R E S E A R C H
Open Access
Essential components of the set of solutions
for the system of vector quasi-equilibrium
problems
Xun-Hua Gong
1*and Jian-Chen Chen
2*Correspondence:
1Department of Mathematics,
Nanchang University, Nanchang, 330031, China
Full list of author information is available at the end of the article
Abstract
In this paper, we prove that, for every vector quasi-equilibrium problem, there exists at least one essential component of the set of its solutions. As application, we show that, for every system of vector quasi-equilibrium problems, there exists at least one essential component of the set of its solutions in the uniform topological space of objective functions and constraint mappings.
1 Introduction
Essential component has been an important aspect in the study of stability for nonlin-ear problems. 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 ap-proximately closed by a mapping whose fixed points are all essential. Kinoshita [] then introduced the notion of essential components of the set of fixed points of a single-valued map. Jiang [] introduced the notion of essential components of the set of Nash equilib-rium points for an n-person non-cooperative game and proved the existence of essential components of the set of Nash equilibrium points. Kohlberg and Mertens [] studied the stability of Nash equilibrium points and suggested that a satisfactory solution for a non-cooperative game should be set-wise, and they proved that such a solution is just an es-sential component of Nash equilibrium points. Recently, Yu, Xiang [], Yu, Luo [], Isac, Yuan [], Yang, Yu [], Lin [], Chen, Gong [] introduced the notion of essential com-ponents to solution sets of various problems such as Ky Fan point problems, equilibrium problems, coincident point problems, vector optimization problem, and symmetric vec-tor quasi-equilibrium problems. On the other hand, in order to describe the real world and economic behavior better, very recently, much attention has been attracted to multi-criteria equilibrium models. Ansari, Schaible and Yao [] studied the system of general-ized vector equilibrium problems. Ansari, Chan and Yang [] studied the system of vec-tor quasi-equilibrium problems (briefly, SVQEP). Fang, Huang and Kim [] studied the system of vector equilibrium problems. Peng, Lee, Yang [] studied the system of general-ized vector quasi-equilibrium problems with set-valued maps (briefly, SGVOEPS). Lin [] studied the system of generalized vector quasi-equilibrium problems (briefly, SGVQEP) in Banach spaces. Peng, Yang and Zhu [] studied the system of vector quasi-equilibrium. Lin [] established essential components of the solution set for SGVQEP under perturba-tions of the best-reply map. But up to now, no paper has established essential components
of the solution set for SVQEP, SGVQEP or SGVQEPS under perturbations of objective functions and constraint mappings. In this paper, we first give a new result of essential components of the solution set for SVQEP under perturbations of objective functions and constraint mappings.
2 Preliminaries and definitions
LetI={, , . . . ,n}be a finite set which has at least two elements. For eachi∈I, letXiand Yibe real Hausdorff topological vector spaces andKia nonempty subset ofXi. For each i∈I, letCibe a closed, convex and pointed cone ofYiwithintCi=∅, whereintCidenotes the interior ofCi. LetK=ni=Ki. For eachi∈I, letfi:K×Ki→Yi be a vector-valued mapping andSi:K→Kibe a set-valued mapping. The SVQEP consists of findingx¯∈K such that for eachi∈I,
¯
xi∈Si(x¯) and fi(x¯,yi) /∈–intCi for allyi∈Si(x¯),
where x¯idenotes theith component ofx¯, andx¯ is said to be a solution of the SVQEP. For eachi∈I,fiis said to be an objective function of the SVQEP and for eachi∈I,Siis said to be a constraint mapping of the SVQEP. The SVQEP includes, as a special case, the following multiobjective generalized game problem:
For eachi∈I, letgi:K→Yibe a vector-valued mapping andGi:Kˆi→Kibe a feasible strategy mapping, whereKˆi=
j∈I,j=iKj. For eachx∈K, we can writex= (xi,xˆi). The mul-tiobjective generalized game problem consists of findingx¯∈Ksuch that for eachi∈I,
¯
xi∈Gi(x¯ˆi) and
gi(yi,x¯ˆi) –gi(x¯i,x¯ˆi) /∈–intCi for allyi∈Gi(x¯ˆi),
wherex¯is said to be a weakly Pareto-Nash equilibrium point. For eachi∈I, setting
fi(x,yi) =gi(yi,xˆi) –gi(xi,xˆi) and Si(x) =Gi(xˆi),
the SVQEP coincides with the multiobjective generalized game problem, which has been studied by Yu and Luo [] but for real functions and Lin [] but forYi=Rki(≤ki≤n) for anyi∈I.
