On the existence and stability of solutions of a mixed general type of variational relation problems

R E S E A R C H

Open Access

On the existence and stability of solutions of

a mixed general type of 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 a mixed general type of variational relation problems, and establish the existence theorem of solutions of mixed general types of variational relation problems. Moreover, we study the stability of a solution set of mixed general types of variational relation problems. We prove that most of mixed general types of variational relation problems (in the sense of Baire category) are essential and, for any mixed general type of variational relation problems, there exists at least one essential connected component of its solution set.

MSC: 49J53; 49J40

Keywords: mixed general types of variational relation problems; existence; generic stability; essential connected components

1 Introduction

It is well known that the equilibrium problems are unified models of several problems, namely optimization problems, saddle point problems, 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 (VR). The stability of the solution set of variational relation problems was studied in [, ]. Further studies of variational relation problems have been done (see [–]). Recently, Agarwalet al.[] presented a unified approach in studying the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium prob-lems, variational inequalities and variational inclusions investigated in the recent litera-ture. Balaj and Lin [] established existence criteria for the solutions of two very general types of variational relation problems.

Motivated and inspired by research works mentioned above, we introduce mixed gen-eral types of variational relation problems, which is a mixed structure of two gengen-eral types of variational relation problems in []. Moreover, we study the stability of a solution set of mixed general types of variational relation problems.

2 Mixed general types of variational relation problems

In [], letX,Y be convex sets in two Hausdorff topological vector spaces,Zbe a topo-logical space,S,S:X⇒X,T:X⇒Y,P:X⇒Zbe set-valued mappings with nonempty values, andR(x,y,z) be a relation linking elementsx∈X,y∈Y andz∈Z. Balaj and Lin

[] established existence criteria for the solutions of the following variational relation problems:

(VRP) Find(x∗,y∗)∈X×Ysuch thatx∗∈S(x∗),y∗∈T(x∗)and,∀u∈S(x∗),∃z∈P(x∗)

for whichR(u,y∗,z)holds.

(VRP) Find(x∗,y∗)∈X×Y such thatx∗∈S(x∗),y∗∈T(x∗)andR(u,y∗,z)holds∀u∈ S(x∗)and∀z∈P(x∗).

In this paper, we introduce mixed general types of variational relation problems. Let X,Ybe convex sets in two Hausdorff topological vector spaces,Zbe a topological space, S:X×Y⇒X,T:X×Y⇒Y,H:X×Y⇒X,G:X×Y⇒Y,P:X×Y⇒Zbe set-valued mappings with nonempty values, andR(x,y,z),Q(y,x,z) be two relations linking elements x∈X,y∈Y andz∈Z. A mixed general type of variational relation problems (MGVR) consists in finding (x∗,y∗)∈X×Ysuch thatx∗∈S(x∗,y∗),y∗∈T(x∗,y∗) and

∀u∈Hx∗,y∗, ∃z∈Px∗,y∗ s.t. Ru,y∗,zholds,

Qv,x∗,zholds, ∀v∈Gx∗,y∗,∀z∈Px∗,y∗.

Remark . Balaj and Lin [] established existence criteria for the solutions of two very general types of variational relation problems. The mixed general type of variational rela-tion problems is a combinarela-tion of (VRP) and (VRP), and (VRP) and (VRP) are some special cases of (MGVR).

Theorem . Assume that

(i) X,Y,Zare three nonempty,compact and convex subsets of three Hausdorff linear topological spaces;

(ii) C={(x,y)∈X×Y:x∈S(x,y)}andD={(x,y)∈X×Y:y∈T(x,y)}are closed in X×Y;

(iii) Pis continuous with nonempty compact values;

(iv) for any(x,y)∈X×Y,coH(x,y)⊂S(x,y),coG(x,y)⊂T(x,y),H,Ghave open fibers,andR(x,·,·),Q(·,y,·)are closed;

(v) for any fixedy∈Y,any finite subset{u, . . . ,un}ofXand anyx∈co{u, . . . ,un}, there arei∈ {, . . . ,n}andz∈P(x,y)such thatR(ui,y,z)holds;

(vi) for any fixedx∈X,any finite subset{v, . . . ,vn}ofYand anyy∈co{v, . . . ,vn},there isi∈ {, . . . ,n}such thatQ(vi,x,z)holds for anyz∈P(x,y).

