Skip to main content

Essential components of the set of solutions for the system of vector quasi-equilibrium problems

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 nonlinear problems. Fort [1] 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. Kinoshita [2] then introduced the notion of essential components of the set of fixed points of a single-valued map. Jiang [3] introduced the notion of essential components of the set of Nash equilibrium 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 [4] 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 essential component of Nash equilibrium points. Recently, Yu, Xiang [5], Yu, Luo [6], Isac, Yuan [7], Yang, Yu [8], Lin [9], Chen, Gong [10] introduced the notion of essential components to solution sets of various problems such as Ky Fan point problems, equilibrium problems, coincident point problems, vector optimization problem, and symmetric vector 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 [11] studied the system of generalized vector equilibrium problems. Ansari, Chan and Yang [12] studied the system of vector quasi-equilibrium problems (briefly, SVQEP). Fang, Huang and Kim [13] studied the system of vector equilibrium problems. Peng, Lee, Yang [14] studied the system of generalized vector quasi-equilibrium problems with set-valued maps (briefly, SGVOEPS). Lin [15] studied the system of generalized vector quasi-equilibrium problems (briefly, SGVQEP) in Banach spaces. Peng, Yang and Zhu [16] studied the system of vector quasi-equilibrium. Lin [9] established essential components of the solution set for SGVQEP under perturbations 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

Let I={1,2,,n} be a finite set which has at least two elements. For each iI, let X i and Y i be real Hausdorff topological vector spaces and K i a nonempty subset of X i . For each iI, let C i be a closed, convex and pointed cone of Y i with int C i , where int C i denotes the interior of C i . Let K= i = 1 n K i . For each iI, let f i :K× K i Y i be a vector-valued mapping and S i :K 2 K i be a set-valued mapping. The SVQEP consists of finding x ¯ K such that for each iI,

x i ¯ S i ( x ¯ )and f i ( x ¯ , y i )int C i for all  y i S i ( x ¯ ),

where x i ¯ denotes the i th component of x ¯ , and x ¯ is said to be a solution of the SVQEP. For each iI, f i is said to be an objective function of the SVQEP and for each iI, S i is said to be a constraint mapping of the SVQEP. The SVQEP includes, as a special case, the following multiobjective generalized game problem:

For each iI, let g i :K Y i be a vector-valued mapping and G i : K i ˆ 2 K i be a feasible strategy mapping, where K i ˆ = j I , j i K j . For each xK, we can write x=( x i , x i ˆ ). The multiobjective generalized game problem consists of finding x ¯ K such that for each iI, x i ¯ G i ( x ¯ i ˆ ) and

g i ( y i , x ¯ i ˆ ) g i ( x i ¯ , x ¯ i ˆ )int C i for all  y i G i ( x ¯ i ˆ ),

where x ¯ is said to be a weakly Pareto-Nash equilibrium point.

For each iI, setting

f i (x, y i )= g i ( y i , x i ˆ ) g i ( x i , x i ˆ )and S i (x)= G i ( x i ˆ ),

the SVQEP coincides with the multiobjective generalized game problem, which has been studied by Yu and Luo [6] but for real functions and Lin [9] but for Y i = R k i (1 k i n) for any iI.

For each iI, setting G i ( x i ˆ )= K i , the multiobjective generalized game problem coincides with the multiobjective game problem, which has been studied by Yu and Xiang [5] and Yang and Yu [8].

Definition 2.1 Let X be a real Hausdorff topological space and Y a real Hausdorff topological vector space with a convex cone C. Let f:XY be a vector-valued function.

(i) f is said to be C-continuous at x 0 X if, for any open neighborhood V of the zero element θ in Y, there is an open neighborhood N( x 0 ) of x 0 in X such that

f(x)f( x 0 )+V+Cfor all xN( x 0 ).

f is said to be C-continuous on X if it is C-continuous at every element of X.

(ii) f is said to be (−C)-continuous at x 0 X if, for any open neighborhood V of θ in Y, there exists an open neighborhood N( x 0 ) of x 0 in X such that

f(x)f( x 0 )+VCfor all xN( x 0 ).

f is said to be (−C)-continuous on X if it is (−C)-continuous at every point of X.

