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Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces

Abstract

In this paper, a scalarization result of ε-weak efficient solution for a vector equilibrium problem (VEP) is given. Using this scalarization result, the connectedness of ε-weak efficient and ε-efficient solutions sets for the VEPs are proved under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper improve and generalize some known results in the literature.

1 Introduction

Let K be a nonempty subset of a real Hausdorff topological vector space E, and f : K × KR a bifunction such that f(x, x) ≥ 0 for all x K. Then, the scalar equilibrium problem consists in finding x ̄ K such that

f ( x ̄ , y ) 0 , y K .

It provides a unifying framework for many important problems, such as optimization problems, variational inequality problems, complementary problems, minimax inequality problems, Nash equilibrium problems, and fixed point problems, and has been widely applied to study problems arising in economics, mechanics, and engineering science (see [1]). On the other hand, several operations research problems are formulated with a multicriteria consideration. These are vector optimization problems, vector variational inequality and complementarity problems and vector equilibrium problems (VEPs). Recently, the VEP has received much attention by many authors because it provides a unified model including vector optimization problems, vector variational inequality problems, vector complementarity problems and vector saddle point problems as special cases (see, for example, [224] and the references therein).

It is well known that another important problem for VEPs is to study the topological properties of solutions set. Among its topological properties, the connectedness is of interest. Recently, Lee et al. [25], Cheng [26] have studied the connectedness of weak efficient solutions set for vector variational inequalities in finite dimensional Euclidean space. Gong [2729] has studied the connectedness of the various solutions set for VEPs in infinite dimension space. Chen et al. [30] studied the connectedness and the compactness of the weak efficient solutions set for set-valued VEPs and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space. Gong and Yao [31] have studied the connectedness of the set of efficient solutions for generalized systems. Zhong et al. [32] have studied the connectedness and path-connectedness of solutions set for symmetric VEPs. However, the connectedness of approximate solutions set for VEPs remained unstudied.

In this paper, we show a scalarization result of ε-weak efficient solution for a VEP. Using this scalarization result, we discuss the connectedness of ε-weak efficient and ε-efficient solutions sets for VEPs under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper generalize some known results due to Gong [27] and Gong and Yao [31].

2 Preliminaries

Throughout this paper, let X and Y be two real Hausdorff topological vector spaces and A a nonempty subset of X. Let F : A × AY be a mapping and C be a closed convex pointed cone in Y. The cone C induces a partial ordering in Y, defined by

z 1 z 2 i f a n d o n l y i f z 2 - z 1 C .

Let

C * = { f Y * : f ( y ) 0 f o r a l l y C }

be the dual cone of C.

Denote the quasi-interior of C* by C#, that is,

C # = { f Y * : f ( y ) > 0 f o r a l l y C \ { 0 } } .

Let D be a nonempty subset of Y. The cone hull of D is defined as

c o n e ( D ) = { t d : t 0 , d D } .

A nonempty convex subset B of the convex cone C is called a base of C if

C = c o n e ( B ) a n d 0 c l ( B ) .

It is easy to see that C# if and only if C has a base.

Throughout this paper, we always assume that intC. Let e be a fixed point in intC and we set

C = { f C * \ { 0 } : f ( e ) = 1 } , C = { f C # : f ( e ) = 1 } .

Now, we give the concepts of ε-weak efficient solution, ε-efficient solution, and ε-f efficient solution for the VEP.

Definition 2.1. A vector x A satisfying

F ( x , y ) - i n t C - ε e f o r a l l y A ,

is called a ε-weak efficient solution to the VEP. Denote by Vε-W(A, F) the set of all ε-weak efficient solutions to the VEP.

Definition 2.2. A vector x A satisfying

F ( x , y ) - C \ { 0 } - ε e f o r a l l y A ,

is called a ε-efficient solution to the VEP. Denote by V ε (A, F) the set of all ε-efficient solutions to the VEP.

Definition 2.3. Let f C*\{0}. A vector x A is called a ε-f efficient solution to the VEP if

f ( F ( x , y ) ) - ε f o r a l l y A ,

Denote by Vε-f(A, F) the set of all ε-f efficient solutions to the VEP.