Then(MGVR)has at least one solution.

Proof DefineA:X×Y⇒XandB:X×Y⇒Yas follows:

A(x,y) =u∈X:R(u,y,z) does not hold∀z∈P(x,y),

B(x,y) =v∈Y:Q(v,x,z) does not hold∃z∈P(x,y).

AsR(x,·,·),Q(·,y,·) are closed for any (x,y)∈X×Y, andPis continuous with nonempty compact values, by Propositions ., . of [],A,Bhave open fibers.

z∈P(x,y) such thatR(ui,y,z) holds, which contradicts the fact thatui∈A(x,y) for any i∈ {, . . . ,n},i.e.,R(ui,y,z) does not hold for anyz∈P(x,y) and anyi∈ {, . . . ,n}. Hence x∈/coA(x,y) for any (x,y)∈X×Y.

Suppose that there exists (x,y)∈X×Ysuch thaty∈coB(x,y), then there is a finite sub-set{v, . . . ,vn}ofB(x,y) such thaty∈co{v, . . . ,vn}. By (vi), there isi∈ {, . . . ,n}such that Q(vi,x,z) holds, which contradicts the fact thatvi∈B(x,y) for anyi∈ {, . . . ,n},i.e., there isz∈P(x,y) such thatQ(vi,x,z) does not hold for anyi∈ {, . . . ,n}. Hencey∈/coB(x,y) for any (x,y)∈X×Y.

DefineA:X×Y⇒XandB:X×Y⇒Yas follows:

A(x,y) =

⎧ ⎨ ⎩

A(x,y)∩H(x,y) if (x,y)∈C,

H(x,y) if (x,y) /∈C,

B(x,y) =

⎧ ⎨ ⎩

B(x,y)∩G(x,y) if (x,y)∈D,

G(x,y) if (x,y) /∈D.

For anyu∈X, A–_{(u) = [H}–_(u)∩A–_{(u)]∪[((X×Y)\C)∩H}–_{(u)] is open inX×Y.} Similarly,B–_{(v) is open for anyv∈Y. Hence,A}–_(u),B–_{(v) are open for any (u,v)∈X×} Y, andx∈/coA(x,y),y∈/coB(x,y) for any (x,y)∈X×Y. By Theorem  of [], there exists (x∗,y∗)∈X×Ysuch thatA(x∗,y∗) =∅andB(x∗,y∗) =∅, which implies thatx∗∈S(x∗,y∗), y∗∈T(x∗,y∗) and

∀u∈Hx∗,y∗, ∃z∈Px∗,y∗ s.t. Ru,y∗,zholds,

Qv,x∗,zholds, ∀v∈Gx∗,y∗,∀z∈Px∗,y∗.

Theorem . Assume that

(i) X,Y,Zare three nonempty,compact and convex subsets of three normed linear topological spaces;

(ii) S,Tare continuous with nonempty convex compact values; (iii) Pis continuous with nonempty compact values;

(iv) R(·,·,·)andQ(·,·,·)are closed;

(v) for any fixedy∈Y,any finite subset{u, . . . ,un}ofXand anyx∈co{u, . . . ,un}, there arei∈ {, . . . ,n}andz∈P(x,y)such thatR(ui,y,z)holds;

(vi) for any fixedx∈X,any finite subset{v, . . . ,vn}ofYand anyy∈co{v, . . . ,vn},there isi∈ {, . . . ,n}such thatQ(vi,x,z)holds for anyz∈P(x,y).

Then there exists(x∗,y∗)∈X×Y such that x∗∈S(x∗,y∗),y∗∈T(x∗,y∗)and

∀u∈Sx∗,y∗, ∃z∈Px∗,y∗ s.t. Ru,y∗,zholds,

Qv,x∗,zholds, ∀v∈Tx∗,y∗,∀z∈Px∗,y∗.

