R E S E A R C H
Open Access
On existence and essential stability of
solutions of symmetric variational relation
problems
Zhe Yang
**Correspondence:
[email protected] School of Economics, Shanghai University of Finance and Economics, Shanghai, 200433, China
Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, 200433, China
Abstract
In this paper, we introduce symmetric variational relation problems and establish the existence theorem of solutions of symmetric variational relation problems. As the special cases, symmetric (vector) quasi-equilibrium problems and symmetric variational inclusion problems are obtained. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. We prove that most of symmetric variational relation problems (in the sense of Baire category) are essential and, for any symmetric variational relation problem, there exists at least one essential component of its solution set.
MSC: 49J53; 49J40
Keywords: symmetric variational relation problems; existence; essential stability
1 Introduction
LetXandYbe real locally convex Hausdorff spaces, and letCandDbe nonempty subsets ofXandY, respectively. LetS:C×D⇒CandT:C×D⇒Dbe set-valued mappings, and
f,g:C×D−→Rbe real functions. According to Noor and Oettli [], the symmetric quasi-equilibrium problem (SQEP) consists in finding (x∗,y∗)∈C×Dsuch thatx∗∈S(x∗,y∗),
y∗∈T(x∗,y∗), and
fx,y∗≥fx∗,y∗, ∀x∈Sx∗,y∗,
gx∗,y≥gx∗,y∗, ∀y∈Tx∗,y∗.
The problem is a generalization of equilibrium problem proposed by Blum and Oettli []. The equilibrium problem contains as special cases, for instance, optimization problems, problems of Nash equilibria, fixed point problems, variational inequalities, and comple-mentarity problems.
Fu [] introduced symmetric vector quasi-equilibrium problems (SVQEP). LetZbe a real Hausdorff topological vector space, and letP⊂Zbe a closed convex, pointed cone with apex at the origin and withintP=∅, whereintPdenotes the interior ofP. LetX,Y,
C,D,S,T be as above. Let vector mappingsf,g:C×D−→Zbe given. The symmetric vector quasi-equilibrium problem (SVQEP) consists in finding (x∗,y∗)∈C×Dsuch that
x∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
fx,y∗–fx∗,y∗∈/–intP, ∀x∈Sx∗,y∗,
gx∗,y–gx∗,y∗∈/–intP, ∀y∈Tx∗,y∗.
Farajzadeh [] considered symmetric vector quasi-equilibrium problems in the Haus-dorff topological vector space by means of a particular technique and answered an open question raised by Fu. The stability of the set of solutions for symmetric vector quasi-equilibrium problems is discussed in [–]. Fakhar and Zafarani [] introduced general-ized symmetric vector quasi-equilibrium problems (GSVQEP). LetX,Y,C,D,S,T,f,g
be as above. LetPbe the set-valued mapping fromZtoZsuch that for everyz∈Z,P(z) is a pointed, closed and convex cone ofZwith a nonempty interiorintP(z) andθ:X−→Z,
η:Y −→Z. The generalized symmetric vector quasi-equilibrium problem (GSVQEP) consists in finding (x∗,y∗)∈C×Dsuch thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
fx,y∗–fx∗,y∗∈/–intPηy∗, ∀x∈Sx∗,y∗,
gx∗,y–gx∗,y∗∈/–intPθx∗, ∀y∈Tx∗,y∗.
It is well known that the equilibrium problems are unified models of several prob-lems, namely, optimization probprob-lems, saddle point probprob-lems, variational inequalities, fixed point problems, Nash equilibrium problemsetc.Recently, Luc [] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [, ]. Further studies of variational relation problems have been done. Lin and Wang [] studied simultaneous variational relation problems (SVR) and related applications. Balaj and Luc [] introduced mixed variational relation problems (MR), and established existence of solutions to a general inclusion problem. Particular cases of variational inclu-sions and intersections of set-valued mappings were also discussed in []. Balaj and Lin [] brought forward generalized variational relation problems (GVR), and obtained an ex-istence theorem of the solutions for a variational relation problem. An exex-istence theorem for a variational inclusion problem, a KKM theorem and an extension of the well-known Ky Fan inequality have been established as particular cases. Lin and Ansari [] introduced a system of quasi-variational relations and established the existence of solutions of SQVP by means of the maximal element theorem for a family of multivalued mappings.