Definition 2.4. [33] Let G be a set-valued map from a topological space W to another topological space Q. We say that G is

  1. (i)

    upper semicontinuous at x0 W if, for any neighborhood U(G(x0)) of G(x0), there is a neighborhood U (x0) of x0 such that G(x) U (G(x0)) for all x U(x0);

  2. (ii)

    upper semicontinuous on W if it is upper semicontinuous at each x W;

  3. (iii)

    lower semicontinuous at x0 W if, for any net {xα : α I} converging to x0 and for any y0 G(x0), there exists a net y α G(x α ) that converges to y0;

  4. (iv)

    lower semicontinuous on W if it is lower semicontinuous at each x W;

  5. (v)

    continuous on W if it is both upper semicontinuous and lower semicontinuous on W;

  6. (vi)

    closed if Graph(G) = {(x, y) : x W, y G(x)} is a closed subset in W × Q.

3 Scalarization

Lemma 3.1. Suppose F (x, A) + C is a convex set for each x A. Then,

V ε - W ( A , F ) = f C V ε - f ( A , F ) .

Proof. We first prove that

f C V ε - f ( A , F ) V ε - W ( A , F ) .

In fact, letting x 0 f C V - f ( A , F ) , then there exists f C',

f ( F ( x 0 , y ) ) - ε , f o r a l l y A .
(3.1)

We claim that x0 Vε-W(A, F). If not, then there exists y0 A such that F (x0, y0) -intC - εe. Thus, we have

f ( F ( x 0 , y 0 ) + ε e ) < 0

and so

f ( F ( x 0 , y 0 ) ) < - f ( ε e ) = - ε ,

which is a contradiction to (3.1).

We next prove that

V ε - W ( A , F ) f C V ε - f ( A , F ) .

Let x Vε-W(A, F). Then, F (x, A) ∩ (-intC - εe) = . Since C is a pionted convex cone, we have

( F ( x , A ) + C + ε e ) ( - i n t C ) = .

By assumption, we know that F (x, A) + C is a convex set. Using the separation theorem for convex sets, there exists some g Y *\{0}, such that

i n f { g ( F ( x , y ) + c + ε e ) : y A , c C } s u p { g ( - c ) : c C } .
(3.2)

From (3.2), we get g C*\{0} and so

g ( F ( x , y ) ) - g ( ε e ) f o r a l l y A ,

where e intC with g (e) > 0. Letting f= g g ( e ) , then f (F (x, y)) ≥ -ε, for all y A and f (e) = 1. Thus, f C' and so

x f C V ε - f ( A , F ) .

This completes the proof.

Remark 3.1 When ε = 0, it is easy to see that

V ε - W ( A , F ) = f C V ε - f ( A , F ) = f C * \ { 0 } V ε - f ( A , F ) .

Therefore, Lemma 3.1 generalizes Lemma 2.1 in [27].

4 Existence of the solutions

Definition 4.1. The bifunction F : A × AY is concave-like with respect to the first variable, if for t [0, 1], the following condition is satisfied: For x1, x2 A, there exists x3 A, such that

F ( x 3 , y ) t F ( x 1 , y ) + ( 1 - t ) F ( x 2 , y ) + C , f o r a l l y A .

The bifunction F : A×AY is convex-like with respect to the second variable, if for t [0, 1], the following condition is satisfied: for y1, y2 A, there exists y3 A, such that

F ( x , y 3 ) t F ( x , y 1 ) + ( 1 - t ) F ( x , y 2 ) - C , f o r a l l x A .

Theorem 4.1. Let A be a nonempty compact subset of X, and f C'. Assume that the following conditions are satisfied:

  1. (i)

    F (x, x) C - εe, for all x A;

  2. (ii)

    F : A × AY is concave-like with respect to the first variable and convex-like with respect to the second variable;

  3. (iii)

    For each fixed y A, the function x f (F (x, y)) is upper semicontinuous on A.

Then, Vε-f(A, F) ≠ .