Proof For anyn, defineSn_:X×Y⇒X,Tn_:X×Y⇒Y,Hn_{:X×Y⇒XandG}n_:X×Y⇒ Yby

Sn(x,y) =S(x,y) +clVn∩X, Tn(x,y) =T(x,y) +clV_n∩Y,

whereVn={x∈X:x< _n},V_n ={y∈Y:y< _n}. SinceS,T are continuous,Hn_,Gn have open fibers, andGraph(Sn_),Graph(Tn_{) are closed inX×Y, by Theorem ., there} existxn_∈Sn_(xn_,yn_),yn_∈Tn_(xn_,yn_{) and}

∀u∈Hnxn,yn, ∃z∈Pxn,yn s.t. Ru,yn,zholds,

Qv,xn_{,zholds, ∀v∈G}n_xn_,yn_,∀z∈Pxn_,yn_.

Since X, Y are nonempty and compact, without loss of generality, we assume that (xn_,yn_{)→(x,y). Letm>  be arbitrarily fixed andn}

>  such thatVn+Vn+clVn⊂Vm. Since S is continuous and (xn_,yn_{)→(x,y), there isN}

 >  such thatx–xn∈Vn and S(xn_,yn_{)⊂S(x,y) +Vnfor anyn>N}

. Therefore, for anyn>max{N,n},

x=x–xn+xn∈x–xn+Sxn,yn+clVn

⊂Vn+S(x,y) +Vn+clVn⊂S(x,y) +Vm.

Hencex∈ _m>(S(x,y) +Vm) =clS(x,y) =S(x,y). Similarly,y∈T(x,y).

Suppose that there existsu∈S(x,y) such thatR(u,y,P(x,y)) does not hold. SinceR(·,·,·) is closed, there existsk>  such thatR(u+Vk,y+Vk,P(x,y) +Vk) does not hold. SinceSis continuous, there exists a sequence{un_{}convergent touwithu}n_∈S(xn_,yn_)⊂S(xn_,yn_{) +} Vn⊂Hn(xn,yn). SinceP is continuous, there exists N>  such that, for any n>N, un_∈u+V

k,yn∈y+Vk,P(xn,yn)⊂P(x,y) +Vk, which implies thatun∈Hn(xn,yn) and R(un_,yn_,P(xn_,yn_{)) does not hold. It is a contradiction.}

Suppose that there existv∈T(x,y) andz∈P(x,y) such thatQ(v,x,z) does not hold. Since T,Pare continuous, there exist two sequences{vn_}and{zn_{}convergent tovandzwith} vn_∈T(xn_,yn_)⊂T(xn_,yn_{) +V}

n⊂Gn(xn,yn) andzn∈P(xn,yn). AsQ(·,·,·) is closed, there existsN>  such that, for anyn>N,Q(vn,xn,zn) does not hold, which is a contradiction.

This completes the proof.

3 Generic stability analysis

LetX,Y,Zbe three nonempty, compact and convex subsets of three normed linear topo-logical spaces. Denote byMthe collection of all (MGVR) such that all conditions of Theo-rem . hold. For eachq∈M, denote byF(q) the solution set ofq. Thus, a correspondence F:M⇒X×Yis well defined. For eachq,q∈M, define the distance onMby

ρq,q= sup

(x,y)∈X×Y

hXS(x,y),S(x,y)+ sup

(x,y)∈X×Y

hYT(x,y),T(x,y)

+ sup

(x,y)∈X×Y

hZP(x,y),P(x,y)+HGr(R),GrR+HGr(Q),GrQ,

where Gr(R) ={(x,y,z)∈X×Y ×Z :R(x,y,z) holds}, Gr(Q) ={(x,y,z)∈X×Y ×Z : Q(y,x,z) holds},hX (hY,hZ) is the Hausdorff distance defined onX(Y,Z), andHis the Hausdorff distance defined onX×Y×Z.

Definition . Letq∈M. A nonempty closed subsete(q) ofF(q) is said to be an essential set ofF(q) if, for any open setU,e(q)⊂U, there is a positiveδsuch thatU∩F(q)=∅for anyq∈Mwithρ(q,q) <δ.

Definition . Letq∈M. An essential subsetm(q)⊂F(q) is said to be a minimal essential set ofF(q) if it is a minimal element of the family of essential sets inF(q) ordered by set inclusion. A connected componentC(q) ofF(q) is said to be an essential component of F(q) ifC(q) is essential.

Remark . () It is easy to see that the problemq∈M is essential if and only if the mappingF:M⇒X×Y is lower semicontinuous atq. () For two closede(q)⊂e(q)⊂ F(q), ife(q) is essential, thene(q) is also essential.

