# Stable Analysis of Solution Set for System of Quasivariational Relations with Applications

Authors: Lefeng Shi; Zhe Yang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The essential stability of solutions for system of quasivariational relations is studied. We show that most of systems of quasivariational relations are essential (in the sense of Baire category) and that, for any system of quasivariational relations, there exists at least one essential component of its solution set. As applications, the existence of essential components of solution set for systems of KKM problems and systems of quasivariational inclusions is obtained.


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## Body

## 1. Introduction

It is well known that the equilibrium problem is a unified model of several problems, namely, optimization problem, saddle point problem, variational inequality problem, fixed point problem, Nash equilibrium problem, and so forth. Recently, several people focused on the study of equilibrium problems and their generalizations. Luc [1] 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 [2, 3]. Various types of sufficient conditions for the existence of solutions of variational relation problems have been investigated in many recent papers (see [4–12]). Furthermore, Agarwal et al. [13] presented a unified approach for studying the existence of solutions for two types of variational relation problems, and Balaj and Lin [14] established the existence criteria for the solutions of two very general types of variational relation problems.However,One could also argue that a sensible equilibrium should be stable against slight perturbations in the payoffs of the game (van Damme [15]). The notation of an essential solution for fixed points was firstly introduced by Fort [16], which means that, for a fixed point x of a mapping f, if each mapping sufficiently near to f has a fixed point arbitrarily near to x, x is said to be essential. The method of essential solution has been widely used in various fields recently. It plays a crucial role in the study of stability of solutions including optimal solutions, Nash equilibria, and fixed points (see [17–30]).Motivated and inspired by research works mentioned above, in this paper, we study the notions of essential stability of solutions for system of quasivariational relations. The results of this paper improve and generalize several known results on the stability of solution set for variational relation problems.