Agarwalet al.[] presented a unified approach in studying the existence of solutions for two types of variational relation problems, which encompass several generalized equi-librium problems, variational inequalities and variational inclusions investigated in the recent literature. Balaj and Lin [] established existence criteria for the solutions of two very general types of variational relation problems. Moreover, Lucet al.[] established two main existence conditions for solutions of variational relation problems without con-vexity, and Pu and Yang [–] studied variational relation problems without the KKM property and on Hadamard manifolds.
and generalize several known results on variational relation problems and symmetric (vec-tor) quasi-equilibrium problems.
2 Symmetric variational relation
LetX,Y be two nonempty, compact and convex subsets of two normed linear spaces, respectively. LetS:X×Y⇒XandT:X×Y⇒Ybe set-valued mappings, andR(x,y,u),
Q(x,y,v) be two relations linkingx,u∈Xandy,v∈Y. The symmetric variational relation problem consists in finding (x∗,y∗)∈X×Ysuch thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
Rx∗,y∗,xholds, ∀x∈Sx∗,y∗,
Qx∗,y∗,yholds, ∀y∈Tx∗,y∗.
We recall first some known results concerning set-valued mappings for later use.
Lemma .(., . of []) (i)Assume that X is a topological space,Y is a locally convex space,and a correspondence f :X⇒Y is upper semicontinuous at some point x∈
X.If the set cl(cof(x))is compact,then the closed convex hull correspondence clcof :X⇒ Y is upper semicontinuous at x. (ii)In a completely metrizable locally convex space,the
closed convex hull of a compact set is compact.
Lemma .(. of []) Let A, . . . ,Anbe nonempty and compact subsets of the Hausdorff
topological linear space.Then co(ni=Ai)is compact.
Lemma . Let A be a nonempty and compact subset of the compact normed linear space E,and let B⊃A be open in E.Then clcoA⊂coB.
Proof SinceB⊃Ais open inE,coBis also open. Then, for anyx∈A, there exists an open convex neighborhoodV(x) ofxsuch thatx∈V(x)⊂clV(x)⊂coB. SinceAis nonempty and compact, there is a finite subset{x, . . . ,xn}ofAsuch that
A⊂
n
i=
V(xi)⊂ n
i=
clV(xi)⊂coB,
which implies that
coA⊂co
n
i=
clV(xi)
⊂coB.
SinceclV(xi) is compact for anyi∈ {, . . . ,n}, by Lemma .,co(
n
i=clV(xi)) is compact.
Hence
clcoA⊂clco
n
i=
clV(xi)
=co
n
i=
clV(xi)
⊂coB.
The following theorem is the main result of this paper.
Theorem . Assume that
(iii) R(·,·,·)andQ(·,·,·)are closed;
(iv) for anyy∈Y,any finite subset{x, . . . ,xn}ofXand anyx∈co{x, . . . ,xn},there is
i∈ {, . . . ,n}such thatR(x,y,xi)holds;
(v) for anyx∈X,any finite subset{y, . . . ,yn}ofYand anyy∈co{y, . . . ,yn},there is
i∈ {, . . . ,n}such thatQ(x,y,yi)holds.
Then the symmetric variational relation problem has at least one solution.
Proof Firstly, denote
Gr(R) =(x,y,u)∈X×Y×X|R(x,y,u) holds ,
Gr(Q) =(x,y,v)∈X×Y×Y|Q(x,y,v) holds .