Proof. Define the set-valued map G : A → 2A by

G ( y ) = { x A , f ( F ( x , y ) ) - ε }

By assumption, y G (y), for all y A, so G (y) ≠ . By assumption, we can see that G (y) is a closed subset of A. Next, we prove that ∩{G (y) : y A} ≠ . Since A is a compact, we need to show that i = 1 n G ( y i ) for any arbitrary chosen y1,..., y n in A. Suppose it is not true. Then, there exists a set B = {y1,..., y n } A such that i = 1 n G ( y i ) . Thus, for any x A, there exists y i B such that x G (y i ). It follows that

f ( F ( x , y i ) ) + ε < 0

and so there exists η i > 0 such that

f ( F ( x , y i ) ) + ε < - η i , i = 1 , 2 , , n .

Since x f (F (x, y)) is upper semicontinuous on A, we can choose η > 0 such that, for any x A, there exists y i B satisfying

f ( F ( x , y i ) ) + ε + η < 0 .

Define g : ARn by

g ( x ) = ( - f ( F ( x , y 1 ) ) - ε - η , - f ( F ( x , y 2 ) ) - ε - η , . . . , - f ( F ( x , y n ) ) - ε - η ) ,

where x A. We get

g ( x ) i n t R + n , f o r a l l x A .
(4.1)

Since f C' and F (x, y) is concave-like with respect to the first variable, we can see that, for t [0, 1], x1, x2 A, there exists x3 A such that

g ( x 3 ) t g ( x 1 ) + ( 1 - t ) g ( x 2 ) - C .

This shows that g ( A ) + R + n is a convex set. It follows from (4.1) that

0 g ( A ) + i n t R + n .

By the separation theorem of convex sets (see, for example, [34]), we can find t1,..., t n ≥ 0 with i = 1 n t i =1 such that

0 i = 1 n t i ( - f ( F ( x , y i ) ) - ε - η ) , f o r a l l x A .

It follows that

i = 1 n t i f ( F ( x , y i ) ) - ε - η , f o r a l l x A .

By assumption, there exists y A such that

F ( x , y ) i = 1 n t i ( F ( x , y i ) ) - C , f o r a l l x A .

Since f C', we have

f ( F ( x , y ) ) f i = 1 n t i ( F ( x , y i ) ) , f o r a l l x A .

So f (F (x, y)) ≤ -ε - η < -ε, for all x A. Setting x = y, it follows that

f ( F ( y , y ) ) < - ε .

On the other hand, by the assumption,

f ( F ( y , y ) ) - ε .

This is a contradiction. Therefore, ∩{G (y) : y A} ≠ , and so there exists x ∩{G (y) : y A}. This means that Vε-f(A, F) ≠ . This completes the proof.

5 Connectedness of the solutions set

In this section, we discuss the connected results of ε-weak efficient solutions set and ε-efficient solutions set.

Definition 5.1. Let A be a convex set. The bifunction F : A × AY is C-concave with respect to the first variable, if for t [0, 1], x1, x2 A,

F ( t x 1 + ( 1 - t ) x 2 , y ) t F ( x 1 , y ) + ( 1 - t ) t F ( x 2 , y ) + C .

It is clear that when A is a convex set, F : A×AY is C-concave with respect to the first variable, then it is concave-like about the first variable.

Theorem 5.1 Let A be a nonempty compact convex subset of X, and f C'. Assume that the following conditions are satisfied.

(i) F (x, x) C - εe, for all x A;

(ii) F : A × AY is C-concave with respect to the first variable and convex-like with respect to the second variable;

(iii) For each fixed y A, the function x f (F (x, y)) is upper semicontinuous on A;

(iv) D = {F (x, y) : x, y A} is a bounded set of Y.

Then, Vε-W(A, F) is a connected set.

Proof. Define a set-valued mapping H : C' → 2A by

H ( f ) = V - f ( A , F ) , f C .