Lemma .(. Lemma, . Closed Graph Theorem of []) (i)The image of a com-pact set under a comcom-pact-valued upper semicontinuous set-valued mapping is comcom-pact. (ii)A correspondence with compact Hausdorff range space is closed if and only if it is upper hemicontinuous and closed-valued.

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.

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, that is, for anyn,m>N,}

sup

(x,y)∈X×Y hX

Sn(x,y),Sm(x,y)+ sup

(x,y)∈X×Y hY

Tn(x,y),Tm(x,y)

+ sup

(x,y)∈X×Y

hZPn(x,y),Pm(x,y)

+HGrRn,GrRm+HGrQn,GrQm≤ε.

() Clearly, we consult Proposition . of []. There areS:X×Y⇒X,T:X×Y⇒Y andP:X×Y⇒Zsuch thatS,T are continuous with nonempty convex compact values, andPis continuous with nonempty compact values.

() There exist two closed subsets A, B of X×Y ×Z such that Gr(Rn_{)→A and}

Gr(Qn_{)→B. Denoteq= (S,T,P,R,Q), where}

R(x,y,z) holds iff (x,y,z)∈A, Q(y,x,v) holds iff (x,y,z)∈B.

ClearlyR(·,·,·) andQ(·,·,·) are closed.

() Suppose the existence ofy∈Y, finite subset{u, . . . ,un}ofXandx∈co{u, . . . ,un} such thatR(ui,y,P(x,y)) does not hold for anyi∈ {, . . . ,n}, which implies (ui,y,P(x,y))∩

Gr(R) =∅,∀i∈ {, . . . ,n}. Sinceqm_{→qfor enough largem, (u}

i,y,Pm(x,y))∩Gr(Rm) =∅,

Suppose the existence of x∈X, finite subset {v, . . . ,vn} of Y, y∈co{v, . . . ,vn} and z∈P(x,y) such thatQ(vi,x,z) does not hold for anyi∈ {, . . . ,n}, which implies (x,vi,z) /∈

Gr(Q), ∀i ∈ {, . . . ,n}. Since qm _{→ q, there exists a sequence {z}m_{} convergent to z} with zm_∈Pm_{(x,y). Hence, for enough large m, (x,vi,z}m_{) /∈Gr(Q}m_{), ∀i∈ {, . . . ,n}, i.e.,} Qm(vi,x,zm) does not hold for anyi∈ {, . . . ,n}, which is a contradiction. Henceq∈M

and (M,ρ) is complete.

Theorem . The mapping F:M⇒X×Y is upper semicontinuous with nonempty com-pact values.

Proof The desired conclusion follows from Lemma . as soon as we show thatGraph(F) is closed. Denoteqn= (Sn,Tn,Pn,Rn,Qn) andq= (S,T,P,R,Q). Let{(qn,xn,yn)∈M×X× Y}∞_n=be a sequence converging to (q,x,y) such that (xn,yn)∈F(qn) for anyn. Thenxn∈ Sn(xn,yn) andyn∈Tn(xn,yn),

∀u∈Snxn,yn, ∃z∈Pnxn,yn s.t. Rnu,yn,zholds,

Qnv,xn,zholds, ∀v∈Tnxn,yn,∀z∈Pnxn,yn.

Clearly,x∈S(x,y) andy∈T(x,y).

Suppose the existence of u ∈ S(x,y) such that R(u,y,P(x,y)) does not hold, then (u,y,P(x,y))∩Gr(R) =∅. SinceS,Pare continuous,qn_→qand

hXSnxn,yn,S(x,y)

≤hX

Snxn,yn,Sxn,yn+hXSxn,yn,S(x,y)→,

hZ

Pnxn,yn,P(x,y)

≤hZ

Pnxn,yn,Pxn,yn+hZ

Pxn,yn,P(x,y)→,

thenSn(xn,yn)→S(x,y) andPn(xn,yn)→P(x,y). Thus, there exists a sequence{un} con-vergent touwithun∈Sn(xn,yn) such that, for enough largen, (un,yn,Pn(xn,yn))∩Gr(Rn) =