ThenGr(R) andGr(Q) are closed inX×Y×XandX×Y×Y, respectively. Next, define two mappingsf:X×Y×X−→Randg:X×Y×Y−→Rby
f(x,y,u) =d
(x,y,u),Gr(R),
g(x,y,v) =d
(x,y,v),Gr(Q),
wheredanddare the distance onX×Y×XandX×Y×Y, respectively. Then (i)f,g are continuous; (ii)f(x,y,u) = if and only ifR(x,y,u) holds, andg(x,y,v) = if and only ifQ(x,y,v) holds; (iii)f(x,y,u)≥ for any (x,y,u)∈X×Y×X, andg(x,y,v)≥ for any (x,y,v)∈X×Y×Y.
Now, define the mappingsH:X×Y⇒XandF:X×Y⇒Yby
H(x,y) =u∈S(x,y)|f(x,y,u) = max
u∈S(x,y)f
x,y,u,
F(x,y) =
v∈Q(x,y)|g(x,y,v) = max
v∈Q(x,y)g
x,y,v.
SinceS,T are continuous with nonempty convex compact values, andf,g are contin-uous, it follows from the well-known Berge maximum theorem thatHandF are upper semicontinuous with nonempty compact values. By Lemma .,clcoHandclcoFare up-per semicontinuous with nonempty convex compact values. Thus, define the mapping
ϕ:X×Y⇒X×Yby
ϕ(x,y) =clcoH(x,y)×clcoF(x,y).
By the well-known Fan-Glicksberg fixed point theorem, there exists (x∗,y∗)∈X×Ysuch that (x∗,y∗)∈ϕ(x∗,y∗),i.e.,
x∗∈clcoHx∗,y∗⊂Sx∗,y∗,
y∗∈clcoFx∗,y∗⊂Tx∗,y∗.
Suppose that the result is false, without loss of generality, there isu∈S(x∗,y∗) such that
R(x∗,y∗,u) does not hold. Thenf(x∗,y∗,u) > . By the definition ofH, we have
fx∗,y∗,u≥fx∗,y∗,u
for anyu∈H(x∗,y∗),i.e.,
Hx∗,y∗⊂u∈X:Rx∗,y∗,udoes not hold .
SinceR(·,·,·) is closed,{u∈X:R(x∗,y∗,u) does not hold}is open inX. By Lemma .,
clcoHx∗,y∗⊂cou∈X:Rx∗,y∗,udoes not hold .
Sincex∗∈clcoH(x∗,y∗),x∗∈co{u∈X:R(x∗,y∗,u) does not hold},i.e., there is a finite sub-set{u, . . . ,un} ⊂ {u∈X:R(x∗,y∗,u) does not hold}such thatx∗∈co{u, . . . ,un}. By
con-dition (iv), there isi∈ {, . . . ,n}such thatR(x∗,y∗,ui) holds, which contradicts the fact thatR(x∗,y∗,ui) does not hold for anyi∈ {, . . . ,n}. This completes the proof.
Remark . By Theorem ., we obtain the following typical examples.
() Letf :X×Y−→Randg:X×Y −→Rbe two real-valued functions. Define the variational relationsR,Qas follows:
R(x,y,u) holds if and only iff(x,y)≤f(u,y);
Q(x,y,v) holds if and only ifg(x,y)≤g(x,v).
The symmetric quasi-equilibrium problem (SQEP) consists in finding (x∗,y∗)∈X×Y
such thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
fx,y∗≥fx∗,y∗, ∀x∈Sx∗,y∗,
gx∗,y≥gx∗,y∗, ∀y∈Tx∗,y∗.
By virtue of Theorem ., problem (SQEP) has at least one solution.
() LetX,Y,S,Tbe as above,Zbe a real Hausdorff topological vector space, andP⊂Z
be a closed convex, pointed cone with apex at the origin and withintP=∅, whereintP
denotes the interior ofP. Let vector mappings f,g:C×D−→Z be given. Define the variational relationsR,Qas follows:
R(x,y,u) holds if and only iff(u,y) –f(x,y) /∈–intP;
Q(x,y,v) holds if and only ifg(x,v) –g(x,y) /∈–intP.
