0267 upper semicontinuity and closedness of the solution sets to parametric quasiequilibrium problems

UPPER SEMICONTINUITY AND CLOSEDNESS OF THE SOLUTION SETS TO PARAMETRIC QUASIEQUILIBRIUM PROBLEMS NGUYEN VAN HUNG*, PHAN THANH KIEU** ABSTRACT In this paper we establish sufficient conditions for the s[.]

UPPER SEMICONTINUITY AND CLOSEDNESS OF THE SOLUTION SETS

TO PARAMETRIC QUASIEQUILIBRIUM PROBLEMS

NGUYEN VAN HUNG * , PHAN THANH KIEU **

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 upper semicontinuity and closedness Our results improve recent existing ones in the literature.

Keywords: parametric quasiequilibrium problems, upper semicontinuity, closedness.

TÓM TẮT

Tính chất nửa liên tục trên và tính đóng của các tập nghiệm của các bài toán tựa cân bằng tổng quát phụ thuộc tham số

Trong bài báo này, chúng tôi thiết lập điều kiện đủ cho các tập nghiệm của các bài toán tựa cân bằng tổng quát phụ thuộc tham số có các tính chất ổn định như: tính nửa liên tục trên và tính đóng Kết quả của chúng tôi là cải thiện một số kết quả tồn tại gần đây trong danh sách tài liệu tham khảo.

Từ khóa: các bài toán tựa cân bằng tổng quát phụ thuộc tham số, tính nửa liên tục trên, tính

đóng

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 K : A ×Λ → 2 A,

K : A×Λ → 2 A, T : A × A×Γ → 2 B, C : A ×Λ→ 2 B and F : A × B × A× M → 2 Z be multifunctions with C( x, λ ) is a proper convex cone values and closed.

Now, 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

(u, v) w U ×V

(u, v) m U ×V

(u, v) s U ×V

means means means

∀u ∈U , ∃v ∈V ,

∃v ∈V , ∀u ∈U ,

∀u ∈U , ∀v ∈V ,

* MSc., Dong Thap University

** BA., Dong Thap University

Nguyen Van Hung et al.

Tạp chí KHOA HỌC ĐHSP TPHCM

1

1

2

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

(QEP λγµ ):

Find x ∈ K1(x , λ) such that ( y, t)α

K2

(x , λ) × T (x ,

y,γ ) statisfying

F (x , t, y, µ)

For λ ∈Λ, γ ∈Γ, µ∈

M consider the following parametric extended quasiequilibrium problem (in short, (QEEP λγµ )):

(QEEP λγµ ):

Find x ∈ K1(x , λ) such that ( y, t)α

K2

(x , λ) × T (x ,

y,γ ) statisfying

F (x , t, y, µ) ∩−intC(x, λ)=∅

For each λ ∈Λ, γ ∈Γ, µ∈ M , we let E(λ) := {x ∈ A | x ∈ K1(x,

Σα, Ξα : Λ× Γ× M → 2

A be set-valued mappings such that Σα(λ,γ ,

µ)

and Ξα(λ,γ , µ)

are the solution sets of (QEP λγµ ) and (QEEP λγµ ), respectively.

Throughout the paper we assume that Σα(λ,γ , µ) ≠∅

and

Ξα(λ,γ , µ) ≠

each (λ,γ , µ

neighborhoods

(λ0 ,γ 0 , µ0 ) ∈Λ×Γ× M

By the definition, the following relations are clear:

Σs ⊆Σm ⊆Σw and Ξs ⊆Ξm ⊆Ξw

Special cases of the problems (QEP λγµ ) and (QEEP λγµ ) are as follows:

(a) If T (x, y,γ ) = {t}, Λ= Γ= M , A = B, X = Y , K1 = K2

= K

and α = m , then (QEP λγµ ) and (QEEP λγµ ) become to (PGQVEP) and (PEQVEP), respectively in

α

Kimura-Yao [8]

(PGQVEP): Find

α

x ∈ K (x , λ ) such that and

F (x , y, λ)

⊂/ − int C( x , λ)), for all y ∈ K (x , λ ).

(PEQVEP): Find x ∈ K (x , λ ) such that

α α

α α

F (x , y, λ) ∩− int C(x , λ) = ∅, for all y ∈ K (x , λ).