By Theorem 4.1, we know that, for each f C', Vε-f(A, F) ≠ . It is easy to see that C' is a convex set and so is connected. Next, we prove that, for each f C', H (f) is a connected set. Let x1, x2 H (f), we have x1, x2 A, and

f ( F ( x i , y ) ) - ε , y A , i = 1 , 2 .
(5.1)

Because F : A ×AY is C-concave with respect to the first variable, for each fixed y A, t [0, 1],

F ( t x 1 + ( 1 - t ) x 2 , y ) t F ( x 1 , y ) + ( 1 - t ) F ( x 2 , y ) + C .

Hence

t f ( F ( x 1 , y ) ) + ( 1 - t ) f ( F ( x 2 , y ) ) f ( F ( t x 1 + ( 1 - t ) x 2 , y ) ) .

It follows from (5.1) that

f ( F ( t x 1 + ( 1 - t ) x 2 , y ) ) - ε .

Hence

t x 1 + ( 1 - t ) x 2 H ( f )

and so H (f) is a convex set. Thus, it is a connected set.

Next, we show that H is upper semicontinuous on C'. Since A is compact, we only need to show that H is closed. Let {(f α , x α ) : α I} Graph (H) be a net such that (f α , x α ) → (f, x), where f α f means that {f α } converges to f with respect to the strong topology β (Y*, Y) in Y*. Since C' is a closed set and A is a compact set, we know that (f, x) C' × A. From x α H (f α ) = Vε- (A, F), we have

f α ( F ( x α , y ) ) - ε .
(5.2)

For any δ > 0,

U = y * Y * : sup u D | y * ( u ) | < δ

is a neighborhood of 0 with respect with to β (Y *, Y). Since f α f, there exists α0 I such that f α - f U, for all αα0. It follows that

sup u D | ( f α - f ) ( u ) | < δ , w h e n e v e r α α 0 .

Therefore, for any y A,

| ( f α - f ) ( F ( x α , y ) ) | < δ , w h e n e v e r α α 0

and so

lim α ( f α ( F ( x α , y ) ) - f ( F ( x α , y ) ) ) = 0 , f o r a l l y A .
(5.3)

Because x f (F (x, y)) is upper semicontinuous on A, then

l i m s u p f ( F ( x α , y ) ) f ( F ( x , y ) ) .
(5.4)

From (5.2), (5.3) and (5.4), we have

- ε l i m s u p f α ( F ( x α , y ) ) (1) lim α ( f α ( F ( x α , y ) ) - f ( F ( x α , y ) ) ) + l i m s u p f ( F ( x α , y ) ) (2) f ( F ( x , y ) ) . (3) (4)

Hence, x H (f) = Vε-f(A, F). By Theorem 3.1 in [35], f C V ε - f ( A , F ) is a connected set. Because F : A × AY is convex-like with respect to the second variable, we have F (x, A) + C is a convex set, by Lemma 3.1,

V ε - W ( A , F ) = f C V ε - f ( A , F )

is a connected set. This completes the proof.

Next, we give an example to illustrate Theorem 5.1.

Example 5.1 Let X = R, Y = R2, C= R + 2 = { ( x 1 , x 2 ) : x 1 0 , x 2 0 } , ε = 2, e = (2, 3), and A = [0, 2]. Let

F ( x , y ) = ( y - x 2 - 2 , y - x 2 - 4 ) , x , y A .

Then, F satisfies all conditions of Theorem 5.1. It is easy to see that V ε - W ( A , F ) = 0 , 2 . Clearly, Vε-W(A, F) is a nonempty connected set.

Definition 5.2. The bifunction F : A ×AY is ε-C strictly monotone if, for any x, y A, xy,

F ( x , y ) + F ( y , x ) - i n t C - 2 ε e .

Theorem 5.2 Suppose that all conditions of Theorem 5.1 are satisfied and F : A × AY is ε-C strictly monotone. Then, Vε-W(A, F) is a path connected set.

Proof. Define the set-valued mapping H : C' → 2A by

H ( f ) = V ε - f ( A , F ) , f C .

By Theorem 4.1, we know that, for each f C', Vε-f(A, F) ≠ . Furthermore, because F : A × AY is ε-C strictly monotone, it is easy to see that, for each f C', H (f) = Vε-f(A, F) is a single point set. From the proof of Theorem 5.1, we know that H is upper semicontinuous on C' and so it is continuous on C'. Since C' is a convex set, so it is a path connected set. Hence,

V ε - W ( A , F ) = H ( C ) = f C H ( f ) = f C V ε - f ( A , F )

is a path connected set. This completes the proof.