The symmetric vector quasi-equilibrium problem (SVQEP) consists in finding (x∗,y∗)∈
X×Ysuch thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
fx,y∗–fx∗,y∗∈/–intP, ∀x∈Sx∗,y∗,
gx∗,y–gx∗,y∗∈/–intP, ∀y∈Tx∗,y∗.
By virtue of Theorem ., problem (SVQEP) has at least one solution.
() LetX,Y,S,T be as above,Zbe a real Hausdorff topological vector space,A,A:
variational relationsR,Qas follows:
R(x,y,u) holds if and only ifA(x,y,u)⊂B(x,y,u);
Q(x,y,v) holds if and only ifA(x,y,v)⊂B(x,y,v).
The symmetric variational inclusion of type (I) consists in finding (x∗,y∗)∈X×Y such thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
Ax∗,y∗,u⊂Bx∗,y∗,u, ∀u∈Sx∗,y∗,
Ax∗,y∗,v⊂Bx∗,y∗,v, ∀v∈Tx∗,y∗.
By virtue of Theorem ., the symmetric variational inclusion of type (I) has at least one solution.
() Define the variational relationsR,Qas follows:
R(x,y,u) holds if and only ifA(x,y,u)∩B(x,y,u)=∅;
Q(x,y,v) holds if and only ifA(x,y,v)∩B(x,y,v)=∅.
The symmetric variational inclusion of type (II) problem consists in finding (x∗,y∗)∈X× Y such thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
Ax∗,y∗,u∩Bx∗,y∗,u=∅, ∀u∈Sx∗,y∗,
Ax∗,y∗,v∩Bx∗,y∗,v=∅, ∀v∈Tx∗,y∗.
By virtue of Theorem ., the symmetric variational inclusion of type (II) has at least one solution.
() Define the variational relationsR,Qas follows:
R(x,y,u) holds if and only if∈A(x,y,u);
Q(x,y,v) holds if and only if∈A(x,y,v).
The symmetric variational inclusion of type (III) problem consists in finding (x∗,y∗)∈
X×Ysuch thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗), and
∈Ax∗,y∗,u, ∀u∈Sx∗,y∗,
∈Ax∗,y∗,v, ∀v∈Tx∗,y∗.
By virtue of Theorem ., the symmetric variational inclusion of type (III) has at least one solution.
3 Essential stability
all conditions of Theorem . hold. For eachq∈M, denote by(q) the solution set ofq. Thus, a set-valued correspondence:M⇒X×Yis well defined. To analyze the stability of (q) inM, some topological structure in the collectionMis also needed. For each
q,q∈M, define the distance onMby
ρq,q= sup (x,y)∈X×Y
hX
S(x,y),S(x,y)+ sup (x,y)∈X×Y
hY
T(x,y),T(x,y)
+hGr(R),GrR+hGr(Q),GrQ,
wherehXis the Hausdorff distance defined onX,hYis the Hausdorff distance defined on
Y,his the Hausdorff distance defined onX×Y ×X, andh is the Hausdorff distance defined onX×Y×Y. Clearly, (M,ρ) is a metric space.
Definition . Letq∈M. An (x,y)∈(q) is said to be an essential point of(q) if, for any
open neighborhoodN(x,y) of (x,y) inX×Y, there is a positiveδsuch thatN(x,y)∩(q)=
∅for anyq∈Mwithρ(q,q) <δ.qis said to be essential if each (x,y)∈(q) is essential.
Definition . Letq∈M. A nonempty closed subsete(q) of(q) is said to be an essential set of(q) if, for any open setU,e(q)⊂U, there is a positiveδsuch thatU∩(q)=∅for anyq∈Mwithρ(q,q) <δ.
Definition . Letq∈M. An essential subsetm(q)⊂(q) is said to be a minimal es-sential set of(q) if it is a minimal element of the family of essential sets ordered by set inclusion. A componentC(q) is said to be an essential component of(q) ifC(q) is essen-tial.