For eachi∈I, settingGi(xˆi) =Ki, the multiobjective generalized game problem coincides with the multiobjective game problem, which has been studied by Yu and Xiang [] and Yang and Yu [].
Definition . LetXbe a real Hausdorff topological space andYa real Hausdorff topo-logical vector space with a convex coneC. Letf :X→Ybe a vector-valued function.
(i) f is said to beC-continuous atx∈Xif, for any open neighborhoodVof the zero
elementθinY, there is an open neighborhoodN(x)ofxinXsuch that
f(x)∈f(x) +V+C for allx∈N(x).
(ii) f is said to be (–C)-continuous atx∈Xif, for any open neighborhoodVofθinY,
there exists an open neighborhoodN(x)ofxinXsuch that
f(x)∈f(x) +V–C for allx∈N(x).
f is said to be (–C)-continuous onXif it is (–C)-continuous at every point ofX.
Definition . LetKbe a nonempty convex subset of a vector spaceX, letYbe a vector space with a convex pointed coneC. Letf :K→Ybe a mapping.f is said to beC-convex if, for anyx,y∈Kandt∈[, ],
tf(x) + ( –t)f(y) –ftx+ ( –t)y∈C.
Definition . LetXandYbe two Hausdorff topological spaces, letF:X→Y be a set-valued mapping.Fis said to be upper semicontinuous (in short, u.s.c.) atx∈Xif, for any
neighborhoodN(F(x)) ofF(x), there exists a neighborhoodN(x) ofxsuch that
F(x)⊂NF(x)
for allx∈N(x).
Fis said to be upper semicontinuous onXifFis u.s.c. at every pointx∈X.
Fis said to be lower semicontinuous (in short, l.s.c.) atx∈Xif, for anyy∈F(x) and
any neighborhoodN(y) ofy, there exists a neighborhoodN(x) ofxsuch that
F(x)∩N(y)=∅ for allx∈N(x).
Fis said to be lower semicontinuous onXif it is lower semicontinuous at everyx∈X. Fis said to be continuous onXif it is both u.s.c. and l.s.c. onX.
Fis said to be a closed mapping ifGraphF={(x,y)∈X×Y:y∈F(x)}is a closed set in X×Y.
Fis an usco mapping ifFis u.s.c. onXandF(x) is compact for everyx∈X.
Let (X,d) be a linear metric space. Denote byCK(X) all nonempty convex compact sub-sets ofX. Define the Hausdorff metrichonCK(X) as follows.
For anyS,S∈CK(X), let
h(S,S) =max
h◦(S,S),h◦(S,S)
,
where
h◦(S,S) =sup
d(b,S) :b∈S
and
d(b,S) =inf
d(b,s) :s∈S
.
of a real locally convex Hausdorff topological vector space X. Let the set-valued mapping S:K→Kbe continuous with nonempty compact convex values. Ifψ:K×K→Y satisfies the following conditions:
(i) ψ(·,·)is (–C)-continuous;
(ii) for any fixedx∈K,ψ(x,·)isC-convex; (iii) for anyx∈K,ψ(x,x) /∈–intC.
Then, there exists an element x*∈K such that x*∈S(x*)and
ψx*,y∈/–intC for all y∈Sx*.
3 Essential components of the solution set for the system of vector quasi-equilibrium problems
Throughout this section, letI={, , . . . ,n}be a finite set which has at least two elements. For eachi∈I, letXi be a real normed linear space andYia Banach space withYi⊂Yn; letKibe a nonempty compact convex subset ofXi, and letCibe a closed convex pointed cone ofYiwithCi=Cn∩YiandintCi=∅. LetK=
n
i=KiandX=
n i=Xi.
Letbe the collection of all vector-valued functions such thatψ:K×K→Ynsuch that: (i)ψ(x,y) is (–Cn)-continuous onK×K; (ii) for each fixedx∈K,ψ(x,·) isCn-convex; (iii) for anyx∈K,ψ(x,x) =θ, whereθis the zero element ofYn; (iv)sup(x,y)∈K×Kψ(x,y)< +∞.