## 2. Definitions and Preliminaries

Lin and Ansari [8] introduced a system of quasivariational relations (SQVP) and established the existence of solutions of SQVP by means of maximal element theorem for a family of multivalued mappings. Let I be any index set. For each i∈I, let Xi be nonempty set, let X=∏i∈I‍Xi, Si,Qi:X⇉Xi be multivalued mappings with nonempty values, and let Ri(x,yi) be a relation linking x∈X and yi∈Xi. A system q=(I,Xi,Qi,Si,Ri) of quasivariational relations consists in finding x*∈X such that, for each i∈I, xi*∈Si(x*) and Ri(x*,yi) holds for any yi∈Qi(x*).Definition 1.
LetI be a finite set, and z0∈X. Let M be the set of all systems of quasivariational relations q=(I,Xi,Qi,Si,Ri) such that the following hold: (i) for each i∈I, Xi is nonempty convex compact subset of a normed linear space; (ii) for each i∈I, Si is upper semicontinuous with nonempty convex compact values; (iii) for each i∈I, Qi-1(yi) is open in X for any yi∈Xi; (iv) zi0∈Qi(x)⊂Si(x) for any x∈X and any i∈I; (v) for each i∈I, any finite set {xi1,…,xin}⊂Xi, and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that Ri(x,xij) holds, where x=(xi)i∈I; (vi) Ri(·,yi) is closed for any yi∈Xi.Clearly, for anyq∈M, we have that (1) {x∈X:xi∈Si(x)} is closed; (2) Qi(x)≠∅ for all x∈X; (3) since Si(x) is convex and Qi(x)⊂Si(x), then co⁡Qi(x)⊂Si(x). By Theorem 3.1 of [8], the problem q has at least one solution. Here, for any q∈M and each i∈I, define two closed-valued mappings Ai,Bi:Xi⇉X by Ai(yi)={x∈X:yi∉Qi(x)} and Bi(yi)={x∈X:Ri(x,yi)holds}. Denote by F(q) the solution set of q. Thus, a correspondence F:M⇉X is well defined. For each q,q′∈M, define the distance on M by
(1)ρ(q,q′)=sup⁡i∈Isup⁡x∈Xhi(Si(x),Si′(x))+sup⁡i∈Isup⁡yi∈Xih(Ai(yi),Ai′(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi(yi),Bi′(yi)),
where hi is the Hausdorff distance on Xi and h is the Hausdorff distance on X. Clearly, (M,ρ) is a metric space.In what follows, the notions of essential solutions, essential problems are introduced (see [25]). Let (X,d), (Y,ρ) be two metric spaces, and let h be the Hausdorff distance defined on Y. A set-valued mapping F:X⇉Y is said to be (1) upper semicontinuous at x∈X if, for any open subset O of Y with O⊃F(x), there exists an open neighborhood U(x) of x such that O⊃F(x′) for any x′∈U(x); (2) upper semicontinuous on X if F is upper semicontinuous on each x∈X; (3) an usco mapping if F is upper semicontinuous on X and F(x) is compact for each x∈X; (4) lower semicontinuous at x∈X if, for any open subset O of Y with O∩F(x)≠∅, there exists an open neighborhood U(x) of x such that O∩F(x′)≠∅ for any x′∈U(x); (5) lower semicontinuous on X if F is lower semicontinuous on each x∈X; (6) closed if Graph⁡(F)={(x,y)∈X×Y∣y∈F(x)} is closed. Let x∈X. A point y∈F(x) is said to be an essential point of F(x) if, for any open neighborhood N(y) of y in Y, there is a δ>0 such that N(y)∩F(x′)≠∅ for any x′∈X with d(x,x′)<δ. If all y∈F(x) are essential, then x is said to be essential. A nonempty closed subset e(x) of F(x) is said to be an essential set of F(x) if, for any open set U, e(x)⊂U, there is a δ>0 such that U∩F(x′)≠∅ for any x′∈X with d(x,x′)<δ. An essential subset m(x)⊂F(x) is said to be a minimal essential set of F(x) if it is a minimal element of the family of essential sets ordered by set inclusion. A component C(x) is called an essential component of F(x) if C(x) is essential.Remark 2 (see [25]).
(1) It is easy to see that the problemx∈X is essential if and only if the mapping F:X⇉Y is lower semicontinuous at x. (2) Let two nonempty closed sets be e1, e2 of F(x), if e1⊂e2, and e1 is essential, so is e2.Lemma 3 (see [31]).
IfX is a complete metric space and F:X⇉Y is usco, then the set of points, where F is lower semicontinuous, is a dense residual set in X.Lemma 4 (see [22]).
LetC,D be two nonempty, convex, and compact subsets of linear normed space Y. Then h(C,λC+μD)≤h(C,D), where λ,μ≥0, λ+μ=1.Lemma 5 (see [24]).
Let(Y,ρ) be a metric space, let K1 and K2 be two nonempty compact subsets of Y, and let V1 and V2 be two nonempty disjoint open subsets of Y. If h(K1,K2)<ρ(V1,V2), then
(2)h(K1,(K1∖V2)∪(K2∖V1))≤h(K1,K2).Lemma 6 (see [32]).
LetX and Y be two Hausdorff topological spaces with Y compact. If F is a closed set-valued mapping from X to Y, then F is upper semicontinuous.Lemma 7 (see [23]).
LetX, Y, Z be three metric spaces, and let S1:Y⇉X and S2:Z⇉X be two set-valued mappings. Suppose that there exists at least one essential component of S1(y) for each y∈Y and there exists a continuous single-valued mapping T:Z→Y such that S2(z)⊃S1(T(z)) for each z∈Z. Then, there exists at least one essential component of S2(z) for each z∈Z.


