Stability of the Solution Sets of Parametric Generalized Quasiequilibrium Problems

Abstract: In this paper we establish sufficient conditions for the solution mappings of parametric generalized vector quasiequilibrium problems to have the stability properties [r]

44

Stability of the Solution Sets of Parametric Generalized

Quasiequilibrium Problems

Nguyen Van Hung1, Phan Thanh Kieu2,*

1

Department of Mathematics, Dong Thap University, 783 Phạm Hữu Lầu, Cao Lãnh, Vietnam

2

Department of Personnel, Dong Thap University, 783 Phạm Hữu Lầu, Cao Lãnh, Vietnam

Received 15 January 2013 Revised 20 February 2013; accepted 08 March 2013

Abstract: In this paper we establish sufficient conditions for the solution mappings of parametric

generalized vector quasiequilibrium problems to have the stability properties such as lower semicontinuity, upper semicontinuity, Hausdorff lower semicontinuity, continuity, Hausdorff continuity and closedness The results presented in the paper improve and extend the main results

of Kimura-Yao [J Global Optim 138, (2008) 429 443], Kimura-Yao [Taiwanese J Math., 12, (2008) 649 669] and Anh-Khanh [J Math Anal Appl., 294, (2004) 699 711] Some examples are given to illustrate our results

Keywords: Parametric generalized quasiequilibrium problems, lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, continuity, Hausdorff continuity closedness

1 Introduction and preliminaries∗∗∗∗

Let X Y, , , ,Λ Γ M be Hausdorff topological spaces, let Z be a Hausdorff topological vector space, A⊆X and B⊆Y be nonempty sets Let K1:A× Λ →2A, K2:A× Λ →2A,

T A A× × Γ → , C A: × Λ →2B and F A B: × × ×A M →2Z be multifunction with C x( ) is closed with nonempty interiors different from Z

For the sake of simplicity, we adopt the following notations Letters w, m and s are used for a weak, middle and strong, respectively, kinds of considered problems For subsets U and V under consideration we adopt the notations

( , ) w u v U V× means ∀ ∈u U,∃ ∈v V ,

( , ) m u v U V× means ∃ ∈v V,∀ ∈u U ,

( , ) s u v U V× means ∀ ∈u U,∀ ∈v V,

_

∗

Corresponding author Tel.: 84- 918569966

E-mail: ptkieu@dthu.edu.vn

ρ1( , ) means U V U∩V ≠ ∅ ,

ρ2( , ) means U V U⊆V u v, ( , ),

( , )u v wU V× means ∃ ∈u U,∀ ∈v V and similarly for m s, ,

ρ1( , )U V means U∩V = ∅ and similarly for ρ2

Let α∈{w, m, s} and α∈{ , , }w m s We consider the following parametric quasiequilibrium problem (in short, (QEPαρ))

(QEPαρ): Find x∈K x1( , )λ such that ( , )y t αK x2( , )λ ×T x y( , , )γ satisfying

( ( , , , ); ( , )).F x t y C x

For each λ∈ Λ,γ ∈ Γ,µ∈M, we let E( ) : {λ = x∈A x| ∈K x1( , )}λ and let

αρ

Σ Λ × Γ × → be a set-valued mapping such that Σαρ( , , )λ γ µ is the solution set of (QEPαρ), i.e.,

2

( , , ) {x E( ) | ( , )y t K x( , ) T x y( , , ) : ( ( , , , ); ( , ))}.F x t y C x

Throughout the paper we assume that Σαρ( , , )λ γ µ ≠ ∅ for each ( , , )λ γ µ in the neighborhoods (λ γ µ0, 0, 0)∈ Λ × Γ ×M

Special cases of the problem (QEPαρ) are as follows:

(a) If T x y ( , , ) γ = { }, x Λ = Γ = M A , = B X , = Y K , 1= K2 = K , ρ = ρ2 and replace C x( , )λ

by −int ( , )C x λ , replace F by f be a vector function, then (QEP

2

α ρ ) become to (PVQEP) in Kimura-Yao [1]

(PQVEP): Find x∈K x( , )λ such that

( , , ) int ( , ), for all ( , )

f x y λ ∈ −/ C x λ y∈K x λ

(b) If T x y ( , , ) γ = { }, x Γ = M A , = B X , = Y K x , 1( , ) λ = K x2( , ) λ = K ( ), λ ρ = ρ2 and replace

