(x). Then∃yi ∈Ti(x) such that
Ri
ai,xˆi,yi
does not hold. (.)
By the lower semicontinuity ofTi, there is a net{yα
i}withyαi ∈Ti(xα) such thatyαi →yi. Sinceaα
i ∈i(xα),
Ri
aαi,xαˆ
i,y α i
holds. (.)
By condition (v) and (.),
Rai,xˆi,y i
holds. (.)
This is a contradiction between (.) and (.). Thus,a
i ∈i(x). Hence,iis upper semi-continuous inKwith nonempty compact values.
By the definition of the mappingis upper semicontinuous with nonempty compact
values. By Lemma ., there exists (x¯i,x¯ˆi)∈Ksuch that, for eachi∈I,x¯i∈Si(x¯) and
Let a net{xα,α∈I} ∈(R):xα→x. We need to prove thatx∈(R). Indeed, by the lower semicontinuity ofTi, for anyy
i ∈Ti(x), there existsyαi ∈Ti(xα) such thatyαi →yi. Asxα∈(R),
Ri
xα i,xαˆi,y
α i
holds.
SinceSiis upper semicontinuous with nonempty and closed values, by Lemma .(i), we find thatSiis closed. Thus,x∈Si(x). By condition (v),
Ri
xi,xˆi,yiholds.
This means thatx∈(R). Thus(R) is a closed set.
If we letSi(x) =Ti(x) =Kifor eachi∈Iandx∈X, then (SQVR) becomes the following system of variational relation problems (in short, (SVR)). So, we obtain following result.
Corollary . For each i∈I,assume that (i) for all(xi,xˆi)∈Ki×Kˆi,Ri(xi,xˆi,xi)holds;
(ii) the set{(xˆi,yi)∈Kˆi×Ki:Ri(·,xˆi,yi)does not hold}is convex inKi;
(iii) the relationRiis convex in the first variable and closed.
Then the(SVR)has a solution,i.e.,there exists(x¯i,x¯ˆi)∈K such that,for each i∈I,
Ri(x¯i,x¯ˆi,yi)holds, ∀yi∈Ki.
Moreover,the solution set of the(SVR)is closed.
IfIis a singleton,Si(x) =S(x),Ti(x) =T(x), then (SQVR) reduces to the following quasi-variational relation problem (in short, (QVR)). So, we also obtain the following result.
Corollary . Assume that
(i) Sis upper semicontinuous inKwith nonempty compact convex values; (ii) Tis lower semicontinuous inKwith nonempty convex values;
(iii) for allx∈K,R(x,x)holds;
(iv) the set{y∈K:R(·,y)does not hold}is convex inK; (v) the relationRis convex in the first variable and closed.
Then the(QVR)has a solution,i.e.,there existsx¯∈K such thatx¯∈S(x¯)and R(x¯,y)holds, ∀y∈T(x¯).
Moreover,the solution set of the(QVR)is closed.
Remark .
(i) For eachi∈I, letXi,YiandKi,Kˆi,K,Ti=Sibe as in (SQVR). Let
Fi:Ki×Kˆi×Ki→Yibe a set-valued mapping,Ci⊂Yibe a nonempty, closed, and
convex cone, and the relationRibe defined as follows:
Then (SQVR) becomes the system of strong vector quasi-equilibrium problem (in short, (SSVQEP)) studied in [].
(ii) LetSi(x) =Ti(x) =Kifor eachi∈Iandx∈Xand letFi,Cias in Remark .(i). Then
(SVR) becomes the system of strong vector equilibrium problems (in short, (SSVEP)) studied in [].
(iii) IfIis a singleton,Si(x) =S(x),Ti(x) =T(x), and letF:K×K→Ybe a set-valued mapping,C⊂Ybe a nonempty, closed, and convex cone, and the relationRbe defined as follows:
R(x,y)holds iff F(x,y)⊂C.
Then (QVR) becomes the strong vector quasi-equilibrium problem (in short, (SQVEP)) studied in [].
(iv) Yang and Pu [] obtained some existence results for the system of strong vector quasi-equilibrium problems. However, the assumptions of theorems in [] are different from the assumptions of Theorem ., Corollary ., and Corollary .. (v) Corollary . is a particular case of Theorems . and . in []. However, the
assumptions and proof methods in Theorems . and . in [] are different from the assumptions and proof methods of Corollary ..