## 3. Main Results

Theorem 8.
(
M
,
ρ
) is a complete metric space.Proof.
Let{qn}n=1∞ be any Cauchy sequence in M, and then, for any ε>0, there is N>0 such that ρ(qn,qm)<ε for any n,m>N; that is,
(3)sup⁡i∈Isup⁡x∈Xhi(Si(x),Si′(x))+sup⁡i∈Isup⁡yi∈Xih(Ai(yi),Ai′(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi(yi),Bi′(yi))≤ε,
for any n,m>N. (1)
Easily, for eachi∈I, there exists a set-valued mapping Si:X⇉Xi such that Sin(x)→Si(x) for any x∈X and Si is upper semicontinuous with nonempty compact convex values.(2)
For eachi∈I, there exist two closed-valued mappings Ai,Bi:Xi⇉X such that, for any yi∈Xi,
(4)Ain(yi)⟶Ai(yi),Bin(yi)⟶Bi(yi).Further, for eachi∈I, define the mapping Qi:X⇉Xi and relation Ri by the following:
(5)Qi(x)={yi∈Xi∣x∉Ai(yi)},Ri(x,yi)holdsifandonlyifx∈Bi(yi).We need to prove thatq=(Si,Qi,Ri)i∈I∈M.(3)
Easily, for eachi∈I and any yi∈Xi, Qi-1(yi)=X∖Ai(yi) is open in X. And, since Bi(yi) is closed, it follows that Ri(·,yi) is closed.(4)
If there existi0∈I and x∈X such that zi00∉Qi0(x), then x∈Ai0(zi00). It follows from Ai0n(zi00)→Ai0(zi00) that there exists a sequence {xn} of X such that xn∈Ai0n(zi00) and xn→x, which implies that zi00∉Qi0n(xn). It is a contradiction. Thus zi0∈Qi(x) for any x∈X and any i∈I.(5)
Suppose the existence ofi∈I and x∈X such that Qi(x)⊄Si(x), and then there exist yi∈Qi(x) such that yi∉Si(x). From yi∈Qi(x), that is, x∉Ai(yi), it follows that x∉Ain(yi) for enough large n; that is, yi∈Qin(x). Moreover, since yi∉Si(x), then yi∉Sin(x) for enough large n. Thus, for enough large n, yi∈Qin(x) and yi∉Sin(x), which implies that Qin(x)⊄Sin(x). It is a contradiction. Thus Qi(x)⊂Si(x) for any i∈I and any x∈X.(6)
Suppose that there existi0∈I, finite set {xi1,…,xim}, for all i∈I, and x=(xi)i∈I∈X with xi∈co⁡{xi1,…,xim}, for all i∈I such that Ri0(x,xi0j) does not hold for any j∈{1,…,m}, and then x∉Bi0(xi0j) for any j∈{1,…,m}. Since Bin(yi)→Bi(yi) for any yi∈Xi and any i∈I, then, for enough large n, x∉Bi0n(xi0j) for any j∈{1,…,m}, which implies that Ri0n(x,xi0j) does not hold for any j∈{1,…,m}. It is a contradiction. Thus, for each i∈I, any finite set {xi1,…,xin}⊂Xi, and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that Ri(x,xij) holds, where x=(xi)i∈I. Hence q=(Si,Qi,Ri)i∈I∈M. This completes the proof.Theorem 9.
The mappingF:M⇉X is upper semicontinuous with compact values.Proof.
By Lemma6, we only prove that Graph⁡(F) is closed. For any {qn∈M}n=1∞ with qn→q and any xn∈F(qn) with xn→x, we will prove x∈F(q).
It follows fromxn∈F(qn) for any n that, for each i∈I, xin∈Sin(xn) and Rin(xn,yi) holds for any yi∈Qi(xn). Since qn→q and xn→x, then xi∈Si(x) for any i∈I. Next, if there exist i0∈I and yi0∈Qi0(x) such that Ri0(x,yi0) does not hold, then x∉Ai0(yi0) and x∉Bi0(yi0). Then, for enough large n, xn∉Ai0n(yi0) and xn∉Bi0n(yi0), which implies that yi0∈Qi0n(xn) and Ri0n(xn,yi0) does not hold for enough large n. It is a contradiction. Thus, for each i∈I, xi∈Si(x) and Ri(x,yi) holds for any yi∈Qi(x); that is, x∈F(q). This completes the proof.Theorem 10.
There exists a dense residual subsetG of M such that, for each q∈G, q is essential.Proof.