( , )

C x λ by − int C, replace F by f be a vector function, then (QEP

2

α ρ ) become to (PVEP) in Kimura-Yao [2]

(PVEP): Find x∈K( )λ such that

( , , ) int , for all ( )

f x y γ ∈ −/ C y∈K λ (c) If T x y( , , )γ ={ },x Λ = Γ =M A, =B X, =Y K, 1=K2 =K, replace F by f be a vector function, then (QEP

2

αρ ) becomes (QEP) in Anh-Khanh [3]

(QEP): Find x∈K x( , )λ such that

( , , ) ( , ), ( , )

f x y λ ∈C x λ ∀ ∈y K x λ

(d) If T x y( , , )γ ={ },x Λ = Γ,A=B X, =Y K, 1 =clK K, 2 =K,ρ =ρ ρ1, =ρ2 and replace

( , )

C x λ by Z
−intC with C⊆Z be closed and int C ≠ ∅, then (QEP

1

αρ ) and (QEP

2

αρ ) become to (QEP) and (SQEP), respectively in Anh-Khanh [4]

(QEP): Find x∈clK x( , )λ such that

( , , ) ( int ) , for all ( , )

F x y λ ∩ Z
− C ≠ ∅ y∈K x λ

and

(SQEP): Find x∈K x( , )λ such that

( , , ) int , for all ( , )

F x y λ ⊆Z
− C y∈K x λ

In this paper we establish sufficient conditions for the solution sets Σαρ to have the stability properties such as the upper semicontinuity, the lower semicontinuity and the Hausdorff lower semicontinuity, continuity and Hausdorff continuity with respect to parameter λ γ µ, ,

The structure of our paper is as follows In the remaining part of this section we recall definitions for later uses Section 2, we establish sufficient conditions for the lower semicontinuity and the Hausdorff lower semicontinuity of solution sets of problems (QEPαρ), and Section 3 is devoted to the upper semicontinuity, continuity and Hausdorff continuity of solution sets of problems (QEPαρ) Now we recall some notions

Definition 1.1[5, 6]

Let X and Y be topological vector spaces and G X: →2Y be a multifunction

(i) G is said to be lower semicontinuous (lsc) at x0∈X if G x( 0)∩U ≠ ∅ for some open set

U ⊆ Y implies the existence of a neighborhood N of x0 such that G x( )∩U ≠ ∅ ∀ ∈, x N G is said to be lower semicontinuous in X if it is lower semicontinuous at each x0∈X

(ii) G is said to be upper semicontinuous (usc) at x0∈X if for each open set U ⊇G x( 0), there

is a neighborhood N of x0 such that U ⊇G x( ),∀ ∈x N G is said to be upper semicontinuous in

X if it is upper semicontinuous at each x0∈X

(iii) G is said to be Hausdorff upper semicontinuous (H-usc) at x0∈X if for each neighborhood

B of the origin in Z, there exists a neighborhood N of x0such that, G x( )⊆G x( 0)+B,∀ ∈x N

G is said to be Hausdorff upper semicontinuous in X if it is Hausdorff upper semicontinuous at each x0∈X

(iv) G is said to be Hausdorff lower semicontinuous (H-lsc) at x0∈X if for each neighborhood

B of the origin in Y , there exists a neighborhood N of x0 such that G x( 0)⊆G x( )+B,∀ ∈x N

G is said to be Hausdorff lower semicontinuous in X if it is Hausdorff lower semicontinuous at each x0∈X

(v) G is said to be continuous at x0∈X if it is both lsc and usc at x0 and to be H-continuous at

0

x ∈X if it is both H-lsc and H-usc at x0 G is said to be continuous in X if it is both lsc and usc

at each x0∈X and to be H-continuous in X if it is both H-lsc and H-usc at each x0∈X

(vi) G is said to be closed at x0∈X if and only if ∀x n →x0,∀y n → y0 such that y n∈G x( n),

we have y0∈G x( 0) G is said to be closed in X if it is closed at each x0∈X

Lemma 1.2 ([7, 8]) LetX and Y be topological vector spaces and : 2Y

multifunction

(i) If G is usc at x0 then G is H-usc at x0 Conversely if G is H-usc at x0 and if G x( )0

compact, then G is usc at x0;