In the next example all assumptions of Corollary . are satisfied, but Theorem . in [] is not applicable. The reason is thatF is neither upperC-continuous nor properly
C-quasiconvex.
Example . LetI={},Xi=Yi=R,Ki= [, ],C=R+, and letSi,Ti:Ki×Ki→Kiand
F:Ki×Ki→Yibe defined by
Si(x,y) =Ti(x,y) = [, ],
Fi(xi,xˆi,yi) =F(x,y) = ⎧ ⎨ ⎩
[, ] ifx=,
[
, ] otherwise.
We let the relationRibe defined byRi(xi,xˆi,yi) holding iffFi(xi,xˆi,yi) =F(x,y)⊆R+. We show that all assumptions of Corollary . are satisfied. However, F is neither upper
C-continuous nor properlyC-quasiconvex atx=.
Firstly, we prove thatFis not upperC-continuous atx=. Indeed, we let a neighbor-hoodU= [–,] of the origin inZ, then for any neighborhoodV= [–ε,+ε] ofx=, whereε> , we choose =x∗∈Vandy=. Then
Fx∗,y=F
x∗,
=
,
⊂F(x,y) +U+C
=F
,
+
–
,
+R+
= [, ] +
– ,
+R+=
,
Next, we show thatFis not properlyC-quasiconvex atx=. Indeed, we lety=,λ=, andx= ,x= . Then
F(x,y) =F
,
=
,
⊂Fxλ+ ( –λ)x,y
+C
=F
,
+R+= [, ] +R+,
F(x,y) =F
,
=
,
⊂Fxλ+ ( –λ)x,y
+C
=F
,
+R+= [, ] +R+.
Thus, it gives also the case where Corollary . can be applied, but Theorem . in [] does not work.
3 Essential components
In this section, we discuss the existence of essential components for (SQVR).
First, we recall some notions; see [, , ]. Let A be a nonempty and compact subset of a linear normed X. Denote byMthe set of all upper semicontinuous maps
R:A→Awith nonempty convex compact values. For anyR,P∈M, we defineξ(R,P) =
supx∈AH(R(x),P(x)), whereHis the Hausdorff metric defined inA. It is easy to verify that (M,ξ) is a metric space. For eachR∈M, we denote by(R) the set of all fixed points ofR. By Kakutani-Fan-Glicksberg’s fixed point theorem,(R) is a nonempty compact set.
For eachR∈M, the connected component of a pointx∈(R) is the union of all the con-nected subsets of(R) containingx. Note that the connected components are connected closed subsets of(R), and, sinceAis compact, thus, all the connected components are connected compact. It is easy to see that the connected components of two distinct points of(R) either coincide or are disjoint, so that all connected components constitute a de-composition of(R) into connected pairwise disjoint compact subsets,i.e.,
(R) =
α∈ α(R),
whereis an index set. For eachα∈,α(R) is a nonempty connected compact subset of(R), and for anyα,β∈(α=β),α(R)∩β(R) =∅.
Definition .([]) LetR∈MandEbe a nonempty and closed subset of(R).Eis said to be an essential set of(R) if, for each open setO⊃E, there exists an open neighborhood
UofRinMsuch that for anyR∈Uwith(R)∩O=∅. If a connected componentα(R) of(R) is an essential set with respect toM, thenα(R) is said be an essential component of(R) with respect toM.
Lemma .([]) For any R∈M,there is at least one essential component of(R)with respect to M.
Letbe the collection of all (SQVR) satisfying the conditions of Theorem .. For each
ω∈, denote by(ω) the solution set ofω. It is easy to see that(ω) =(), where
is the best-reply map ofω. By the proof of Theorem ., we know∈M.
Definition . Letω∈andαa connected component of(ω).α is said to be es-sential if it, as a connected component of(), is an essential component of() with respect toM, whereis the best-reply map ofω.
By Lemma ., we obtain the following result.
Theorem . For anyω∈,there is at least one essential component of(ω).
Remark .
(i) In the special case of Remark ., Theorem . improves and extends Theorems . and . in [].
(ii) In , Khanh and Quan [] studied the existence of essential components for generalized KKM points. Moreover, Khanh and Quan also applied these results to optimization-related problems. However, the assumptions and proof methods are very different from the assumptions and proof methods in Theorem ..