Since(M,ρ) is complete by Theorem 8, and the mapping F:M⇉X is upper semicontinuous with nonempty compact values by Theorem 9, by Lemma 3, there is a dense residual subset G of M, where F is lower semicontinuous; thus, q is essential for each q∈G by Remark 2(1).Theorem 11.
For eachq∈M, there exists at least one minimal essential subset of F(q).Proof.
By Theorem9, F:M⇉X is upper semicontinuous with nonempty compact values; that is, for each open set O⊃F(q), there exists δ>0 such that, for any q′∈M with ρ(q,q′)<δ, O⊃F(q′). Hence F(q) is an essential set of itself. Let Θ denote the family of all essential sets of F(q) ordered by set inclusion. Then Θ is nonempty and every decreasing chain of elements in Θ has a lower bound (because by the compactness the intersection is in Θ); therefore, by Zorn’s lemma, Θ has a minimal element and this minimal element is a minimal essential set of F(q).Theorem 12.
For eachq∈M, there exists at least one connected minimal essential subset of F(q).Proof.
For eachq∈M, let m(q)⊂F(q) be a minimal essential subset of F(q). Suppose that m(q) was not connected, then there exist two nonempty compact subsets c1(q), c2(q) with m(q)=c1(q)∪c2(q), and there exist two disjoint open subsets V1, V2 in X such that V1⊃c1(q) and V2⊃c2(q). Since m(q) is a minimal essential set of F(q), neither c1(q) nor c2(q) is essential. There exist two open sets O1⊃c1(q) and O2⊃c2(q) such that, for any δ>0, there exist q1,q2∈M with
(6)ρ(q,q1)<δ,ρ(q,q2)<δ,F(q1)∩O1=∅,F(q2)∩O2=∅.
Now, we choose two open sets W1, W2 such that
(7)c1(q)⊂W1⊂W¯1⊂O1∩V1,c2(q)⊂W2⊂W¯2⊂O2∩V2,
and inf⁡⁡{d(a,b)∣a∈W1,b∈W2}=ε>0.
Sincem(q) is essential, then, for m(q)⊂(W1∪W2), there exists 0<δ*<ε such that F(q′)∩(W1∪W2)≠∅ for any q′∈M with ρ(q,q′)<δ*. Since m(q) is a minimal essential set of F(q), then neither c1(q) nor c2(q) is essential. Thus, for δ*/16>0, there exist two q1,q2∈M such that
(8)F(q1)∩W1=∅,F(q2)∩W2=∅,ρ(q1,q)<δ*16,ρ(q2,q)<δ*16.
Thus ρ(q1,q2)<δ*/8.
Now, define the following system of quasivariational relationsq′=(Si′,Qi′,Ri′)i∈I by
(9)Si′(x)=λ(x)Si1(x)+μ(x)Si2(x),∀x∈X,∀i∈I,Ai′(yi)=(Ai1(yi)∖W2)∪(Ai2(yi)∖W1),∀yi∈Xi,∀i∈I,Bi′(yi)=(Bi1(yi)∖W2)∪(Bi2(yi)∖W1),∀yi∈Xi,∀i∈I,Qi′(x)={yi∈Xi:x∉Ai(yi)},∀x∈X,∀i∈I,Ri′(x,yi)holdsifandonlyifx∈Bi′(yi),∀i∈I,
where
(10)λ(x)=d(x,W¯2)d(x,W¯1)+d(x,W¯2),μ(x)=d(x,W¯1)d(x,W¯1)+d(x,W¯2),∀x∈X,Ai1(yi)={x∈X:yi∉Qi1(x)},Ai2(yi)={x∈X:yi∉Qi2(x)},Bi1(yi)={x∈X:Ri1(x,yi)holds},Bi2(yi)={x∈X:Ri2(x,yi)holds}.
Easily, we can check the following.(i)
Si′ is upper semicontinuous with nonempty compact convex values for each i∈I.(ii)
Qi′-1(yi)=X∖Ai1(yi) is open for any yi∈Xi and any i∈I.(iii)
For anyi∈I and any yi∈Xi, since Bi1(yi) is closed, then Ri(·,yi) is closed.(iv)
For eachi∈I, since zi0∈Qi1(x) and zi0∈Qi2(x) for any x∈X, then x∉Ai1(zi0) and x∉Ai2(zi0) for any x∈X, which implies that x∉Ai1(zi0) for any x∈X; that is, zi0∈Qi′(x) for any x∈X.(v)
Next, we showQi′(x)⊂Si′(x) for any x∈X and any i∈I. For any fixed i∈I, we have three cases.
Ifx∈W1, then Qi′(x)=Qi1(x) and Si′(x)=Si1(x). It follows from q1∈M that
(11)Qi′(x)=Qi1(x)⊂Si1(x)=Si′(x).Ifx∈W2, then Qi′(x)=Qi2(x) and Si′(x)=Si2(x). It follows from q2∈M that
(12)Qi′(x)=Qi2(x)⊂Si2(x)=Si′(x).Ifx∈X∖(W1∪W2), then Qi′(x)=Qi1(x)∩Qi2(x) and Si′(x)=λ(x)Si1(x)+μ(x)Si2(x). For any zi∈Qi′(x)=Qi1(x)∩Qi2(x), we have zi∈Qi1(x)⊂Si1(x) and zi∈Qi2(x)⊂Si2(x). Then