∅,i.e.,un∈Sn(xn,yn) andRn(un,yn,Pn(xn,yn)) does not hold, which is a contradiction. Suppose the existence ofv∈T(x,y) andz∈P(x,y) such thatQ(v,x,z) does not hold, then (x,v,z) /∈Gr(Q). Similarly,Tn(xn,yn)→T(x,y) andPn(xn,yn)→P(x,y). Thus, there exist two sequences{vn}and{zn}convergent touandzwithvn∈Tn(xn,yn) andzn∈Pn(xn,yn). Hence, for enough large n, (xn_,vn_,zn_{) /∈Gr(Q}n_),i.e.,vn_∈Tn_(xn_,yn_),zn_∈Pn_(xn_,yn_{) and} Qn_(vn_,xn_,zn_{) does not hold, which is a contradiction. Hence (x,y)∈F(q).}

Theorem . (i)There exists a dense residual subsetGofMsuch that q is essential for each q∈G. (ii)For any q∈M,there exists at least one minimal essential subset of F(q).

Proof The proofs are similar to those of Theorems . and . of []. Here, we do not

repeat the process.

4 Existence of essential connected components

problems such that all conditions of Theorem . hold. ClearlyM⊂M. For convenience in the later presentation, for any subsetAofX, denoteAc_{={x∈X:x∈/A}.}

Lemma .([]) Let C, D be two nonempty, convex and compact subsets of a linear normed space E.Then h(C,λC+μD)≤h(C,D),where h is the Hausdorff distance defined on E,andλ,μ≥,λ+μ= .

Lemma .([]) Let(Y,ρ)be a metric space,Kand Kbe two nonempty compact sub-sets of Y,Vand Vbe 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 . For any q∈M,every minimal essential subset of F(q)is connected.

Proof For each fixedq∈M, letm(q)⊂F(q) be a minimal essential subset ofF(q). Ifm(q) is not connected, then there are two nonempty compact subsetsc(q),c(q) and two dis-joint open subsetsV,VofX×Ysuch thatm(q) =c(q)∪c(q) andV⊃c(q),V⊃c(q). Sincem(q) is a minimal essential set ofF(q), neitherc(q) norc(q) is essential. There exist two open setsO⊃c(q),O⊃c(q) such that, for anyδ> , there existq,q∈Mwith

ρq,q<δ, ρq,q<δ, Fq∩O=∅, F

q∩O=∅.

DenoteW=V∩O,W=V∩O, we know thatW,Ware open,W⊃c(q),W⊃c(q) and we may assume thatV⊃W,V⊃W. Denote

G=X×Y×

P

W_cc, G=X×Y×

P

W_cc.

SincePis continuous with nonempty compact values, andWc,Wcare nonempty com-pact inX×Y, by Lemma .,P(Wc),P(Wc) are nonempty compact inZ. ThusG,G are open inX×Y×Z.

To prove by contraposition that (P(Wc))c∩(P(Wc))c=∅, suppose the existence of z∈Zsuch thatz∈(P(Wc))c∩(P(Wc))c, which implies thatz∈/P(Wc) andz∈/P(Wc), i.e.,Wc

 ∩P–(z) =∅,Wc∩P–(z) =∅. It follows thatP– (z)⊂WandP–(z)⊂W, which contradicts the fact thatW∩W=∅.

Denoteinf{d(a,b)|a∈G,b∈G}=ε> . Sincem(q) is essential, andm(q)⊂(W∪W), there exists  <δ∗<εsuch thatF(q)∩(W∪W)= ∅for anyq∈Mwithρ(q,q) <δ∗. Sincem(q) is a minimal essential set ofF(q), neitherc(q) norc(q) is essential. Thus, for

δ∗

> , there exist twoq

_,q_∈M

such that

Fq∩W=∅, F

q∩W=∅, ρ

q,q< δ

∗

, ρ

q,q< δ

∗

.