(iii) IfIis a singleton, then (SQVR) becomes the variational relation problem (in short, (VR)) studied by Khanh and Luc in []. However, Khanh and Luc discussed some kinds of semicontinuous sets as outer-continuous, inner-continuous, inner-open, and outer-open solution sets for (VR), while our Theorem . discusses the existence of essential components for (SQVR) by using Kakutani-Fan-Glicksberg’s fixed point theorem.
4 Applications (I): system of quasi-variational inclusion problems
For eachi∈I, letXi,YiandKi,Kˆi,K,Ti=Sibe as in (SQVR). LetFi:Ki×Kˆi×Ki→Yi be a set-valued mapping. We consider the following system of quasi-variational inclusion problems (in short, (SQIP)): Find (x¯i,x¯ˆi)∈Ksuch that, for eachi∈I,x¯i∈Si(x¯) and
∈Fi(x¯i,x¯ˆi,yi), ∀yi∈Si(x¯),
wherex¯is a solution of (SQIP).
Definition . LetX,Y,Zbe topological vector spaces. SupposeF:X×Y×X→Zis a multifunction.Fis said to begeneralized{}-quasiconvex(in the first variable) in a convex setA⊂X, if, whenever ∈F(x,y,z) and ∈F(x,y,z), then ∈F(λx+ ( –λ)x,y,z),
∀x,x∈A,∀λ∈[, ].
Theorem . For each i∈I,assume that
(i) Siis continuous inKwith nonempty compact convex values;
(ii) for all(xi,xiˆ)∈Ki×Kˆi,∈Fi(xi,xˆi,xi);
(iii) the set{(xˆi,yi)∈Kˆi×Ki: /∈Fi(·,xˆi,yi)}is convex inKi;
(iv) for all(xˆi,yi)∈Kˆi×Ki,Fi(·,xˆi,yi)is generalized{}-quasiconvex inKi;
Then the(SQIP)has a solution,i.e.,there exists(x¯i,x¯ˆi)∈K such that,for each i∈I,x¯i∈Si(x¯)
and
∈Fi(x¯i,x¯ˆi,yi), ∀yi∈Si(x¯).
Moreover,the solution set of the(SQIP)is closed.
Proof Let the relationRibe defined as follows:
Ri(xi,xˆi,yi) holds iff ∈Fi(xi,xˆi,yi).
Then the problem (SQIP) becomes a particular case of (SQVR) and Theorem . is a direct
consequence of Theorem ..
Next, we discuss the existence of essential components for (SQIP).
Letbe the collection of all (SQIP) satisfying the conditions of Theorem .. For each θ∈, denote by(θ) the solution set ofθ. It is easy to see that(θ) =(), whereis the best-reply map ofθ. By the proof of Theorem ., we know∈M.
Definition . Letθ ∈andαbe a connected component of(θ).αis said to be
essentialif it, as a connected component of(), is an essential component of() with respect toM, whereis the best-reply map ofθ.
By Lemma ., we also obtain the following result.
Theorem . For anyθ∈,there is at least one essential component of(θ).
5 Applications (II): system of weak vector quasi-equilibrium problems
For eachi∈I, letXi,YiandKi,Kˆi,K,Ti=Sibe as in (SQVR). Letfi:Ki×Kˆi×Ki→Yibe a vector function andCi⊂Yibe a nonempty, closed, and convex cone. We consider the following weak system of quasi-equilibrium problems (in short, (WSQVEP)).
(WSQVEP): Find(x¯i,x¯ˆi)∈Ksuch that, for eachi∈I,x¯i∈Si(x¯)and
fi(x¯i,x¯ˆi,yi) /∈–intCi, ∀yi∈Si(x¯).
Definition . LetX,Y,Zbe topological vector spaces andC⊂Zbe a nonempty, closed, and convex cone. Supposef :X×Y×X→Zis a vector function.f is said to beweakly C-quasiconvex(in the first variable) in a convex setA⊂X, if wheneverf(x,y,z) /∈–intC
andf(x,y,z) /∈–intC, thenf(λx+ ( –λ)x,y,z) /∈–intC,∀x,x∈A,∀λ∈[, ].
Theorem . For each i∈I,assume that
(i) Siis continuous inKwith nonempty compact convex values;
(ii) for all(xi,xˆi)∈Ki×Kˆi,fi(xi,xˆi,xi) /∈–intCi;
(iii) the set{(xˆi,yi)∈Kˆi×Ki:fi(·,xˆi,yi)∈–intCi}is convex inKi;
(iv) for all(xˆi,yi)∈Kˆi×Ki,fi(·,xˆi,yi)is weaklyCi-quasiconvex inKi;
Then the(WSQVEP)has a solution,i.e.,there exists(x¯i,x¯ˆi)∈K such that,for each i∈I,
¯
xi∈Si(x¯)and
fi(x¯i,x¯ˆi,yi) /∈–intCi, ∀yi∈Si(x¯).