(ii) If G is H-lsc at x0 then G is lsc at x0 The converse is true if G x( 0) is compact;

(iii) If G is usc at x0 and if G x( )0 is closed, then G is closed at x0;

(iv) If Z is compact and G is closed at x0 then G is usc at x0;

(v) If G has compact values, then G is usc at x0 if and only if, for each net { }xα ⊆ X which converges to x0 and for each net {yα}⊆G x( α), there are y∈G x( 0) and a subnet {yβ} of {yα}

such that yβ → y

2 Lower semicontinuity of solution set

In this section, we discuss the lower semicontinuity and the Hausdorff lower semicontinuity of solution sets for parametric generalized quasiequilibrium problems (QEPαρ)

Theorem 2.1 Assume for problem (QEPαρ) that

(i) E is lsc at λ0, K2 is usc and compact-valued in K A1( , ) { }Λ × λ0 ;

(ii) in K A1( , )Λ ×K K A2( 1( , ), ) { }Λ Λ × γ0 , T is usc and compact-valued if α = s, and lsc if

w

(iii )the set {( , , , , )x t y µ λ ∈K A1( , )Λ ×T K A( 1( , ),Λ K K A2( 1( , ), ), )Λ Λ Γ ×

2( 1( , ), ) { } { }: ( ( , , , ); ( , ))}0 0

K K A Λ Λ × µ × λ ρ F x t y µ C x λ is closed

Then Σαρ is lower semicontinuous at (λ γ µ0, 0, 0)

Proof Since α ={ , , }w m s and ρ ={ ,ρ ρ1 2}, we have in fact six cases However, the proof techniques are similar We consider only the cases α =s,ρ =ρ2 Suppose to the contrary that

2

sρ

Σ is not lsc at (λ γ µ0, 0, 0), i.e.,

2

0 s ( 0, 0, 0)

∃ ∈ Σ , ∃(λ γ µn, n, n)→(λ γ µ0, 0, 0),

n s n n n n

∀ ∈ Σ →/ Since E is lsc at λ0, there is a net x n′ ∈E(λn), x n′ →x0 By the

above contradiction assumption, there must be a subnet x′ m of x′ n such that, ∀ m,

m s m m m

x′ ∈ Σ/ ρ λ γ µ , i.e., ∃y m∈K x2( m′,λm),∃ ∈t m T x( m′,y m,γm) such that

(F x t m′, ,m y m,µm)⊆/ C x( m′,λm) (2.1)

As K2 is usc at (x0,λ0) and K x2( ,0 λ0) is compact, one has y0∈K x2( 0,λ0) such that 0

m

y → y (taking a subnet if necessary) By the lower semicontinuity of T at (x y0, 0,γ0), one has

( , , )

m m m m

t ∈T x y γ such that t m→t0 Since (x t m′, ,m y m,λ γm, m,µm)→( , ,x t y0 0 0,λ γ µ0, 0, 0)and by condition (iii) and (2.1) yields that

( , , , ) ( , )

F x t y µ ⊆/ C x λ

which is impossible since

2

0 s ( 0, 0, 0)

x ∈ Σ ρ λ γ µ Therefore,

2

sρ

Σ is lsc at (λ γ µ0, 0, 0)

The following example shows that the lower semicontinuity of E is essential

Example 2.1 Let A=B= X =Y =,Λ = Γ =M =[0,1],λ0 =0, ( , )C x λ =[0, )∞ ,

2

( , , , ) 2 , ( , , ) { }, ( , ) [0,1]

F x t y λ = λ T x y λ = x K x λ = and

1

( , )

K x

oth wise

λ λ

λ



=

− −



We have E(0)= −[ 1,1],∀ ∈λ (0,1], E( )λ = − −[ λ 1, 0],∀ ∈λ (0,1] Hence K2 is usc and the conditions (ii) and (iii) of Theorem 2.1 are easily seen to be fulfilled But Σαρ is not upper semicontinuous at λ0 =0 The reason is that E is not lower semicontinuous In fact

(0, 0, 0) [ 1,1]

αρ

Σ = − and Σαρ( , , )λ γ µ = − −[ λ 1, 0],∀ ∈λ (0,1]

The following example shows that in this the special case, assumption (iii) of Theorem 2.1 may

be satistied even in cases, but both assumptions (ii1) and (iii1) of Theorem 2.1 in Anh-Khanh [4] are not fulfilled