Definition 2.2 Let K be a nonempty convex subset of a vector space X, let Y be a vector space with a convex pointed cone C. Let f:KY be a mapping. f is said to be C-convex if, for any x,yK and t[0,1],

tf(x)+(1t)f(y)f ( t x + ( 1 t ) y ) C.

Definition 2.3 Let X and Y be two Hausdorff topological spaces, let F:X 2 Y be a set-valued mapping. F is said to be upper semicontinuous (in short, u.s.c.) at x 0 X if, for any neighborhood N(F( x 0 )) of F( x 0 ), there exists a neighborhood N( x 0 ) of x 0 such that

F(x)N ( F ( x 0 ) ) for all xN( x 0 ).

F is said to be upper semicontinuous on X if F is u.s.c. at every point xX.

F is said to be lower semicontinuous (in short, l.s.c.) at x 0 X if, for any y 0 F( x 0 ) and any neighborhood N( y 0 ) of y 0 , there exists a neighborhood N( x 0 ) of x 0 such that

F(x)N( y 0 )for all xN( x 0 ).

F is said to be lower semicontinuous on X if it is lower semicontinuous at every xX.

F is said to be continuous on X if it is both u.s.c. and l.s.c. on X.

F is said to be a closed mapping if GraphF={(x,y)X×Y:yF(x)} is a closed set in X×Y.

F is an usco mapping if F is u.s.c. on X and F(x) is compact for every xX.

Let (X,d) be a linear metric space. Denote by CK(X) all nonempty convex compact subsets of X. Define the Hausdorff metric h on CK(X) as follows.

For any S 1 , S 2 CK(X), let

h( S 1 , S 2 )=max { h ( S 1 , S 2 ) , h ( S 2 , S 1 ) } ,

where

h ( S 1 , S 2 )=sup { d ( b , S 2 ) : b S 1 }

and

d(b, S 2 )=inf { d ( b , s ) : s S 2 } .

Theorem 2.1[17]

Let Y be a real Hausdorff topological vector space, andCYbe a closed convex pointed cone withintC. Let K be a nonempty compact convex subset of a real locally convex Hausdorff topological vector space X. Let the set-valued mappingS:K 2 K be continuous with nonempty compact convex values. Ifψ:K×KYsatisfies the following conditions:

(i) ψ(,)is (−C)-continuous;

(ii) for any fixedxK, ψ(x,)is C-convex;

(iii) for anyxK, ψ(x,x)intC.

Then, there exists an element x Ksuch that x S( x )and

ψ ( x , y ) intCfor allyS ( x ) .

3 Essential components of the solution set for the system of vector quasi-equilibrium problems

Throughout this section, let I={1,2,,n} be a finite set which has at least two elements. For each iI, let X i be a real normed linear space and Y i a Banach space with Y i Y n ; let K i be a nonempty compact convex subset of X i , and let C i be a closed convex pointed cone of Y i with C i = C n Y i and int C i . Let K= i = 1 n K i and X= i = 1 n X i .

Let Φ be the collection of all vector-valued functions such that ψ:K×K Y n such that: (i) ψ(x,y) is ( C n )-continuous on K×K; (ii) for each fixed xK, ψ(x,) is C n -convex; (iii) for any xK, ψ(x,x)=θ, where θ is the zero element of Y n ; (iv) sup ( x , y ) K × K ψ(x,y)<+.

Let M be the collection of all set-valued mappings S:K 2 K such that: (i) for each xK, S(x) is convex and closed; (ii) S is continuous on K.

Let H=Φ×M. For any u 1 =( ψ , S ), u 2 =( ψ , S )H, define

ρ 1 ( u 1 , u 2 )= sup ( x , y ) K × K ψ ( x , y ) ψ ( x , y ) + sup x K h ( S ( x ) S ( x ) ) ,

where is the norm on Y n and h is the Hausdorff metric defined on CK(X). Clearly, (H, ρ 1 ) is a metric space.

For any u=(ψ,S)H, by Theorem 2.1, there exists a solution x K to the vector quasi-equilibrium problem: x S( x ) and

ψ ( x , y ) int C n for all yS ( x ) .

For each u=(ψ,S)H, define

F(u)= { x K : x S ( x )  and  ψ ( x , y ) int C n  for all  y S ( x ) } .
(1)

Thus F(u) for any uH and uF(u) indeed defines a set-valued mapping from H to K.

The following lemma can be found in [18].