LetMbe the collection of all set-valued mappingsS:K→K such that: (i) for each x∈K,S(x) is convex and closed; (ii)Sis continuous onK.
LetH=×M. For anyu= (ψ,S),u= (ψ,S)∈H, define
ρ(u,u) = sup (x,y)∈K×K
ψ(x,y) –ψ(x,y)+sup
x∈K
hS(x) –S(x),
where · is the norm onYnandhis the Hausdorff metric defined onCK(X). Clearly, (H,ρ) is a metric space.
For anyu= (ψ,S)∈H, by Theorem ., there exists a solutionx*∈Kto the vector
quasi-equilibrium problem:x*∈S(x*) and
ψx*,y∈/–intCn for ally∈S
x*.
For eachu= (ψ,S)∈H, define
F(u) =x∈K:x∈S(x) andψ(x,y) /∈–intCnfor ally∈S(x)
. ()
ThusF(u)=∅for anyu∈Handu→F(u) indeed defines a set-valued mapping fromH toK.
The following lemma can be found in [].
Lemma . F:H→K is an usco mapping.
Proof SinceKis compact, by [], it suffices to prove thatGraph(F) ={(u,x)∈H×K:x∈ F(u)}is closed. Let{(un,xn)} ⊂Graph(F) with (un,xn)→(u,x¯)∈H×K, whereun= (ψn,Sn) andu= (ψ,S). Sincexn∈F(un), we have
xn∈Sn(xn) and ψn(xn,y) /∈–intCn for ally∈Sn(xn).
For any open neighborhoodOofS(x¯) inK, sinceS(x¯) is compact, by [, p.], there is
ε> such that
x∈K:dx,S(x¯)<ε
⊂O,
whered(x,S(x¯)) =infa∈S(¯x)x–a. Sinceρ((ψn,Sn), (ψ,S))→,xn→ ¯x, andSis u.s.c. at
¯
x, there isNsuch that for anyn>N, we have
sup
x∈K
hSn(x),S(x)<ε/,
and
S(xn)⊂
x∈K:dx,S(x¯)<ε/
.
So whenevern>N, we have
Sn(xn)⊂
x∈K:dx,S(xn)
<ε/
⊂x∈K:dx,S(x¯)<ε
⊂O.
Sincexnbelongs toSn(xn), andS(x¯) andSn(xn) are compact, by Lemma ., there exists a subsequence{xnk}of{xn}such thatxnk→x∈S(x¯). Sincexnk→ ¯x, we have
¯
x=x∈S(x¯). ()
SinceSis l.s.c. atx¯∈K, for anyz∈S(x¯), by [], there existsan∈S(xn) such thatan→z. Sinceρ((ψn,Sn), (ψ,S))→, there exists a subsequence{Snk}of{Sn}such that
sup
x∈K
hSnk(x),S(x)
< /k.
Thus, there exists a subsequence{xnk}of{xn}such that
hSnk(xnk),S(xnk)
< /k,
which implies that there existsan
k∈Snk(xnk) such that
a
nk–ank< /k,
where {ank} is a subsequence of{an}. Sinceank–z ≤ a
nk –ank+ank –z< /k+ ank–zandank→z(k→+∞), we have thatank →z(k→+∞). Asa
have
ψnk
xnk,a
nk
/
∈–intCn for allk. ()
Now we need to show that
ψ(x¯,z) /∈–intCn. ()
If the conclusion is false, thenψ(x¯,z)∈–intCn, which implies that there isε¯> such that
ψ(x¯,z) +ε¯B⊂–intCn, ()
whereBdenotes the open unit ball inYn. Sinceψis (–Cn)-continuous onK×K,xnk→ ¯x
andank→z, for aboveε¯> , there is a positive integerksuch that
ψxnk,a
nk
∈ψ(x¯,z) + (/)ε¯B–Cn for allk≥k. ()
On the other hand, sinceρ((ψnk,Snk), (ψ,S))→, there is a positive integerkwithk≥
k, such that
ψnk(x,y)∈ψ(x,y) + (/)ε¯B for any (x,y)∈K×Kand allk≥k. ()
By (), () and (), we have
ψnk(xnk,a
nk)∈ψ(xnk,a
nk) + (/)ε¯B⊂ψ(x¯,z) + (/)ε¯B+ (/)ε¯B–Cn ⊂ψ(x¯,z) +ε¯B–Cn⊂–intCn–Cn⊂–intCn for allk≥k.