(b) If T (x, y,γ ) = {t}, Λ= Γ, A = B, X = Y , K1 = clK , K2 = K ,α = m, C(x, λ)

replace "

⊆/ −λ int C(x, )" by " ⊆ Z ‚ − int C " with C ⊆

Z

be closed and int C ≠∅, then (QEP λγµ ) become to (SQEP) in Anh-Khanh [1].

(SQEP): Find x ∈ K (x , λ ) such that

F (x , y, λ ) ⊆ Z ‚ − int C, for all y ∈ K (x , λ)

(c) If T (x, y,γ ) = {t}, Λ= Γ= M , A = B, X = Y , K1 = K2 = K ,α

= m

and replace F

by

f be a vector function, then (QEP λγµ ) become to (PVQEP) in Kimura-Yao [7].

α

α

(PQVEP): Find x ∈ K (x , λ ) such that

f (x , y, λ) ∈/ − int C( x , λ)), for all y ∈ K (x, λ)

The parametric generalized quasiequilibrium problems include many rather general problems as particular cases as vector minimization, variational inequalities, Nash equilibria, fixedpoint and coincidence-point problems, complementarity problems, minimax inequalities, etc Stability properties of solutions have been investigated even in models for vector quasiequilibrium problems [1, 3, 4, 7, 8, 9], variational problems [5, 6, 10, 11] and the references therein

In this paper we establish sufficient conditions for the solution sets Σα,

the stability properties such as the upper semicontinuity and closedness 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 is devoted to the upper semicontinuity and closedness of solution sets for parametric quasiequilibrium problems (QEP λγµ ) and

(QEEP λγµ ).

Now we recall some notions in [1, 2, 12] Let X and Z be as above and

G : X → 2 Z

be a multifunction

G

is said to be lower semicontinuous (lsc) at x0 if

G(x0 ) ∩U ≠ ∅ for some open set U ⊆ Z implies the existence of a neighborhood N of

x0 such that, for all x ∈ N , G(x) ∩U ≠∅ An equivalent formulation is that: G is lsc at

x0

if ∀xα→ x0 , ∀z0 ∈ G(x0 ), ∃zα ∈ G(xα), zα → z0 G is called upper semicontinuous (usc) at x0 if for each open set U ⊇ G(x0 ) , there is a neighborhood N of x0 such that

U ⊇ G(N ) G is said to be Hausdorff upper semicontinuous (H-usc in short; Hausdorff lower semicontinuous, H-lsc, respectively) at x0 if for each neighborhood B of the origin in Z , there exists a neighborhood

N of x0 such that, G( x) ⊆ G( x0 ) + B,∀x ∈ N

( G( x0 ) ⊆ G( x) + B,∀x ∈ N )

G

is said to be continuous at x0 if it is both lsc and usc at

x0 and to be H-continuous at x0 if it is both H-lsc and H-usc at x0 We say that G satisfies a certain property in a

G

satisfies it at all points of A

Proposition 1.1 (See [1, 2, 12]) Let A and Z be as above and

(i) If

at

x0 then G is H -usc at x0 Conversely if G is H -usc

at

x0 and if

α α

G(x0 ) compact, then G is usc at x0 ;

(ii If G is usc

at

x0 ;

(iv) If G has compact values, then G is usc at x0 if and only if, for each net

{xα} ⊆

are

y ∈ G( x0 )

and a subnet {yβ} of {yα} such

that

yβ → y.

In this section, we discuss the upper semicontinuity and closedness of solution sets for parametric quasiequilibrium problems (QEP λγµα ) and (QEEP λγµα ).

Theorem 2.1.

(i) E is usc at

λ0 and E(λ0

) is compact, and

K2 is lsc in K1( A, Λ) ×{λ0} ;

1 ( A, Λ) × K2 (K1 ( A, Λ), Λ) ×{γ 0} ,

T

is usc and compact-valued

α = m ), and lsc if α = s ;

(iii) the set {(x, t, y, µ, λ) ∈ K1 ( A, Λ) ×T (K1 ( A, Λ), K2 (K1 ( A, Λ), Λ), Γ) ×

K2 (K1 ( A, Λ), Λ) ×{µ0}×{λ0}: F (x, t, y,

Then Σα is both upper semicontinuous and closed at (λ0 ,γ 0 , µ0 )