Next, we give an example to illustrate Theorem 5.2.

Example 5.2 Let X = R, Y = R2, C= R + 2 = { ( x 1 , x 2 ) : x 1 0 , x 2 0 } , ε = -1, e = (1, 2), and A = [-1, 1]. Let

F ( x , y ) = ( x ( y - x ) + 1 , x ( y - x ) + 2 ) , x , y A .

Then, F satisfies all the conditions of Theorem 5.2. It is easy to see that Vε-W(A, F) = {0}. Clearly, Vε-W(A, F) is a nonempty path connected set.

Next, we give a lemma before we give the connectedness theorem of ε-efficient solutions set.

Lemma 5.1. Suppose that all conditions of Theorem 5.2 are satisfied, then

f C V ε - f ( A , F ) V ε ( A , F ) c l f C V ε - f ( A , F ) .

Proof. By Theorem 4.1, we know that, for each f C', Vε-f(A, F) ≠ . By definition, we have

f C V ε - f ( A , F ) V ε ( A , F ) V ε - W ( A , F ) .
(5.5)

From Lemma 3.1, we have

V ε - W ( A , F ) = f C V ε - f ( A , F ) .
(5.6)

By (5.5) and (5.6), we have

f C V ε - f ( A , F ) V ε ( A , F ) f C V ε - f ( A , F ) .

Next, we prove that

f C V ε - f ( A , F ) c l f C V ε - f ( A , F ) .

Define the set-valued mapping: H : C' → 2A by

H ( f ) = V ε - f ( A , F ) , f C .

From Theorem 5.2, we know that H (f) is a single-valued mapping, and H is continuous on C'.

Let x f C V ε - f ( A , F ) . Then, there exists f C', such that

{ x } = V ε - f ( A , F ) = H ( f ) .

Let g C", and set

f n = n - 1 n f + 1 n g = f + 1 n ( g - f ) .

Then, f n C# and f n (e) = 1. Thus, f n C".

Next, we show that {f n } converges to f with respect to the topology β (Y*, Y). For any neighborhood of 0 with respect to β (Y*, Y), there exist bounded subsets B i Y (i = 1, 2,..., m) and δ > 0 such that

i = 1 m f Y * : sup y B i | f ( y ) | < δ U .

Since B i is bounded and g - f Y *, it is easy to see that |(g - f) (B i )| is bounded for i = 1, 2,..., m. This implies that there exists N such that

sup y B i 1 n ( g - f ) ( y ) < δ , i = 1 , 2 , . . . , m ; n N .

Hence, 1 n ( g - f ) U, that is f n - f U. Hence, {f n } converges to f with respect to β (Y*, Y).

Since H is continuous on f, we have H (f n ) → H (f). Set {x n } = H (f n ), then

{ x n } = H ( f n ) = V ε - f n ( A , F ) f C V ε - f ( A , F ) .

Because {x} = H (f), we have x n x. This implies that

x c l f C V ε - f ( A , F ) .

Since x f C V ε - f ( A , F ) is arbitrary, we have

f C V ε - f ( A , F ) c l f C V ε - f ( A , F ) .

Therefore,

f C V ε - f ( A , F ) V ε ( A , F ) c l f C V ε - f ( A , F ) .

This completes the proof.

Theorem 5.3. Suppose that all the conditions of Theorem 5.2 are satisfied. Then, V ε (A, F) is a connected set.

Proof. By Lemma 5.1, we have

f C V ε - f ( A , F ) V ε ( A , F ) c l f C V ε - f ( A , F ) .
(5.7)

From Theorem 5.2, we can get

f C V ε - f ( A , F )

is connected set and so (5.7) implies that V ε (A, F) is a connected set. This completes the proof.

Remark 5.1 When ε = 0, we can get

f C V ε - f ( A , F ) = f c V ε - f ( A , F ) .

Therefore, Theorem 5.3 generalizes Theorem 2.2 in [31].