This contradicts (). Hence () holds. Then by the arbitrariness ofz∈S(x¯), we obtain that
ψ(x¯,z) /∈–intCn for allz∈S(x¯). ()
By () and (), we have that ((ψ,S),x¯)∈Graph(F). Hence,Graph(F) is closed.F(u) is also closed, for allu∈H. By the compactness ofK, we know thatFis a set-valued mapping with compact values. Hence,Fis an usco mapping. The proof is completed.
For each u∈H, the component of a pointx∈F(u) is the union of all the connected subsets ofF(u) containingx. Note that the components are connected closed subsets of F(u), and thus are connected and compact, see []. It is easy to see that the components of two distinct points ofF(u) either coincide or are disjoint, so that all components constitute a decomposition ofF(u) into connected pairwise disjoint compact subsets,i.e.,
F(u) = α∈
Fα(u),
Definition . Letu∈Handmbe a nonempty closed subset ofF(u).mis said to be an essential set ofF(u) if, for each open setO⊃m, there existsδ> such that for anyu∈H withρ(u,u) <δ,F(u)∩O=∅. If a componentFα(u) ofF(u) is an essential set, thenFα(u)
is said to be an essential component ofF(u). An essential setmofF(u) is said to be a minimal essential set ofF(u) ifmis a minimal element of the family of essential sets in F(u) ordered by set inclusion.
Lemma .[] Let A, B and C be nonempty convex compact subsets of a normed linear space X. Then h(A,λB+μC)≤λh(A,B) +μh(A,C), where h is the Hausdorff metric defined on CK(X),λ≥,μ≥, andλ+μ= .
The following theorem is the exist theorem of an essential component of the set of so-lutions for the vector quasi-equilibrium problem.
Theorem .
(a) For anyu∈H, there exists at least one minimal essential set ofF(u), and every minimal essential set ofF(u)is connected;
(b) For anyu∈H, there exists at least one essential connected component ofF(u).
Proof
(a) SinceF is upper semicontinuous, following the idea of Lemma . in [], we can easily obtain that there exists one minimal essential set ofF(u) for eachu∈H. Now, for each minimal essential set ofF(u), as Yang and Yu did in [], we prove that each minimal essential set ofF(u) is connected. Letm(u) be one minimal essential set ofF(u). Ifm(u) is not connected, then there exist two nonempty closed setsc(u) andc(u) ofF(u) and two
open setsVandVinKsuch that
m(u) =c(u)∪c(u), c(u)⊂V, c(u)⊂V, V∩V=∅.
Sincem(u) is a minimal essential set ofF(u), neitherc(u) norc(u) is essential. Thus,
there exist two open setsO⊃c(u) andO⊃c(u) such that, for anyδ> , there exist
u,u∈Hwith
ρ(u,u) <δ, ρ(u,u) <δ, but F(u)∩O=∅, F(u)∩O=∅. ()
SetW=V∩O, andW=V∩O. Then bothWandWare open sets andc(u)⊂W,
andc(u)⊂W. Sincec(u) andc(u) are a closed subset of the compact setF(u),c(u)
andc(u) are a compact set, there exist two open setsUandUsuch that
c(u)⊂U⊂U⊂W, c(u)⊂U⊂U⊂W, and W∩W=∅.
Sincem(u) is essential andm(u)⊂U∪U, there existsδ> such that for anyuwith ρ(u,u) <δ, we have
SinceU⊂OandU⊂O, for aboveδ/ > , by (), there existu,u∈Hsuch that
ρ(u,u) <δ/, ρ(u,u) <δ/, F(u)∩U=∅, F(u)∩U=∅. ()
Sinceu,u∈H, we haveu= (ψ,S) andu= (ψ,S). Now we defineS:K→K and ψ:K×K→Ynby
S(x) =λ(x)S(x) +μ(x)S(x) for allx∈K
and
ψ(x,y) =λ(x)ψ(x,y) +μ(x)ψ(x,y) for all (x,y)∈K×K,
respectively, where
λ(x) = d(x,U) d(x,U) +d(x,U)
, μ(x) = d(x,U) d(x,U) +d(x,U)
for allx∈K.
It is obvious that λandμare continuous functions onK withλ(x)≥,μ(x)≥, and
λ(x) +μ(x) = for anyx∈K.