(13)zi=λ(x)zi+μ(x)zi∈λ(x)Si1(x)+μ(x)Si2(x)=Si′(x).ThusQi′(x)⊂Si′(x).(vi)
Suppose that there existi0∈I, finite set {xi1,…,xim}, for all i∈I, and x=(xi)i∈I∈X with xi∈co⁡{xi1,…,xim}, for all i∈I such that Ri0′(x,xi0j) does not hold for any j∈{1,…,m}, and then x∉Bi0′(xi0j) for any j∈{1,…,m}. Since W1∩W2=∅, without loss of generality, we assume x∉W1. For any j∈{1,…,m}, since x∉Bi0′(xi0j)=(Bi01(xi0j)∖W2)∪(Bi02(xi0j)∖W1), then x∉Bi02(xi0j)∖W1, for all j∈{1,…,m}. It follows from x∉W1 that x∉Bi02(xi0j), ∀j∈{1,…,m}; that is, Ri02(x,xi0j) does not hold for any j∈{1,…,m}. It is a contradiction.(vii)
By Lemmas4 and 5, we have
(14)ρ(q′,q)=sup⁡i∈Isup⁡x∈Xhi(Si(x),Si′(x))+sup⁡i∈Isup⁡yi∈Xih(Ai(yi),Ai′(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi(yi),Bi′(yi))≤sup⁡i∈Isup⁡x∈Xhi(Si(x),Si1(x))+sup⁡i∈Isup⁡x∈Xhi(Si1(x),Si′(x))+sup⁡i∈Isup⁡yi∈Xih(Ai(yi),Ai1(yi))+sup⁡i∈Isup⁡yi∈Xih(Ai1(yi),Ai′(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi(yi),Bi1(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi1(yi),Bi′(yi))≤sup⁡i∈Isup⁡x∈Xhi(Si(x),Si1(x))+sup⁡i∈Isup⁡x∈Xhi(Si1(x),Si2(x))+sup⁡i∈Isup⁡yi∈Xih(Ai(yi),Ai1(yi))+sup⁡i∈Isup⁡yi∈Xih(Ai1(yi),Ai2(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi(yi),Bi1(yi))+sup⁡i∈Isup⁡yi∈Xih(Bi1(yi),Bi2(yi))≤3(116+216)δ*<δ*.Thusq′∈M and ρ(q′,q)<δ*.
Since(F(q′)∩W1)∪(F(q′)∩W2)=F(q′)∩(W1∪W2)≠∅ and W1∩W2=∅, then we assume F(q′)∩W1≠∅ without loss of generality. Then there exists x¯∈F(q′)∩W1 such that, for each i∈I, x¯∈W1, x¯i∈Si′(x¯), and Ri′(x¯,yi) holds for any yi∈Qi′(x¯). When x¯∈W1, we have that, for each i∈I,
(15)Si′(x¯)=λ(x¯)Si1(x¯)+μ(x¯)Si2(x¯)=Si1(x¯),x¯∈Bi′(yi)=(Bi1(yi)∖W2)∪(Bi2(yi)∖W1)⟹x¯∈Bi1(yi),Qi′(x¯)=Qi1(x¯);
that is, for each i∈I, x¯i∈Si1(x¯) and Ri1(x¯,yi) holds for any yi∈Qi1(x¯), which implies that x¯∈F(q1). Hence F(q1)∩W1≠∅, which contradicts F(q1)∩W1=∅. Thus m(q) is connected.Theorem 13.
For eachq∈M, there exists at least one essential component of F(q).Proof.
By Theorem12, there exists at least one connected minimal essential subset m(q) of F(q). Thus, there is a component C of F(q) such that m(q)⊂C. It is obvious that C is essential by Remark 2(2). Thus C is an essential component.Remark 14.
IfI is a singleton, and Si(x)=Qi(x)=Xi for any x∈X and any i∈I, the system of quasivariational relations coincides with the variational relation problem considered in [3]. Thus, the results of [3] are obtained as a special case of this paper’s results. There, M′ is the set of variational relations R(x,y) linking x,y∈X such that (i) X is nonempty convex compact subset of a normed linear space; (ii) for any finite set {x1,…,xn}⊂X, and any x∈co⁡{x1,…,xn}, there is j∈{1,…,n} such that R(x,xj) holds; (iii) R(·,y) is closed for any y∈X. Then we have the following results, which are Lemma 3.2, Theorems 3.1–3.4 in [3].Theorem 15.
The mappingF:M′⇉X is upper semicontinuous with compact values.Theorem 16.
There exists a dense residual subsetG′ of M′ such that, for each q∈G′, q is essential.Theorem 17.
For eachq∈M′, there exists at least one minimal essential subset of F(q).Theorem 18.
For eachq∈M′, there exists at least one minimal essential subset of F(q).Theorem 19.
For eachq∈M′, there exists at least one essential component of F(q).