Lemma 3.1 Let X be a metric space andK(X)be the family of all nonempty compact subsets of X. LetA, A n K(X) (n=1,2,) satisfy the condition that for each open set O containing A, there exists an integer N such that whenevern>N, we have A n O. Then for any sequence{ x n }with x n A n (n=1,2,), there exists a subsequence which converges to a point in A.

Lemma 3.2F:H 2 K is an usco mapping.

Proof Since K is compact, by [19], it suffices to prove that Graph(F)={(u,x)H×K:xF(u)} is closed. Let {( u n , x n )}Graph(F) with ( u n , x n )(u, x ¯ )H×K, where u n =( ψ n , S n ) and u=(ψ,S). Since x n F( u n ), we have

x n S n ( x n )and ψ n ( x n ,y)int C n for all y S n ( x n ).

For any open neighborhood O of S( x ¯ ) in K, since S( x ¯ ) is compact, by [[19], p.108], there is ε 0 >0 such that

{ x K : d ( x , S ( x ¯ ) ) < ε 0 } O,

where d(x,S( x ¯ ))= inf a S ( x ¯ ) xa. Since ρ 1 (( ψ n , S n ),(ψ,S))0, x n x ¯ , and S is u.s.c. at x ¯ , there is N such that for any n>N, we have

sup x K h ( S n ( x ) , S ( x ) ) < ε 0 /2,

and

S( x n ) { x K : d ( x , S ( x ¯ ) ) < ε 0 / 2 } .

So whenever n>N, we have

S n ( x n ) { x K : d ( x , S ( x n ) ) < ε 0 / 2 } { x K : d ( x , S ( x ¯ ) ) < ε 0 } O.

Since x n belongs to S n ( x n ), and S( x ¯ ) and S n ( x n ) are compact, by Lemma 3.1, there exists a subsequence { x n k } of { x n } such that x n k x 0 S( x ¯ ). Since x n k x ¯ , we have

x ¯ = x 0 S( x ¯ ).
(2)

Since S is l.s.c. at x ¯ K, for any zS( x ¯ ), by [19], there exists a n S( x n ) such that a n z. Since ρ 1 (( ψ n , S n ),(ψ,S))0, there exists a subsequence { S n k } of { S n } such that

sup x K h ( S n k ( x ) , S ( x ) ) <1/k.

Thus, there exists a subsequence { x n k } of { x n } such that

h ( S n k ( x n k ) , S ( x n k ) ) <1/k,

which implies that there exists a n k S n k ( x n k ) such that

a n k a n k <1/k,

where { a n k } is a subsequence of { a n }. Since a n k z a n k a n k + a n k z<1/k+ a n k z and a n k z (k+), we have that a n k z (k+). As a n k S n k ( x n k ), we have

ψ n k ( x n k , a n k ) int C n for all k.
(3)

Now we need to show that

ψ( x ¯ ,z)int C n .
(4)

If the conclusion is false, then ψ( x ¯ ,z)int C n , which implies that there is ε ¯ >0 such that

ψ( x ¯ ,z)+ ε ¯ Bint C n ,
(5)

where B denotes the open unit ball in Y n . Since ψ is ( C n )-continuous on K×K, x n k x ¯ and a n k z, for above ε ¯ >0, there is a positive integer k 0 such that

ψ ( x n k , a n k ) ψ( x ¯ ,z)+(1/2) ε ¯ B C n for all k k 0 .
(6)

On the other hand, since ρ 1 (( ψ n k , S n k ),(ψ,S))0, there is a positive integer k 1 with k 1 k 0 , such that

ψ n k (x,y)ψ(x,y)+(1/2) ε ¯ Bfor any (x,y)K×K and all k k 1 .
(7)

By (7), (6) and (5), we have

ψ n k ( x n k , a n k ) ψ ( x n k , a n k ) + ( 1 / 2 ) ε ¯ B ψ ( x ¯ , z ) + ( 1 / 2 ) ε ¯ B + ( 1 / 2 ) ε ¯ B C n ψ ( x ¯ , z ) + ε ¯ B C n int C n C n int C n for all  k k 1 .