We can see that: (i) ψ(x,y) is (–Cn)-continuous on K × K; (ii) for each fixed x ∈ K, ψ(x,y) is Cn-convex in y; (iii) ψ(x,x) = θ ∈/ –intCn for all x ∈ K; (iv)sup(x,y)∈K×Kψ(x,y)< +∞; (v) for eachx∈K,S(x) is convex and compact; (vi)S is continuous onK. Hencev:= (ψ,S)∈H. By Lemma ., we have
hS(x),S(x)≤λ(x)hS(x),S(x)
+μ(x)hS(x),S(x)
≤hS(x),S(x)
+hS(x),S(x)
for allx∈K,
and
ψ(x,y) –ψ(x,y)=ψ(x,y) –λ(x)ψ(x,y) –μ(x)ψ(x,y)
≤λ(x)ψ(x,y) –ψ(x,y)+μ(x)ψ(x,y) –ψ(x,y) ≤ψ(x,y) –ψ(x,y)
+ψ(x,y) –ψ(x,y) for all (x,y)∈K×K.
Thus, by (), we have
ρ(u,v) = sup (x,y)∈K×K
ψ(x,y) –ψ(x,y)+sup
x∈K
hS(x),S(x)
≤ sup
(x,y)∈K×K
ψ(x,y) –ψ(x,y)+ sup (x,y)∈K×K
ψ(x,y) –ψ(x,y)
+sup
x∈K
hS(x),S(x)
+sup
x∈K
hS(x),S(x)
Using (), we have
F(v)∩(U∪U)=∅. ()
Ifx∈U, thenλ(x) = ,μ(x) = ,S(x) =S(x) andψ(x,y) =ψ(x,y) for ally∈K. Ifx∈
F(v), thenx∈S(x) andψ(x,y) /∈–intCnfor ally∈S(x). SinceS(x) =S(x) andψ(x,y) = ψ(x,y) for ally∈K, we havex∈F(u). This contradicts (). Thus, we haveF(v)∩U=∅.
Similarly, we can show thatF(v)∩U=∅. This contradicts (). Hence,m(u) is connected,
so the conclusion (a) holds.
(b) For anyu∈H, by (a), there exists at least one essential connected setmofF(u). There exists a componentFα(u) ofF(u) such thatm⊂Fα(u). It is obvious thatFα(u) is
essential.
Now, for eachi∈I, letfi:K×Ki→Yibe a vector-valued mapping andSi:K→Kia set-valued mapping. Let
D={(f, . . . ,fi, . . . ,fn): for eachi∈I={, , . . . ,n},fi(·,·) is (–Ci)-continuous on
K×Ki; for eachi∈Iand each fixedx∈K,fi(x,·) isCi-convex; for eachi∈I and eachx∈K,fi(x,xi) =θ, wherexiis theith component ofxandθis the zero element ofYi; for eachi∈I,sup(x,yi)∈K×Kifi(x,yi)< +∞, where · is the norm
onYn}.
Let
Q={(S, . . . ,Si, . . . ,Sn): for eachi∈I={, , . . . ,n} and each x∈K,Si(x) is a
nonempty compact convex subset ofKi; for eachi∈I,Siis continuous onK}.
LetP=D×Q. For any
p=
(f, . . . ,fn), (S, . . . ,Sn)
, p=
(f, . . . ,fn), (S, . . . ,Sn)
∈P,
define
ρ(p,p) =
n
i=
sup
(x,yi)∈K×Ki
fi(x,yi) –fi(x,yi)+sup
x∈K h
n
i=
Si(x),
n
i=
Si(x)
,
wherehis the Hausdorff metric defined onCK(X). Clearly, (P,ρ) is a metric space.
LetK=ni=Ki. It is clear thatKis a nonempty compact convex subset ofX=ni=Xi. For any (f, . . . ,fn)∈D, and (S, . . . ,Sn)∈Q, define the mappingψ:K×K→Ynby
ψ(x,y) = n
i=
fi(x,yi), x= (x, . . . ,xn),y= (y, . . . ,yi, . . . ,yn)∈K,
and the mappingS:K→Kby
S(x) = n
i=
Since (f, . . . ,fn)∈Dand (S, . . . ,Sn)∈Q, we can see thatS:K→K is continuous with
nonempty convex and compact valued, (i)ψ(·,·) is (–Cn)-continuous onK×K; (ii) for each fixedx∈K,ψ(x,·) is (–Cn)-convex; and (iii)ψ(x,x) =θ∈/–intCnfor allx∈K. It is clear that (ψ,S)∈H. SinceYi⊂Ynfor eachi∈I={, , . . . ,n}, by Theorem ., there exists an elementx*∈Ksuch thatx*∈S(x*) and
ψx*,y∈/–intCn for ally∈S
x*.