Remark . () It is easy to see that the problemq∈M is essential if and only if the mapping:M⇒X×Y is lower semicontinuous atq. () For two closede(q)⊂e(q)⊂
(q), ife(q) is essential, thene(q) is also essential.
First of all, let us introduce some mathematical tools for the following proof, which can be found in [–].
Lemma .([]) Let X and Y be two topological spaces with Y compact.If F is a closed set-valued mapping from X to Y,then F is upper semi-continuous.
Lemma .([]) If X,Y are two metric spaces,X is complete and F:X⇒Y is upper semicontinuous with nonempty compact values, then the set of points, where F is lower semicontinuous,is a dense residual set in X.
Lemma .([]) Let C,D be two nonempty,convex and compact subsets of linear normed space E.Then
h(C,λC+μD)≤h(C,D),
Lemma .([]) Let(Y,ρ)be a metric space,Kand Kbe two nonempty compact
sub-sets of Y,Vand Vbe two nonempty disjoint open subsets of Y.If h(K,K) <ρ(V,V) := inf{ρ(x,y)|x∈V,y∈V},then
hK, (K\V)∪(K\V)
≤h(K,K),
where h is the Hausdorff metric defined on Y.
Theorem . (M,ρ)is a complete metric space.
Proof Let{qn}∞
n= be any Cauchy sequence inM, then, for anyε> , there isN> such thatρ(qn,qm) <εfor anyn,m>N,i.e.,
sup (x,y)∈X×Y
hX
Sn(x,y),Sm(x,y)+ sup (x,y)∈X×Y
hY
Tn(x,y),Tm(x,y)
+hGrRn,GrRm+hGrQn,GrQm≤ε
for anyn,m>N.
() Clearly, there exist set-valued mappingsS:X×Y⇒X,T:X×Y⇒Y, and a closed subsetA ofX×Y ×X, a closed subsetBof X×Y×Y such thatSn(x,y)−→S(x,y),
Tn(x,y)−→T(x,y) for any (x,y)∈X×Y, andS,Tare continuous with nonempty convex
compact values, andGr(Rn)−→A,Gr(Qn)−→B.
() Further, define the following symmetric variational relationq= (S,T,R,Q) by
R(x,y,u) holds if and only if (x,y,u)∈A,
Q(x,y,v) holds if and only if (x,y,v)∈B.
Clearly,qn−→qunder the distanceρ. We will show thatq∈M. (i) Clearly,R(·,·,·) andQ(·,·,·) are closed.
(ii) Suppose that there existy∈Y, a finite subset{x, . . . ,xn}ofXandx∈co{x, . . . ,xn}
such that R(x,y,xi) does not hold for eachi∈ {, . . . ,n}, then (x,y,xi) /∈A for eachi∈
{, . . . ,n}. Sinceqm−→q, then (x,y,x
i) /∈Gr(Rm) for eachi∈ {, . . . ,n}and for enough large
m, which implies thatRm(x,y,x
i) does not hold for eachi∈ {, . . . ,n}. It is a contradiction.
Thus, for anyy∈Y, any finite subset{x, . . . ,xn}ofXand anyx∈co{x, . . . ,xn}, there is
i∈ {, . . . ,n}such thatR(x,y,xi) holds. Similarly, for anyx∈X, any finite subset{y, . . . ,yn}
ofYand anyy∈co{y, . . . ,yn}, there isi∈ {, . . . ,n}such thatQ(x,y,yi) holds. Henceq∈M
and (M,ρ) is complete.
Theorem . The solution mapping :M ⇒X× Y is upper semicontinuous with nonempty compact values.
Proof The desired conclusion follows from Lemma . as soon as we show thatGraph() is closed. Let {(qn,xn,yn)∈M×X×Y}∞
Qn(xn,yn,v) hold for anyu∈Sn(xn,yn) and anyv∈Tn(xn,yn). Sinceqn−→q, it follows
thatx∈S(x,y) andy∈T(x,y).