This contradicts (3). Hence (4) holds. Then by the arbitrariness of zS( x ¯ ), we obtain that

ψ( x ¯ ,z)int C n for all zS( x ¯ ).
(8)

By (2) and (8), we have that ((ψ,S), x ¯ )Graph(F). Hence, Graph(F) is closed. F(u) is also closed, for all uH. By the compactness of K, we know that F is a set-valued mapping with compact values. Hence, F is an usco mapping. The proof is completed. □

For each uH, the component of a point xF(u) is the union of all the connected subsets of F(u) containing x. Note that the components are connected closed subsets of F(u), and thus are connected and compact, see [20]. It is easy to see that the components of two distinct points of F(u) either coincide or are disjoint, so that all components constitute a decomposition of F(u) into connected pairwise disjoint compact subsets, i.e.,

F(u)= α Λ F α (u),

where Λ is an index set for each αΛ, F α (u) is a nonempty connected compact subset of F(u) and, for any α,βΛ (αβ), F α (u) F β (u)=.

Definition 3.1 Let uH and m be a nonempty closed subset of F(u). m is said to be an essential set of F(u) if, for each open set Om, there exists δ>0 such that for any u H with ρ 1 (u, u )<δ, F( u )O. If a component F α (u) of F(u) is an essential set, then F α (u) is said to be an essential component of F(u). An essential set m of F(u) is said to be a minimal essential set of F(u) if m is a minimal element of the family of essential sets in F(u) ordered by set inclusion.

Lemma 3.3[7]

Let A, B and C be nonempty convex compact subsets of a normed linear space X. Thenh(A,λB+μC)λh(A,B)+μh(A,C), where h is the Hausdorff metric defined onCK(X), λ0, μ0, andλ+μ=1.

The following theorem is the exist theorem of an essential component of the set of solutions for the vector quasi-equilibrium problem.

Theorem 3.1

(a) For anyuH, there exists at least one minimal essential set ofF(u), and every minimal essential set ofF(u)is connected;

(b) For anyuH, there exists at least one essential connected component ofF(u).

Proof(a) Since F is upper semicontinuous, following the idea of Lemma 2.2 in [5], we can easily obtain that there exists one minimal essential set of F(u) for each uH. Now, for each minimal essential set of F(u), as Yang and Yu did in [8], we prove that each minimal essential set of F(u) is connected. Let m(u) be one minimal essential set of F(u). If m(u) is not connected, then there exist two nonempty closed sets c 1 (u) and c 2 (u) of F(u) and two open sets V 1 and V 2 in K such that

m(u)= c 1 (u) c 2 (u), c 1 (u) V 1 ,c(u) V 2 , V 1 V 2 =.

Since m(u) is a minimal essential set of F(u), neither c 1 (u) nor c 2 (u) is essential. Thus, there exist two open sets O 1 c 1 (u) and O 2 c 2 (u) such that, for any δ>0, there exist u 1 , u 2 H with

ρ 1 (u, u 1 )<δ, ρ 1 (u, u 2 )<δ,butF( u 1 ) O 1 =,F( u 2 ) O 2 =.
(9)

Set W 1 = V 1 O 1 , and W 2 = V 2 O 2 . Then both W 1 and W 2 are open sets and c 1 (u) W 1 , and c 2 (u) W 2 . Since c 1 (u) and c 2 (u) are a closed subset of the compact set F(u), c 1 (u) and c 2 (u) are a compact set, there exist two open sets U 1 and U 2 such that

c 1 (u) U 1 U 1 ¯ W 1 , c 2 (u) U 2 U 2 ¯ W 2 ,and W 1 W 2 =.

Since m(u) is essential and m(u) U 1 U 2 , there exists δ >0 such that for any u with ρ 1 (u, u )< δ , we have

F ( u ) ( U 1 U 2 ).
(10)

Since U 1 O 1 and U 2 O 2 , for above δ /2>0, by (9), there exist u 1 , u 2 H such that

ρ 1 (u, u 1 )< δ /2, ρ 1 (u, u 2 )< δ /2,F( u 1 ) U 1 =,F( u 2 ) U 2 =.
(11)

Since u 1 , u 2 H, we have u 1 =( ψ 1 , S 1 ) and u 2 =( ψ 2 , S 2 ). Now we define S :K 2 K and ψ :K×K Y n by

S (x)=λ(x) S 1 (x)+μ(x) S 2 (x)for all xK

and

ψ (x,y)=λ(x) ψ 1 (x,y)+μ(x) ψ 2 (x,y)for all (x,y)K×K,

respectively, where

λ(x)= d ( x , U 2 ¯ ) d ( x , U 1 ¯ ) + d ( x , U 2 ¯ ) ,μ(x)= d ( x , U 1 ¯ ) d ( x , U 1 ¯ ) + d ( x , U 2 ¯ ) for all xK.