That is
f
x*,y
+· · ·+fi
x*,yi
+· · ·+fn
x*,yn
/
∈–intCn for ally∈S
x*, . . . ,yn∈Sn
x*.
For eachi∈I, by the arbitrariness ofyj∈Sj(x*),j∈ {, . . . ,n},j=i, takeyj=x*j, and by assumptionfj(x*,x*j) =θ,j= , . . . ,n, andj=i, we obtain thatx*i∈Si(x*) and
fi
x*,yi
/
∈–intCn for allyi∈Si
x*.
Sincefi(x*,yi)∈YiandCi=Cn∩Yi, it follows that
fi
x*,yi
/
∈–intCi for allyi∈Si
x*.
Thus, there existsx*= (x*
, . . . ,x*n)∈Ksuch that for eachi∈I,x*i∈Si(x*) and
fi
x*,yi
/
∈–intCi for allyi∈Si
x*. ()
For eachp∈P, denote byE(p) all solutions to the SVQEP. By (), there existsx*∈E(p),
thusE(p)=∅. Similar to Definition ., we can define the minimal essential set and essen-tial component ofE(p).
Lemma . For each p= ((f, . . . ,fn), (S, . . . ,Sn))∈P, define the mapping T:P→H by
T(p) = (ψ,S),
where
ψ(x,y) = n
i=
fi(x,yi), x= (x, . . . ,xn),y= (y, . . . ,yi, . . . ,yn)∈K
and
S(x) = n
i=
Si(x), x∈K.
Then T is continuous.
For any p = ((f, . . . ,fn), (S, . . . ,Sn)), p = ((f, . . . ,fn), (S, . . . ,Sn)) ∈ P, if ρ(p,p) <ε, then by the definition ofρ, we have
ρ
T(p),T(p)
= sup
(x,y)∈K×K
n
i=
fi(x,yi) –
n
i=
fi(x,yi)
+sup
x∈K h
n
i=
Si(x),
n
i=
Si(x)
≤ sup
(x,y)∈K×K
f(x,y) –f(x,y)
+· · ·+ sup
(x,y)∈K×K
fn(x,yn) –fn(x,yn)+sup x∈K h
n
i=
Si(x), n
i=
Si(x)
= n
i=
sup
(x,yi)∈K×Ki
fi(x,yi) –fi(x,yi)+sup
x∈K h
n
i=
Si(x),
n
i=
Si(x)
≤ρ(p,p) <ε.
This completes the proof of the lemma.
The following lemma can be found in [].
Lemma . Let U, Y and Z be three metric spaces, F:U→Y be an usco mapping and G:Z→Y be a set-valued mapping. Suppose that there exists a continuous mapping T: Z→U such that G(z)⊃F(T(z))for each z∈Z. Furthermore, suppose that there exists at least one essential component of F(ϕ)for eachϕ∈U. Then there exists at least one essential component of G(z)for each z∈Z.
As application of Theorem ., now we will show that, for every system of vector quasi-equilibrium problems, there exists at least one essential component of the set of its solu-tions in the uniform topological space of objective funcsolu-tions and constraint mappings.
Theorem . For each p∈P, there exists at least one essential component of E(p).
Proof For anyp= ((f, . . . ,fn), (S, . . . ,Sn))∈P, defineT:P→HbyT(p) = (ψ,S), where
ψ(x,y) = n
i=
fi(x,yi), x= (x, . . . ,xn),y= (y, . . . ,yi, . . . ,yn)∈K
and
S(x) = n
i=
Si(x), x∈K.
By Lemma .,Tis continuous. Now we need to prove that for eachp∈P,E(p)⊃F(T(p)), whereFis defined by (). Ifx*= (x*
, . . . ,x*i, . . . ,x*n)∈F(T(p)), thenx*∈K,x*∈S(x*) and
ψx*,y∈/–intCn for ally∈S
That is
f
x*,y
+· · ·+fi
x*,yi
+· · ·+fn
x*,yn
/
∈–intCn for ally∈S
x*, . . . ,yn∈Sn
x*.