Moreover,the solution set of the(WSQVEP)is closed.
Proof Let the relationRibe defined as follows:
Ri(xi,xˆi,yi) holds iff fi(xi,xˆi,yi) /∈–intCi.
Then the problem (WSQVEP) becomes a particular case of (SQVR) and Theorem . is a
direct consequence of Theorem ..
Definition . ([]) Let X and Z be two topological vector spaces and A⊆ X be
nonempty convex set, and C⊂ Z be a nonempty, closed, and convex cone. Suppose
f :A→Zis a vector function.f is calledC-continuous at x∈Aif, for any open neigh-borhoodV of the zero elementθ inZ, there exists an open neighborhoodUofxinA such that
f(x)∈f(x) +V+C, ∀x∈U;
andC-continuous inAif it isC-continuous at every point ofA.
Remark . Linet al.[] obtained some existence results of (WSVQEP). However, the assumptions of Theorems ., ., and . in [] are different from the assumptions in Theorem .. Example . shows that all the assumptions of Theorem . are satisfied. But Theorem . in [] does not work. The reason is thatfiis not-Ci-continuous.
Example . LetI={},Xi=Yi=Zi=R,Ki= [, ],Ci=R+, and letSi:Ki→Ki and
f :Ki→Rbe defined by
Si(x) = [, ],
f(x,y) =f(x) = ⎧ ⎨ ⎩
[,] ifx=,
[, ] otherwise.
We show that all assumptions of Theorem . are satisfied. However,fis not-C-continuous atx=. Thus, it gives the case where Theorem . can be applied but Theorems ., ., and . in [] do not work.
Remark . If we letSi(x) =Ki for eachi∈I. Then (WSQVEP) becomes the system of vector equilibrium problem (in short, (SVEP)). If we letI={}, then (WSQVEP) becomes the strong vector equilibrium problem (in short, (VEP)). The problems (SVEP) and (VEP) are studied in [].
Definition . ([]) Let X andZ be two topological vector spaces and A⊆X be a
f :A→Zis a vector function.f is calledC-quasiconvex-likeif, for anyxi,x∈Aand each λ∈[, ], we have
either fλx+ ( –λ)x
∈f(x) –C
or fλx+ ( –λ)x
∈f(x) –C.
Example . shows that in the special cases of Remarks . and ., all the assumptions of Theorem . are satisfied. But Theorems ., ., and . in [] do not work. The reason is thatfiis not-Ci-quasiconvex-like.
Example . LetXi, Yi,Zi, Ki,Ci be as in Example ., and letSi: [, ]→Kiandf : [, ]→Rbe defined by
Si(x) = [, ],
f(x) = ⎧ ⎨ ⎩
[,] ifx=,
[, ] otherwise.
We show that all assumptions of Theorem . are satisfied. However, f is not-C -quasi-convex-like atx=. Indeed, letλ= andx= ,x=. Then
fλx+ ( –λ)x
=f
=
,
⊂f(x) +C=f() +C
= [, ] + [, +∞),
fλx+ ( –λ)x
=f
=
,
⊂f(x) +C=f
+C
= [, ] + [, +∞).
Thus, Theorem . can be applied but Theorems ., ., and . in [] do not work.
Theorem . Assume for the problem(WSQVEP)assumptions(i), (ii), (iii),and(iv)are as in Theorem.and replace(v)by(v)
(v) fiis continuous inKi×Kˆi×Ki.
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 . with
suit-able modifications.
Next, we discuss the existence of essential components for (WSQVEP).
Letϒbe the collection of all (WSQVEP) satisfying the conditions of Theorem .. For eachγ∈ϒ, denote by(γ) the solution set ofγ. It is easy to see that(γ) =(), where is the best-reply map ofγ. By the proof of Theorem ., we know that∈M.
By Lemma ., we also obtain the following result.
Theorem . For anyγ ∈ϒ,there is at least one essential component of(γ).
Remark . In the special case of Remarks . and ., Theorem . improves and extends Theorems . and . in [].
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.
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.
Received: 14 February 2014 Accepted: 13 June 2014 Published:18 Jul 2014
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