Thusρ(q_,q_{) <}δ∗

. Next, defineq= (S,T,P,R,Q) as follows:

S(x,y) =λ(x,y)S(x,y) +μ(x,y)S(x,y),

A=GrR\G

∪GrR\G

, B=GrQ\G

∪GrQ\G

,

R(u,y,z) holds iff (u,y,z)∈A, Q(v,x,z) holds iff (x,v,z)∈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 check that (i)S,Tare continuous with nonempty compact convex values. (ii) SinceGr(R_{) andGr(R}_{) are closed inX×Y×Z,Ais closed inX×Y ×Z, which} implies thatR(·,·,·) is closed. Similarly,Q(·,·,·) is closed. (iii) Suppose the existence of y∈Y, finite subset{u, . . . ,un} ⊂Xandx∈co{u, . . . ,un}such thatR(ui,y,P(x,y)) does not hold for anyi∈ {, . . . ,n},i.e., (ui,y,P(x,y))∩Gr(R) =∅,∀i∈ {, . . . ,n}. Since

GrR=GrR\G

∪GrR\G

andW∩W=∅, without loss of generality, we may assume that (x,y)∈Wc, which implies P(x,y)⊂P(Wc). Since (ui,y,P(x,y))∩[Gr(R)\G] =∅for alli∈ {, . . . ,n}, that is,

y,ui,P(x,y)

∩GrR∩X×Y×P

W_c=∅, ∀i∈ {, . . . ,n},

then (ui,y,P(x,y))∩Gr(R) =∅,∀i∈ {, . . . ,n},i.e.,R(ui,y,P(x,y)) does not hold for any i∈ {, . . . ,n}, which is a contradiction.

(iv) Suppose the existence ofx∈X, finite subset{v, . . . ,vn} ⊂Y,y∈co{v, . . . ,vn}and z∈P(x,y) such thatQ(vi,x,z) does not hold for anyi∈ {, . . . ,n},i.e., (x,vi,z) /∈Gr(Q),

∀i∈ {, . . . ,n}. Since

GrQ=GrQ\G

∪GrQ\G

andW∩W=∅, without loss of generality, we may assume that (x,y)∈Wc, which implies z∈P(x,y)⊂P(Wc). Since (x,vi,z) /∈[Gr(Q)\G] for alli∈ {, . . . ,n}, that is, (x,vi,z) /∈ [Gr(Q_)∩(X×Y×P

(Wc))] for alli∈ {, . . . ,n}, then

(x,vi,z) /∈GrQ, ∀i∈ {, . . . ,n},

that is, there isz∈P(x,y) such thatQ(vi,x,z) does not hold for anyi∈ {, . . . ,n}, which is a contradiction. Henceq∈M.

(v) By Lemmas ., .,

ρq,q= sup

(x,y)∈X×Y

hXS(x,y),S(x,y)+ sup

(x,y)∈X×Y

hYT(x,y),T(x,y)

+HGr(R),GrR+HGr(Q),GrQ

≤ sup

(x,y)∈X×Y

hXS(x,y),S(x,y)+ sup

(x,y)∈X×Y

+ sup

(x,y)∈X×Y

hYT(x,y),T(x,y)+ sup

(x,y)∈X×Y

hYT(x,y),T(x,y)

+HGr(R),GrR+HGrR,GrR

+HGr(Q),GrQ+HGrQ,GrQ

<δ∗.

Thusq∈Mandρ(q,q) <δ∗.

Since (F(q)∩W)∪(F(q)∩W) =F(q)∩(W∪W)=∅, without loss of generality, we assumeF(q)∩W=∅. Then there exists (x,y)∈F(q)∩Wsuch that (x,y)∈W,x∈ S(x,y),y∈T(x,y), and

∀u∈S(x,y), ∃z∈P(x,y) s.t. R(u,y,z) holds,

Q(v,x,z) holds, ∀v∈T(x,y),∀z∈P(x,y).

It follows from (x,y)∈W thatS(x,y) =S(x,y),T(x,y) =T(x,y),P(x,y)⊂P(W) and P(x,y)∩P(Wc) =∅. Therefore,

∀u∈S(x,y), ∃z∈P(x,y) s.t. R(u,y,z) holds,

Q(v,x,z) holds, ∀v∈T(x,y),∀z∈P(x,y).

Then (x,y)∈F(q)∩W, which is a contradiction. This completes the proof.

Theorem . For any q∈M, there exists at least one essential connected component of F(q).

Proof By Theorem ., there exists at least one connected minimal essential subsetm(q) ofF(q). Thus, there is a componentCofF(q) such thatm(q)⊂C. It is obvious thatCis essential by Remark .(). This completes the proof.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

This research is supported the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 13CG35), and open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (no. 201309KF02).

Received: 11 April 2014 Accepted: 15 August 2014 Published:02 Sep 2014

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