## 4. Applications (I): Systems of KKM Problems

LetI be any index set. For each i∈I, let Xi be a nonempty convex compact subset of topological vector space Ei and X=∏i∈I‍Xi, and let Si,Qi:X⇉Xi and Gi:Xi⇉X be multivalued mappings with nonempty values. A system of KKM problems consists in finding x*=(xi*)i∈I∈X such that, for each i∈I, xi*∈Si(x*) and x*∈⋂yi∈Qi(x*)‍Gi(yi).Definition 20.
LetI be a finite set, and z0∈X. Let M1 be the set of systems of KKM problems η=(I,Xi,Si,Qi,Gi) such that, for each i∈I, (1) Xi is a nonempty convex compact subset of a normed linear space Ei; (2) zi0∈Qi(x)⊂Si(x) for any x∈X and any i∈I, and Qi-1(yi) is open in X for all yi∈Xi; (3) Si is upper semicontinuous with nonempty compact convex values; (4) the multivalued mapping Gi:Xi⇉X holds nonempty closed values; (5) for any finite set {xi1,…,xin}⊂Xi, and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that x∈Gi(xij) holds, where x=(xi)i∈I.For eachη∈M1, denote by F1(η) the solution set of η and qη∈M by qη=(Si,Qi,Riη)i∈I, where
(16)Riη(x,yi)holdsifandonlyifx∈Gi(yi),∀i∈I.
Easily F1(η) is nonempty. Furthermore, for any η,η′∈M1, we define the distance on M1 by ρ1(η,η′)=ρ(qη,qη′). Thus there exists an isometric mapping T1:M1→M such that T1(η)=qη and F1(η)=F(qη)=F(T1(η)).Theorem 21.
For eachη∈M1, there exists at least one essential component of F1(η).Proof.
SinceT1:M1→M is an isometric mapping such that T1(η)=qη, it is continuous. By Theorem 13, there exists at least one essential component of F(q) for each q∈M, and by Lemma 7, there exists at least one essential component of F1(η) for each η∈M1.