It is obvious that λ and μ are continuous functions on K with λ(x)0, μ(x)0, and λ(x)+μ(x)=1 for any xK.

We can see that: (i) ψ (x,y) is ( C n )-continuous on K×K; (ii) for each fixed xK, ψ (x,y) is C n -convex in y; (iii) ψ (x,x)=θint C n for all xK; (iv) sup ( x , y ) K × K ψ (x,y)<+; (v) for each xK, S (x) is convex and compact; (vi) S is continuous on K. Hence v:=( ψ , S )H. By Lemma 3.3, we have

h ( S ( x ) , S ( x ) ) λ ( x ) h ( S ( x ) , S 1 ( x ) ) + μ ( x ) h ( S ( x ) , S 2 ( x ) ) h ( S ( x ) , S 1 ( x ) ) + h ( S ( x ) , S 2 ( x ) ) for all  x K ,

and

ψ ( x , y ) ψ ( x , y ) = ψ ( x , y ) λ ( x ) ψ 1 ( x , y ) μ ( x ) ψ 1 ( x , y ) λ ( x ) ψ ( x , y ) ψ 1 ( x , y ) + μ ( x ) ψ ( x , y ) ψ 2 ( x , y ) ψ ( x , y ) ψ 1 ( x , y ) + ψ ( x , y ) ψ 2 ( x , y ) for all  ( x , y ) K × K .

Thus, by (11), we have

ρ 1 ( u , v ) = sup ( x , y ) K × K ψ ( x , y ) ψ ( x , y ) + sup x K h ( S ( x ) , S ( x ) ) sup ( x , y ) K × K ψ ( x , y ) ψ 1 ( x , y ) + sup ( x , y ) K × K ψ ( x , y ) ψ 2 ( x , y ) + sup x K h ( S ( x ) , S 1 ( x ) ) + sup x K h ( S ( x ) , S 2 ( x ) ) ρ 1 ( u , u 1 ) + ρ 1 ( u , u 2 ) < δ / 2 + δ / 2 = δ .

Using (10), we have

F(v)( U 1 U 2 ).
(12)

If x U 1 , then λ(x)=1, μ(x)=0, S (x)= S 1 (x) and ψ (x,y)= ψ 1 (x,y) for all yK. If xF(v), then x S (x) and ψ (x,y)int C n for all y S (x). Since S (x)= S 1 (x) and ψ (x,y)= ψ 1 (x,y) for all yK, we have xF( u 1 ). This contradicts (11). Thus, we have F(v) U 1 =. Similarly, we can show that F(v) U 2 =. This contradicts (12). Hence, m(u) is connected, so the conclusion (a) holds.

(b) For any uH, by (a), there exists at least one essential connected set m of F(u). There exists a component F α (u) of F(u) such that m F α (u). It is obvious that F α (u) is essential. □

Now, for each iI, let f i :K× K i Y i be a vector-valued mapping and S i :K 2 K i a set-valued mapping. Let

Let

Let P=D×Q. For any

p 1 = ( ( f 11 , , f 1 n ) , ( S 11 , , S 1 n ) ) , p 2 = ( ( f 21 , , f 2 n ) , ( S 21 , , S 2 n ) ) P,

define

ρ 2 ( p 1 , p 2 )= i = 1 n sup ( x , y i ) K × K i f 1 i ( x , y i ) f 2 i ( x , y i ) + sup x K h ( i = 1 n S 1 i ( x ) , i = 1 n S 2 i ( x ) ) ,

where h is the Hausdorff metric defined on CK(X). Clearly, (P, ρ 2 ) is a metric space.

Let K= i = 1 n K i . It is clear that K is a nonempty compact convex subset of X= i = 1 n X i . For any ( f 1 ,, f n )D, and ( S 1 ,, S n )Q, define the mapping ψ:K×K Y n by

ψ(x,y)= i = 1 n f i (x, y i ),x=( x 1 ,, x n ),y=( y 1 ,, y i ,, y n )K,

and the mapping S:K 2 K by

S(x)= i = 1 n S i (x),xK.