For eachi∈I, by the arbitrariness ofyj∈Sj(x*),j∈ {, . . . ,n},j=i, takeyj=x*j, and by assumptionfj(x*,x*
j) =θ,j= , . . . ,n, andj=i, we obtain thatx*i∈Si(x*) and
fi
x*,yi
/
∈–intCn for allyi∈Si
x*.
Sincefi(x*,yi)∈YiandCi=Cn∩Yi, it follows that
fi
x*,yi
/
∈–intCi for allyi∈Si
x*.
Hence x* ∈ E(p) and hence E(p)⊃F(T(p)). Thus, by Lemma ., Theorem . and Lemma ., there exists at least one essential component ofE(p).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This work was carried out in collaboration between all authors. X-HG gave the ideas of the problems in this research and interpreted the results. J-CC proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.
Author details
1Department of Mathematics, Nanchang University, Nanchang, 330031, China.2College of Mathematics and Information
Science, Nanchang Hangkong University, Nanchang, 330063, China.
Acknowledgements
This research was partially supported by the National Natural Science Foundation of China (Grant No. 11061023) and the Natural Science Foundation of Jiangxi Province (2010GZS0176), China.
Received: 11 May 2012 Accepted: 3 August 2012 Published: 30 August 2012 References
1. Fort, MK: Essential and nonessential fixed points. Am. J. Math.72, 315-322 (1950)
2. Kinoshita, S: On essential component of the set of fixed points. Osaka J. Math.4, 19-22 (1952)
3. Jiang, JH: Essential components of the set of fixed points of multivalued mappings and its application to the theory of games. Sci. Sin.12, 951-964 (1963)
4. Kohlberg, E, Mertens, JF: On the strategic stability of equilibria. Econometrica54, 1003-1037 (1986)
5. Yu, J, Xiang, SW: On essential components of the set of Nash equilibrium points. Nonlinear Anal., Theory Methods Appl.38, 259-264 (1999)
6. Yu, J, Luo, Q: On essential components of the solution set of generalized games. J. Math. Anal. Appl.230, 303-310 (1999)
7. Isac, G, Yuan, GXZ: The essential components of coincident points for weakly inward and outward set-valued mappings. Appl. Math. Lett.12, 121-126 (1999)
8. Yang, H, Yu, J: Essential components of the set of weakly Pareto-Nash equilibrium points. Appl. Math. Lett.15, 553-560 (2002)
9. Lin, Z: Essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in two different topological spaces. J. Optim. Theory Appl.124, 387-405 (2005)
10. Chen, JC, Gong, XH: The stability of set of solutions for symmetric vector quasi-equilibrium problems. J. Optim. Theory Appl.136, 359-374 (2008)
11. Ansari, QH, Schaible, S, Yao, JC: The system of generalized vector equilibrium problems with applications. J. Glob. Optim.22, 3-16 (2002)
12. Ansari, QH, Chan, WK, Yang, XQ: The system of vector quasi-equilibrium problems with applications. J. Glob. Optim. 29, 45-57 (2004)
13. Fang, YP, Huang, NJ, Kim, JK: Existence results for system of vector equilibrium problems. J. Glob. Optim.35, 71-83 (2006)
14. Peng, JW, Lee, HWJ, Yang, XM: On system of generalized vector quasi-equilibrium problems with set-valued maps. J. Glob. Optim.36, 139-158 (2006)
16. Peng, JW, Yang, XM, Zhu, DL: System of vector quasi-equilibrium and its applications. Appl. Math. Mech.27, 1107-1114 (2006)
17. Chen, JC, Gong, XH: Generic stability of the solution set for symmetric vector quasi-equilibrium problems under the condition of cone-convexity. Acta Math. Sci.30, 1006-1017 (2010)
18. Yang, H, Yu, J: The generic stability and the existence of essential components of the solution sets for generalized vector variational-like inequalities. J. Syst. Sci. Math. Sci.22, 90-95 (2002) (in Chinese)
19. Aubin, JP, Ekeland, I: Applied Nonlinear Analysis. Wiley, New York (1984) 20. Engelking, R: General Topology. Helderman, Berlin (1989)
doi:10.1186/1029-242X-2012-181