Next, we give an example to illustrate Theorem 5.3.

Example 5.3 Let X = R, Y = R, C = R+, ε = 1, e = 1, and A = [1, 2]. Let F (x, y) = x (y - x) - 1 for all x, y A. Then, it is easy to check that all the conditions of Theorem 5.3 are satisfied and

V ε ( A , F ) = { 1 } .

Clearly, V ε (A, F) is a nonempty connected set.

Abbreviations

VEP:

vector equilibrium problem.

References

  1. Blum B, Oettli W: From optimzation and variational inequalities to equilibrium problems. Math. Stud 1994, 63: 123–145.

    MathSciNet  Google Scholar 

  2. Chen GY, Huang XX, Yang XQ: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin; 2005.

    Google Scholar 

  3. Giannessi F, (ed): Vector Variational Inequilities and Vector Equilibria: Mathematical Theories. Kluwer, Dordrechet; 2000.

  4. Ansari QH, Flores-Bazan F: Recession methods for generalized vector equilibrium problems. J Math Anal Appl 2006, 321: 132–146. 10.1016/j.jmaa.2005.07.059

    Article  MathSciNet  Google Scholar 

  5. Ansari QH, Oettli W, Schläger D: A generalization of vector equilibrium. Math Methods Oper Res 1997, 46: 147–152. 10.1007/BF01217687

    Article  MathSciNet  Google Scholar 

  6. Ansari QH, Yao JC: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett 1999, 12: 53–56.

    Article  MathSciNet  Google Scholar 

  7. Chen B, Gong XH: Continuity of the solution set to parametric set-valued weak vector equilibrium problems. Pacific J. Optim 2010, 6: 511–520.

    MathSciNet  Google Scholar 

  8. Chen CR, Li SJ, Teo KL: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim 2009, 45: 309–318. 10.1007/s10898-008-9376-9

    Article  MathSciNet  Google Scholar 

  9. Konnov IV, Yao JC: Existence of solutions of generalized vector equilibrium problems. J. Math. Anal. Appl 1999, 233: 328–335. 10.1006/jmaa.1999.6312

    Article  MathSciNet  Google Scholar 

  10. Fu JY: Generalized vector quasivariational problems. Math. Methods Oper. Res 2000, 52: 57–64. 10.1007/s001860000058

    Article  MathSciNet  Google Scholar 

  11. Hou SH, Yu H, Chen GY: On vector quasi-equilibrium problems with set-valued maps. J. Optim. Theory Appl 2003, 119: 139–154.

    Article  MathSciNet  Google Scholar 

  12. Tan NX: On the existence of solutions of quasi-variational inclusion problems. J. Optim. Theory Appl 2004, 123: 619–638. 10.1007/s10957-004-5726-z

    Article  MathSciNet  Google Scholar 

  13. Peng JW, Joseph Lee HW, Yang XM: On systems of generalized vector quasi-quilibrium problem with set-valued maps. J. Glob. Optim 2006, 35: 139–158.

    Article  Google Scholar 

  14. Lin LJ, Ansari QH, Huang YJ: Some existence results for solutions of generalized vector quasi-equilibrium problems. Math. Methods Oper. Res 2007, 65: 85–98. 10.1007/s00186-006-0102-4

    Article  MathSciNet  Google Scholar 

  15. Long XJ, Huang NJ, Teo KL: Existence and stability of solutions for generalized strong vector quasi-equilibrium problem. Math. Comput. Model 2008, 47: 445–451. 10.1016/j.mcm.2007.04.013

    Article  MathSciNet  Google Scholar 

  16. Huang NJ, Li J, Yao JC: Gap functions and existence of solutions for a system of vector equilibrium problems. J. Optim. Theory Appl 2007, 133: 201–212. 10.1007/s10957-007-9202-4

    Article  MathSciNet  Google Scholar 

  17. Li J, Huang NJ: Implicit vector equilibrium problems via nonlinear scalarisation. Bull. Aust. Math. Soc 2005, 72: 161–172. 10.1017/S000497270003495X