Since ( f 1 ,, f n )D and ( S 1 ,, S n )Q, we can see that S:K 2 K is continuous with nonempty convex and compact valued, (i) ψ(,) is ( C n )-continuous on K×K; (ii) for each fixed xK, ψ(x,) is ( C n )-convex; and (iii) ψ(x,x)=θint C n for all xK. It is clear that (ψ,S)H. Since Y i Y n for each iI={1,2,,n}, by Theorem 2.1, there exists an element x K such that x S( x ) and

ψ ( x , y ) int C n for all yS ( x ) .

That is

f 1 ( x , y 1 ) ++ f i ( x , y i ) ++ f n ( x , y n ) int C n for all  y 1 S 1 ( x ) ,, y n S n ( x ) .

For each iI, by the arbitrariness of y j S j ( x ), j{1,,n}, ji, take y j = x j , and by assumption f j ( x , x j )=θ, j=1,,n, and ji, we obtain that x i S i ( x ) and

f i ( x , y i ) int C n for all  y i S i ( x ) .

Since f i ( x , y i ) Y i and C i = C n Y i , it follows that

f i ( x , y i ) int C i for all  y i S i ( x ) .

Thus, there exists x =( x 1 ,, x n )K such that for each iI, x i S i ( x ) and

f i ( x , y i ) int C i for all  y i S i ( x ) .
(13)

For each pP, denote by E(p) all solutions to the SVQEP. By (13), there exists x E(p), thus E(p). Similar to Definition 3.1, we can define the minimal essential set and essential component of E(p).

Lemma 3.4 For eachp=(( f 1 ,, f n ),( S 1 ,, S n ))P, define the mappingT:PHby

T(p)=(ψ,S),

where

ψ(x,y)= i = 1 n f i (x, y i ),x=( x 1 ,, x n ),y=( y 1 ,, y i ,, y n )K

and

S(x)= i = 1 n S i (x),xK.

Then T is continuous.

Proof It is easy to check that for each p=(( f 1 ,, f n ),( S 1 ,, S n ))P, T(p)=(ψ,S)H.

For any p 1 =(( f 11 ,, f 1 n ),( S 11 ,, S 1 n )), p 2 =(( f 21 ,, f 2 n ),( S 21 ,, S 2 n ))P, if ρ 2 ( p 1 , p 2 )<ε, then by the definition of ρ 1 , we have

This completes the proof of the lemma. □

The following lemma can be found in [8].

Lemma 3.5 Let U, Y and Z be three metric spaces, F:U 2 Y be an usco mapping andG:Z 2 Y be a set-valued mapping. Suppose that there exists a continuous mappingT:ZUsuch thatG(z)F(T(z))for eachzZ. Furthermore, suppose that there exists at least one essential component ofF(φ)for eachφU. Then there exists at least one essential component ofG(z)for eachzZ.

As application of Theorem 3.1, 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 solutions in the uniform topological space of objective functions and constraint mappings.

Theorem 3.2 For eachpP, there exists at least one essential component ofE(p).

Proof For any p=(( f 1 ,, f n ),( S 1 ,, S n ))P, define T:PH by T(p)=(ψ,S), where

ψ(x,y)= i = 1 n f i (x, y i ),x=( x 1 ,, x n ),y=( y 1 ,, y i ,, y n )K

and

S(x)= i = 1 n S i (x),xK.

By Lemma 3.4, T is continuous. Now we need to prove that for each pP, E(p)F(T(p)), where F is defined by (1). If x =( x 1 ,, x i ,, x n )F(T(p)), then x K, x S( x ) and

ψ ( x , y ) int C n for all yS ( x ) .

That is

f 1 ( x , y 1 ) ++ f i ( x , y i ) ++ f n ( x , y n ) int C n for all  y 1 S 1 ( x ) ,, y n S n ( x ) .

For each iI, by the arbitrariness of y j S j ( x ), j{1,,n}, ji, take y j = x j , and by assumption f j ( x , x j )=θ, j=1,,n, and ji, we obtain that x i S i ( x ) and

f i ( x , y i ) int C n for all  y i S i ( x ) .

Since f i ( x , y i ) Y i and C i = C n Y i , it follows that

f i ( x , y i ) int C i for all  y i S i ( x ) .