## 5. Applications (II): Systems of Variational Inclusions

LetI be any index set. For each i∈I, let Xi be a nonempty subset of Hausdorff topological space, let Zi be a Hausdorff topological vector space, and let Si,Qi:X⇉Xi, Ai,Bi:X×Xi⇉Zi be multivalued mappings with nonempty values.A system of quasivariational inclusions of type (I) consists in findingx*∈X such that, for each i∈I, xi*∈Si(x*) and Ai(x*,yi)⊂Bi(x*,yi) for all yi∈Qi(x*).A system of quasivariational inclusions of type (II) consists in findingx*∈X such that, for each i∈I, xi*∈Si(x*) and Ai(x*,yi)∩Bi(x*,yi)≠∅ for all yi∈Qi(x*).A system of quasivariational inclusions of type (III) consists in findingx*∈X such that, for each i∈I, xi*∈Si(x*) and 0∈Ai(x*,yi) for all yi∈Qi(x*).Definition 22.
LetI be a finite set, and z0∈X. Let M21 be the set of systems of quasivariational inclusions of type (I) η=(I,Xi,Si,Qi,Ai,Bi) such that, for each i∈I, (1) Xi is a nonempty convex compact subset of a normed linear space Ei; (2) zi0∈Qi(x)⊂Si(x) for any x∈X and any i∈I, and Qi-1(yi) is open in X for all yi∈Xi; (3) Si is upper semicontinuous with nonempty compact convex values; (4) for any yi∈Xi, Ai(·,yi) is lower semicontinuous, and Bi(·,yi) is closed; (5) for any {xi1,…,xin} of Xi and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that Ai(x,xij)⊂Bi(x,xij), where x=(xi)i∈I.
For eachη∈M21, denote by F21(η) the solution set of η. For any η∈M21, define qη by qη=(Si,Qi,Riη)i∈I, where
(17)Riη(x,yi)holdsifandonlyifAi(x,yi)⊂Bi(x,yi),∀i∈I.
By Theorem 3.5 of [8], qη∈M. Easily, F21(η) is nonempty. Further, for any η,η′∈M21, we define the distance on M21 by ρ21(η,η′)=ρ(qη,qη′). Thus there exists an isometric mapping T21:M21→M such that T21(η)=qη and F21(η)=F(qη)=F(T21(η)).Theorem 23.
For eachη∈M21, there exists at least one essential component of F21(η).Proof.
SinceT21:M21→M is an isometric mapping such that T21(η)=qη, it is continuous. By Theorem 13, there exists at least one essential component of F(q) for each q∈M, and by Lemma 7, there exists at least one essential component of F21(η) for each η∈M21.It is similar to Definition22 and Theorem 23, and the following results are obtained. Here, we do not repeat the process.Definition 24.
LetI be a finite set, and z0∈X. Let M22 be the set of systems of quasivariational inclusions of type (II) η=(I,Xi,Si,Qi,Ai,Bi) such that, for each i∈I, (1) Xi is a nonempty convex compact subset of a normed linear space Ei; (2) zi0∈Qi(x)⊂Si(x) for any x∈X and any i∈I, and Qi-1(yi) is open in X for all yi∈Xi; (3) Si is upper semicontinuous with nonempty compact convex values; (4) for any yi∈Xi, Ai(·,yi) is upper semicontinuous with nonempty compact values and Bi(·,yi) is closed; (5) for any {xi1,…,xin} of Xi and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that Ai(x,xij)∩Bi(x,xij)≠∅, where x=(xi)i∈I.Definition 25.
LetI be a finite set, and z0∈X. Let M23 be the set of systems of quasivariational inclusions of type (III) η=(I,Xi,Si,Qi,Ai) such that, for each i∈I, (1) Xi is a nonempty convex compact subset of a normed linear space Ei; (2) zi0∈Qi(x)⊂Si(x) for any x∈X and any i∈I, and Qi-1(yi) is open in X for all yi∈Xi; (3) Si is upper semicontinuous with nonempty compact convex values; (4) for any yi∈Xi, Ai(·,yi) is closed; (5) for any {xi1,…,xin} of Xi and any xi∈co⁡{xi1,…,xin}, there is j∈{1,…,n} such that 0∈Ai(x,xij), where x=(xi)i∈I.For eachj=2,3 and each η∈M2j, denote the solution set of η by F2j(η).Theorem 26.
Forj=2,3 and each η∈M2j, there exists at least one essential component of F2j(η).


## 6. Conclusions

In this paper, we study the notions of essential stability of solutions for system of quasivariational relations. We show that most of systems of quasivariational relations (in the sense of Baire category) are essential and that, for any system of quasivariational relations, there exists at least one essential component of its solution set. As applications, the existence of essential components of solution set for systems of KKM problems and systems of quasivariational inclusions is obtained.



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