    Article  Google Scholar 

  18. Li J, Huang NJ, Kim JK: On implicit vector equilibrium problems. J. Math. Anal. Appl 2003, 283: 501–512. 10.1016/S0022-247X(03)00277-4

    Article  MathSciNet  Google Scholar 

  19. Park S: Fixed points of better admissible maps on generalized convex spaces. J. Korea Math. Soc 2000,37(6):885–899.

    Google Scholar 

  20. Park S: Equilibrium existence theorems in KKM spaces. Nonlinear Anal. TMA 2008, 69: 4352–4364. 10.1016/j.na.2007.10.058

    Article  Google Scholar 

  21. Park S: Compact Browder maps and equilibria of abstract economies. J. Appl. Math. Comput 2008, 26: 555–564. 10.1007/s12190-007-0022-3

    Article  MathSciNet  Google Scholar 

  22. Park S: Remarks on fixed points, maximal elements, and equilibria of economies in abstract convex spaces. Taiwanese J. Math 2008, 12: 1365–1383.

    MathSciNet  Google Scholar 

  23. Park S, Kim H: Coincidence theorems for admissible multifunctions on generalized convex spaces. J. Math. Anal. Appl 1996, 197: 173–187. 10.1006/jmaa.1996.0014

    Article  MathSciNet  Google Scholar 

  24. Park S, Kim H: Founditions of the KKM theory on generalized convex spaces. J. Math. Anal. Appl 1997, 209: 551–571. 10.1006/jmaa.1997.5388

    Article  MathSciNet  Google Scholar 

  25. Lee GM, Kim DS, Lee BS, Yen ND: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal.: Theory, Methods Appl 1998, 34: 745–765. 10.1016/S0362-546X(97)00578-6

    Article  MathSciNet  Google Scholar 

  26. Cheng YH: On the connectedness of the solution set for the weak vector variational inequality. J. Math. Anal. Appl 2001, 260: 1–5. 10.1006/jmaa.2000.7389

    Article  MathSciNet  Google Scholar 

  27. Gong XH: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl 2001, 108: 139–154. 10.1023/A:1026418122905

    Article  MathSciNet  Google Scholar 

  28. Gong XH, Fu WT, Liu W: Super efficiency for a vector equilibrium in locally convex topological vector spaces. In Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Edited by: Giannessi F. Kluwer Academic Publishers, Netherlands; 2000:233–252.

    Chapter  Google Scholar 

  29. Gong XH: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl 2007, 133: 151–161. 10.1007/s10957-007-9196-y

    Article  MathSciNet  Google Scholar 

  30. Chen B, Gong XH, Yuan SM: Connectedness and compactness of weak efficient solutions set for set-valued vector equilibrium problems. J. Inequal. Appl 2008, 2008: 15. Article ID 581849

    Article  MathSciNet  Google Scholar 

  31. Gong XH, Yao JC: Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl 2008, 138: 189–196. 10.1007/s10957-008-9378-2

    Article  MathSciNet  Google Scholar 

  32. Zhong RY, Huang NJ, Wong MM: Connectedness and path-connectedness of solution sets for symmetric vector equilibrium problems. Taiwan. J. Math 2009, 13: 821–836.

    MathSciNet  Google Scholar 

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

    Google Scholar 

  34. Robertson AP, Robertson W: Topological Vector Spaces. Cambridge University Press; 1964.

    Google Scholar 

  35. Warburton AR: Quasi-concave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives. J. Optim. Theory Appl 1983, 40: 537–557. 10.1007/BF00933970

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (50874096, 70831005) and the Open Fund (PLN0904) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

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Correspondence to Nan-jing Huang.

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Authors' contributions

BC carried out the study of connectedness of ε-weak efficient and ε- efficient solutions sets for VEPs and drafted the manuscript. Q-YL participated in the design of the study. Z-BL gave some examples to show the main results. N-JH conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

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Chen, B., Liu, Qy., Liu, Zb. et al. Connectedness of approximate solutions set for vector equilibrium problems in Hausdorff topological vector spaces. Fixed Point Theory Appl 2011, 36 (2011). https://doi.org/10.1186/1687-1812-2011-36

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