Suppose that there existsu∈S(x,y) such thatR(x,y,u) does not hold, then there ex-ists un∈Sn(xn,yn) such thatun−→u, and (x,y,u) /∈Gr(R). For enough large n, we
have (xn,yn,un) /∈Gr(Rn), i.e., Rn(xn,yn,un) does not hold. It is a contradiction. There-foreR(x,y,u) holds for anyu∈S(x,y). Similarly,Q(x,y,v) holds for anyv∈T(x,y). Hence
(x,y)∈(q).
Theorem . There exists a dense residual subsetGofMsuch that q is essential for each
q∈G.
Proof Since the metric space (M,ρ) is complete (by Theorem .), and the mapping
:M⇒X×Y is upper semicontinuous with compact values (by Theorem .), by Lemma .,is lower semicontinuous on a dense residual subsetGofM. Thus, by
Re-mark .(),qis essential for eachq∈G.
Theorem . For each q∈M,there exists at least one minimal essential subset of(q).
Proof By Theorem .,:M⇒X×Yis upper semicontinuous with compact values, that is, for each open setO⊃(q), there existsδ> such thatO⊃(q) for anyq∈Mwith
ρ(q,q) <δ. Hence(q) is an essential set of itself. Letdenote the family of all essential sets of(q) ordered by set inclusion. Thenis nonempty and every decreasing chain of elements inhas a lower bound (because by the compactness the intersection is in); therefore, by Zorn’s lemma,has a minimal element and it is a minimal essential set of
(q).
Theorem . For each q∈M,every minimal essential subset of(q)is connected.
Proof For eachq∈M, letm(q)⊂(q) be a minimal essential subset of(q). Suppose that
m(q) is not connected, then there exist two non-empty compact subsetsc(q),c(q) with
m(q) =c(q)∪c(q), and there exist two disjoint open subsetsV,VofX×Ysuch that
V⊃c(q),V⊃c(q). Sincem(q) is a minimal essential set of(q), neitherc(q) norc(q) is essential. There exist two open setsO⊃c(q),O⊃c(q) such that, for anyδ> , there existq,q∈Mwith
ρq,q<δ, ρq,q<δ, q∩O=∅,
q∩O=∅.
DenoteW=V∩O,W=V∩O, we know thatW,Ware open,W⊃c(q),W⊃
c(q), and we may assume thatV⊃W,V⊃W. DenoteG=W×XandG=W×Y, inf{d(a,b)|x∈G,b∈G}=ε> .
Sincem(q) is essential andm(q)⊂(W∪W), there exists <δ∗<εsuch that(q)∩ (W∪W)=∅for anyq∈Mwithρ(q,q) <δ∗. Sincem(q) is a minimal essential set of
(q), neitherc(q) norc(q) is essential. Thus, forδ ∗
> , there exist twoq
,q∈Msuch
that
q∩W=∅,
q∩W=∅, ρ
q,q< δ ∗
, ρ
q,q<δ ∗
Thus
ρq,q< δ ∗
.
Next, define the symmetric variational relation problemq= (S,T,R,Q) by
S(x,y) =λ(x,y)S(x,y) +μ(x,y)S(x,y),
T(x,y) =λ(x,y)T(x,y) +μ(x,y)T(x,y),
A=GrR\G
∪GrR\G
,
B=GrQ\G
∪GrQ\G
,
R(x,y,u) holds if and only if (x,y,u)∈A,
Q(x,y,v) holds if and only if (x,y,v)∈B,
where
λ(x,y) = d((x,y),W)
d((x,y),W) +d((x,y),W), ∀(x,y)∈X×Y,
μ(x,y) = d((x,y),W)
d((x,y),W) +d((x,y),W), ∀(x,y)∈X×Y.
Easily, we can check that
(i)S,Tare continuous with nonempty compact convex values.
(ii) SinceGr(R) andGr(R) are closed inX×Y×X,Ais closed inX×Y×X, which implies thatR(·,·,·) is closed. Similarly,Q(·,·,·) is closed.