Example 2.2 Let A B X Y T, , , , , , ,Λ Γ M,λ0,C as in Example 2.1, and let

1( , ) 2( , ) [0,1]

( , , )

F x y

oth wise

λ λ

λ



=

− −



We shows that the assumptions (i), (ii) and (iii) of Theorem 2.1 are satisfied and

( , , )) [0,1], [0,1]

Σ = ∀ ∈ But both assumptions (ii1) and (iii1) of Theorem 2.1 in

Anh-Khanh [4] are not fulfilled

The following example shows that in this the special case, assumption of Theorem 2.1 may

be satistied, but Theorem 2.1 and Theorem 2.3 in Anh-Khanh [4] are not fulfilled

Example 2.3 Let A B X Y T, , , , , , ,Λ Γ M,λ0,C as in Example 2.2 and let

1( , ) 2( , ) [0, ]

2

[0, 1] if 0, ( , , , )

F x t y

oth wise

λ



We show that the assumptions (i), (ii) and (iii) of Theorem 2.1 are satisfied and

( , , )) [0, ], [0,1]

2

αρ

λ

Σ = ∀ ∈ Theorem 2.1 and Theorem 2.3 in Anh-Khanh [4] are not fulfilled The reason is that F is neither usc nor lsc at ( , , 0)x y

Remark 2.7 In cases as in Section 1 (a), (b) and (c) Then, Theorem 5.1, 5.2 and 5.3 in [1]

Theorem 5.1, 5.2, 5.3 and 5.4 in [2], Theorem 3.1 in [3] are particular cases of Theorem 2.1

Theorem 2.2 Impose the assumption of Theorem 2.1 and the following additional conditions:

(iv) K2 is lsc in K A1( , ) { }Λ × λ0 and E(λ0) is compact;

(v) the set {( , , )x t y ∈K A1( , )Λ ×T K A( 1( , ),Λ K K A2( 1( , ), ), )Λ Λ Γ ×

2( 1( , ), ) : ( ( , , , 0); ( , 0))}

K K A Λ Λ ρ F x t y µ C x λ is closed

Then Σαρ is Hausdorff lower semicontinuous at (λ γ µ0, 0, 0)

Proof We consider only for the cases α =s,ρ =ρ2 We first prove that

2( 0, 0, 0)

sρ λ γ µ

closed Indeed, we let

2( 0, 0, 0)

n s

x ∈ Σ ρ λ γ µ such that x n →x0 If

2

0 s ( 0, 0, 0)

x ∈ Σ/ ρ λ γ µ ,

0 2( 0, 0), 0 ( 0, 0, 0)

F x t y( 0, ,0 0,µ0)⊆/ C x( 0,λ0) (2.2)

By the lower semicontinuity of K2(.,λ0) at x0, one has y n∈K x2( n,λ0) such that y n → y0 Since

2( 0, 0, 0)

n s

x ∈ Σ ρ λ γ µ , we have

F x t y( n, ,n n,µ0)⊆C x( n,λ0) (2.3)

By the condition (v), we see a contradiction between (2,2) and (2.3) Therefore,

2( 0, 0, 0)

sρ λ γ µ

Σ

is closed

On the other hand, since

sρ λ γ µ E λ

2( 0, 0, 0)

sρ λ γ µ

2

sρ

Σ is lower semicontinuous at (λ γ µ0, 0, 0) and

2( 0, 0, 0)

sρ λ γ µ

Σ

is compact Hence

2

sρ

Σ is Hausdorff lower semicontinuous at (λ γ µ0, 0, 0) And so we complete the proof

3 Upper semicontinuity of solution set

In this section, we discuss the upper semicontinuity, continuity and H-continuity of solution sets for parametric generalized quasiequilibrium problems (QEPαρ)

Theorem 3.1 Assume for problem (QEPαρ) that

(i) E is usc at λ0 and E(λ0) is compact, and K2 is lsc in K A1( , ) { }Λ × λ0 ;

(ii) in K A1( , )Λ ×K K A2( 1( , ), ) { }Λ Λ × γ0 , T is usc and compact-valued if α = w (or α = m), and lsc if α = s;

(iii) the set {( , , , , )x t y µ λ ∈K A1( , )Λ ×T K A( 1( , ),Λ K K A2( 1( , ), ), )Λ Λ Γ ×