Hence x E(p) and hence E(p)F(T(p)). Thus, by Lemma 3.2, Theorem 3.1 and Lemma 3.5, there exists at least one essential component of E(p). □

References

  1. Fort MK: Essential and nonessential fixed points. Am. J. Math. 1950, 72: 315–322. 10.2307/2372035

    Article  MathSciNet  Google Scholar 

  2. Kinoshita S: On essential component of the set of fixed points. Osaka J. Math. 1952, 4: 19–22.

    MathSciNet  Google Scholar 

  3. Jiang JH: Essential components of the set of fixed points of multivalued mappings and its application to the theory of games. Sci. Sin. 1963, 12: 951–964.

    Google Scholar 

  4. Kohlberg E, Mertens JF: On the strategic stability of equilibria. Econometrica 1986, 54: 1003–1037. 10.2307/1912320

    Article  MathSciNet  Google Scholar 

  5. Yu J, Xiang SW: On essential components of the set of Nash equilibrium points. Nonlinear Anal., Theory Methods Appl. 1999, 38: 259–264. 10.1016/S0362-546X(98)00193-X

    Article  MathSciNet  Google Scholar 

  6. Yu J, Luo Q: On essential components of the solution set of generalized games. J. Math. Anal. Appl. 1999, 230: 303–310. 10.1006/jmaa.1998.6202

    Article  MathSciNet  Google Scholar 

  7. Isac G, Yuan GXZ: The essential components of coincident points for weakly inward and outward set-valued mappings. Appl. Math. Lett. 1999, 12: 121–126.

    Article  MathSciNet  Google Scholar 

  8. Yang H, Yu J: Essential components of the set of weakly Pareto-Nash equilibrium points. Appl. Math. Lett. 2002, 15: 553–560. 10.1016/S0893-9659(02)80006-4

    Article  MathSciNet  Google Scholar 

  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. 2005, 124: 387–405. 10.1007/s10957-004-0942-0

    Article  MathSciNet  Google Scholar 

  10. Chen JC, Gong XH: The stability of set of solutions for symmetric vector quasi-equilibrium problems. J. Optim. Theory Appl. 2008, 136: 359–374. 10.1007/s10957-007-9309-7

    Article  MathSciNet  Google Scholar 

  11. Ansari QH, Schaible S, Yao JC: The system of generalized vector equilibrium problems with applications. J. Glob. Optim. 2002, 22: 3–16. 10.1023/A:1013857924393

    Article  MathSciNet  Google Scholar 

  12. Ansari QH, Chan WK, Yang XQ: The system of vector quasi-equilibrium problems with applications. J. Glob. Optim. 2004, 29: 45–57.

    Article  MathSciNet  Google Scholar 

  13. Fang YP, Huang NJ, Kim JK: Existence results for system of vector equilibrium problems. J. Glob. Optim. 2006, 35: 71–83. 10.1007/s10898-005-1654-1

    Article  MathSciNet  Google Scholar 

  14. Peng JW, Lee HWJ, Yang XM: On system of generalized vector quasi-equilibrium problems with set-valued maps. J. Glob. Optim. 2006, 36: 139–158. 10.1007/s10898-006-9004-5

    Article  MathSciNet  Google Scholar 

  15. Lin Z: The study of the system of generalized vector quasi-equilibrium problems. J. Glob. Optim. 2006, 36: 627–635. 10.1007/s10898-006-9031-2

    Article  Google Scholar 

  16. Peng JW, Yang XM, Zhu DL: System of vector quasi-equilibrium and its applications. Appl. Math. Mech. 2006, 27: 1107–1114. 10.1007/s10483-006-0811-y

    Article  MathSciNet  Google Scholar 

  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. 2010, 30: 1006–1017.

    MathSciNet  Google Scholar 

  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. 2002, 22: 90–95. in Chinese

    MathSciNet  Google Scholar 

  19. Aubin JP, Ekeland I: Applied Nonlinear Analysis. Wiley, New York; 1984.

    Google Scholar 

  20. Engelking R: General Topology. Helderman, Berlin; 1989.

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xun-Hua Gong.

Additional information

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.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Gong, XH., Chen, JC. Essential components of the set of solutions for the system of vector quasi-equilibrium problems. J Inequal Appl 2012, 181 (2012). https://doi.org/10.1186/1029-242X-2012-181

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-181

Keywords