(iii) Suppose that there existy∈Y, a finite subset{x, . . . ,xn} ⊂Xandx∈co{x, . . . ,xn}
such thatR(x,y,xi) does not hold for anyi∈ {, . . . ,n}, then
x,y,xi∈/A=GrR\G
∪GrR\G
, ∀i∈ {, . . . ,n}.
AsW∩W=∅, without loss of generality, we may assume that (x,y)∈(X,Y)\W. There-fore,
x,y,xi∈/GrR\G, ∀i∈ {, . . . ,n},
i.e.,
x,y,xi∈/GrR∩(X,Y)\W×X
, ∀i∈ {, . . . ,n},
which implies that (x,y,xi)∈Gr(R) for anyi∈ {, . . . ,n}. ThusR(x,y,xi) does not hold for
anyi∈ {, . . . ,n}, which is a contradiction. Hence, for anyy∈Y, any finite subset{x, . . . ,xn}
ofXand anyx∈co{x, . . . ,xn}, there isi∈ {, . . . ,n}such thatR(x,y,xi) holds. Similarly, for
anyx∈Y, any finite subset{y, . . . ,yn}ofYand anyy∈co{y, . . . ,yn}, there isi∈ {, . . . ,n}
(iv) Further, by Lemma . and Lemma ., we have
sup (x,y)∈X×Y
hX
S(x,y),S(x,y)≤ sup (x,y)∈X×Y
hX
S(x,y),S(x,y)
+ sup (x,y)∈X×Y
hX
S(x,y),S(x,y)
≤ δ∗
+ δ∗ ≤ δ ∗, sup (x,y)∈X×Y
hY
T(x,y),T(x,y)≤ sup (x,y)∈X×Y
hY
T(x,y),T(x,y)
+ sup (x,y)∈X×Y
hY
T(x,y),T(x,y)
≤ δ∗
+ δ∗ ≤ δ ∗,
hGr(R),GrR≤hGr(R),GrR+hGrR,GrR
≤ δ∗
+ δ∗ ≤ δ ∗,
hGr(Q),GrQ≤hGr(Q),GrQ+hGrQ,GrQ
≤ δ∗
+ δ∗ ≤ δ ∗. Hence
ρq,q= sup (x,y)∈X×Y
hX
S(x,y),S(x,y)+ sup (x,y)∈X×Y
hY
T(x,y),T(x,y)
+hGr(R),GrR+hGr(Q),GrQ
≤
δ ∗
<δ∗.
Thusq∈Mandρ(q,q) <δ∗. Since
q∩W
∪q∩W
=q∩(W∪W)=∅,
S(x,y) and anyv∈T(x,y). Since (x,y)∈W,
S(x,y) =S(x,y), T(x,y) =T(x,y),
R(x,y,u) and Q(x,y,v) hold∀u∈S(x,y),∀v∈T(x,y),
which implies that (x,y)∈(q). Hence(q)∩W
=∅, which contradicts(q)∩W=∅.
Thusm(q) is connected.
Theorem . For each q∈M,there exists at least one essential component of(q).
Proof By Theorem ., there exists at least one connected minimal essential subsetm(q) of(q). Thus, there is a componentCof(q) such thatm(q)⊂C. It is obvious thatCis essential by Remark .(). ThusCis an essential component.
Remark . Our paper has improved the results of []. (i) The symmetric (vector) quasi-equilibrium problem is a special case of symmetric variational relation problem; (ii) In [], the existence of essential connected components is based on disturbance ofS,Tfor fixed
f,g(see Section in []), but the existence of essential connected components in our paper is based on disturbance ofS,T,R,Q, which are more general.
4 Conclusion
In this paper, symmetric variational relation problems are introduced, and we establish the existence theorem of solutions of symmetric variational relation problems. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. Our paper improves the results of [].
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
Supported by the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (Project Number: 201309KF02).
Received: 25 July 2013 Accepted: 1 December 2013 Published:02 Jan 2014
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10.1186/1029-242X-2014-5
Cite this article as:Yang:On existence and essential stability of solutions of symmetric variational relation problems.