2( 1( , ), ) { 0} { } : ( ( , , , ); ( , ))}0

K K A Λ Λ × µ × λ ρ F x t y µ C x λ is closed

Then Σαρ is both usc and closed at (λ γ µ0, 0, 0)

Proof Similar arguments can be applied to six cases We present only the proof for the cases

where α =w,ρ =ρ2 We first prove that

2

wρ

Σ is upper semicontinuous at (λ γ µ0, 0, 0) Indeed, we suppose to the contrary that

2

wρ

Σ is not upper semicontinuous at (λ γ µ0, 0, 0), i.e.,there is an open set

U of

2( 0, 0, 0)

wρ λ γ µ

Σ such that for all {(λ γ µn, n, n)} convergent to {(λ γ µ0, 0, 0)}, there exists

n w n n n

x ∈ Σ ρ λ γ µ , x n∈/U, ∀ n By the upper semicontinuity of E and compactness of E(λ0), one can assume that x n →x0 for some x0∈E(λ0) If

2

0 w ( 0, 0, 0)

x ∈ Σ/ ρ λ γ µ , then

0 2( 0, 0), 0 ( 0, 0, 0)

F x t y( 0, ,0 0,µ0)⊆/ C x( 0,λ0) (3.1)

By the lower semicontinuity of K2 at (x0,λ0), y n∈K x2( n,λn) such that y n →y0 Since

n w n n n

x ∈ Σ ρ λ γ µ , ∃ ∈t n T x y( n, n,γn) such that

F x t y( n, ,n n,µn)⊆C x( n,λn) (3.2)

Since T is usc and T x y( 0, 0,γ0) is compact, one has a subnet t m∈T x( m,y m,γm) such that 0

m

t →t

for some t0∈T x y( ,0 0,γ0)

By the condition (iii) we see a contradiction between (3.1) and (3.2) Thus,

2

0 w ( 0, 0, 0)

x ∈ Σ ρ λ γ µ ⊆U , this contradicts to the fact x n∈/U, ∀ n Hence,

2

wρ

Σ is upper semicontinuous at (λ γ µ0, 0, 0)

Now we prove that

2

wρ

Σ is closed at (λ γ µ0, 0, 0) Indeed, we suppose that

2

wρ

Σ is not closed at

(λ γ µ, , ), i.e., there is a net (x n,λ γ µn, n, n)→( ,x0 λ γ µ0, 0, 0) with

n w n n n

x ∈ Σ ρ λ γ µ but

2

0 w ( 0, 0, 0)

x ∈ Σ/ ρ λ γ µ The further argument is the same as above And so we have

2

wρ

Σ is closed at

(λ γ µ, , )

The following example shows that the upper semicontinuity and compactness of E are essential

Example 3.1 Let A=B= X =Y =,Λ = Γ =M =[0,1],λ0 =0, ( , )C x λ =[0, )∞ ,

+sinx

( , , , ) 2 , ( , ) ( 1, ], ( , ) { 1}

F x t y λ = λ K x λ = − −λ λ K x λ = − and T x y( , , )λ =[0,e2 +cosx λ]

Then, we have E(0)= −( 1, 0] and E( )λ = − −( λ 1, ],λ ∀ ∈λ (0,1] We show that K2 is lsc and assumption (ii) and (iii) of Theorem 3.1 are fulfilled But Σαρ is neither usc nor closed at λ0 =0 and

(0, 0, 0)

αρ

Σ is not compact The reason is that E is not usc at 0 and E(0) is not compact In fact

(0, 0, 0) ( 1, 0]

αρ

Σ = − and Σαρ( , , )λ γ µ = − −( λ 1, ],λ ∀ ∈λ (0,1]

Remark 3.1

(i) In Theorem 4.1 in Kimura-Yao [1] the same conclusion as Theorem 3.1 was proved in anther way Its assumptions (i)-(iv) derive (i) Theorem 3.1 assumption (v) coincides with (iii) of Theorem 3.1 (ii) The assumption in Theorem 3.1, we have K2 is lsc in K A1( , ) { }Λ × λ0 (which is not imposed in this Theorem 4.1 of [1]) Example 3.2 shows that the lower semicontinuity of K2

needs to be added to Theorem 4.1 of [1]

Example 3.2 Let X Y, , , ,Λ Γ M,λ0, ( , )C x λ as in Example 3.1 and let [ 1 1, ]

2 2

A=B= − ,

1

1

2

2

1 1 , 0, if 0,

2 2 ( , )

1

2

K x

oth wise

λ λ

= 





We have E( )λ =[0,1],∀ ∈λ [0,1] Hence E is usc at 0 and E(0) is compact and condition (ii) and (iii) of Theorem 3.1 are easily seen to be fulfilled But Σαρ is not upper semicontinuous at

λ = The reason is that K2 is not lower semicontinuous

The following example shows a case where the assumed compactness in Theorem 4.1 of [1]

is violated but the assumptions of Theorem 3.1 are fulfilled

Example 3.3 Let X Y, , , ,Λ Γ M T, ,λ0,C, as in Example 3.2 and let A=B=[0, 2),

( , , )

F x y λ =x−y and K x1( , )λ =K x2( , )λ =(x−λ−1]∩A We show that the assumptions of Theorem 3.1 are easily seen to be fulfilled and so Σαρ is usc and closed at (0, 0, 0), although A is not compact

The following example shows that the condition (iii) of Theorem 3.1 is essential

Example 3.4 Let Λ Γ, ,M T, ,λ0,C as in Example 3.2 and let X =Y = A=B=[0,1],

K x1( , )λ =K x2( , )λ =[0,1] and

( , , )

F x y

λ

−



We show that assumptions (i) and (ii) of Theorem 3.1 are easily seen to be fulfilled But Σαρ is not usc at λ0 =0 The reason is that assumption (iii) is violated

Indeed, taking 1

n

λ

= = = → as n→ ∞, then {(x y n, n,λn}→(0,1, 0) and

( n, n, n) (0,1,1 / ) 1 0

F x y λ =F n = > but F(0,1, 0)= − <1 0

The following example shows that all assumptions of Theorem 3.1 are fulfilled But Theorem 3.2-3.4 in Anh and Khanh [4] cannot be applied

Example 3.5 Let A B X Y, , , , , ,Λ Γ M,λ0,C as in Example 3.1 and let

1( , ) 2( , ) [0, 3 ]

K x λ =K x λ = λ , T x y ( , , ) γ = [0, 2cos x+sin x+26 4 ] and

{ }

sin cos 1

( , , , )

er

x x

F x t y

λ λ

=



=



We show that assumptions (i), (ii) and (iii) of Theorem 3.1 are easily seen to be fulfilled But αρ

Σ is usc at (0, 0, 0) But Theorem 3.2-3.4 in Anh and Khanh [4] cannot be applied The reason is that F is neither usc nor lsc

Remark 3.2 In cases as in Section 1 (b) Then, Theorem 4.1 and 4.2 in [2] are particular cases of

Theorem 3.1

Theorem 3.2 Suppose that all conditions in Theorem 2.1 and Theorem 3.1 are satisfied Then,

we have Σαρ is both continuous and closed at (λ γ µ0, 0, 0)

Theorem 3.3 Suppose that all conditions in Theorem 2.2 and Theorem 3.1 are satisfied Then,

we have Σαρ is both H-continuous and closed at (λ γ µ0, 0, 0)

References

[1] K Kimura and J C Yao (2008), Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems, J Glob Optim., 41 pp 187-202

[2] K Kimura and J C Yao (2008), Sensitivity analysis of vector equilibrium problems, Taiwanese J Math., 12, pp 649-669

[3] L Q Anh and P Q Khanh (2010), Continuity of solution maps of parametric quasiequilibrium problems J Glob Optim 46, pp 247-259

[4] L Q Anh and P Q Khanh (2004), Semicontinuity of the solution sets of parametric multivalued vector quasiequilibrium problems, J Math Anal Appl., 294, pp 699-711

[5] N V Hung (2013), Stability of solution set for parametric generalized vector mixed quasivariational inequality problem J Inequal Appl., accepted,

[6] K Kimura and J C Yao (2008), Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J Optim Theory Appl., 138, pp 429– 443

[7] K Kimura and J C Yao (2008), Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J Math., 9, pp 2233-2268

[8] D T Luc (1989), Theory of Vector Optimization: Lecture Notes in Economics and Mathematical Systems, Springer-Verlag Berlin Heidelberg.