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

the cases where α = w We first prove that Σw is upper semicontinuous at (λ0 ,γ 0 , µ0 ) Indeed, we suppose to the contrary that

(λ0 ,γ 0 , µ0 ) , i.e., there is an open set U of Σw (λ0,γ 0, µ0) such that for all {(λn ,γ n , µn

)}

convergent to {(λ0 ,γ 0 , µ0 )}, there

exists x n ∈Σw (λn ,γ n , µn )

,

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 x0 ∈/ Σw (λ0,γ 0, µ0) , then ∃y0 ∈ K2 (x0 , λ0 ), ∀t0 ∈ T (x0 ,

y0 ,γ 0 )

such that

By the lower semicontinuity of K2 at (x0 , λ0 ) , there

exists

y n ∈ K2 (x n , λn

)

such α

that y n → y0 Since x n ∈Σw (λn ,γ n , µn ) , ∃t n ∈T (x n , y n ,γ

n )

such that

F (x n , t n , y n , µn )

)

(2.2)

Since

T is usc and T (x0 , y0 ,γ 0

)

is compact, one has a subnet t m ∈ T (x m , y m ,γ m )

such that t m → t0 for some t0 ∈T (x0 , y0 ,γ 0 )

By the condition (iii) we see a contradiction between (2.1) and (2.2) Thus,

x0 ∈Σw (λ0,γ 0, µ0) ⊆ U , this contradicts to the fact

semicontinuous at (λ0 ,γ 0 , µ0 )

x n ∈/

U ,

∀n Hence, Σw is upper

Now we prove that

Σw is closed at (λ0 ,γ 0 , µ0 ) Indeed, we suppose that

closed at (λ0 ,γ 0 , µ0 ) , i.e., there is

a net

(x n , λn ,γ n , µn ) → (x0 , λ0 ,γ 0 ,

µ0 )

with

x n ∈Σw (λn ,γ n , µn

) but x0 ∈/ Σw (λ0 ,γ 0 , µ0 ) The further argument is the same as

above

And so we have

Σw is closed at (λ0 ,γ 0 , µ0 ) □

The following example shows that the upper semicontinuity and compactness of

E are essential

Example 2.2.

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

F (x, t, y, λ) = 32 λ +sinx , K (x, λ) = (−λ −1, λ], K (x, λ) = {0} and T (x, y, λ) = [0, 23 x +2cosλ ]

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 2.1 are fulfilled But

nor closed at λ0 =

0)

is not compact The reason is that E is not usc at 0

and E (0) is not compact. In fact Σα(0, 0, 0) = (−1,

0]

and

Σα(λ,γ , µ) = (−λ − 1, λ ], ∀λ ∈ (0,1]

Remark 2.3.

The assumption in Theorem 2.1 we have

K2 is lsc in K1 ( A, Λ) ×{λ0} (which is not

imposed in this Theorem 4.1 of [8] and [7]) Example 2.4 shows that the lower semicontinuity of

Example 2.4.

K2 needs to be added to Theorem 4.1 of [8] and [7]

Let

X ,Y , Λ, Γ, M , λ0 , C(x,

λ) as in Example 2.2 and let A = B = [−1 , 1 ] ,

2 2

F (x, t, y, λ) = x + y +λ, K1(x, λ)

=

1

[0, ],T (x, y, 2 λ) = {t} We have

 1 1 

- ,0,  K (x, λ) =  

 2 2

if λ = 0,

 1











otherwise.

We have E(λ) = [0,1], ∀λ ∈[0,1] Hence E is usc at 0

condition (ii) and (iii) of Theorem 2.1 are easily seen to be fulfilled

But Σα is not upper semicontinuous at λ0 = 0 The reason is

semicontinuous In fact



 0, 2

 1







Σ (λ,γ , µ) = = 



if λ = 0,

α







1



2





otherwise.

The following example shows that the condition (iii) of Theorem 2.1 is essential

Example 2.5.

Let Λ, Γ, M ,T , λ0 ,

C as in Example 2.4 and

let

X = Y = A = B = [0,1] ,

K1 (x, λ) = K2 (x, λ) = [0,1] and

 x −

y F (x, t, y, λ) = 

2

if λ = 0,



 −

We show that assumptions (i) and (ii) of Theorem 2.1 are easily seen to be fulfilled

But Σα is not usc at λ0 = 0 The reason is that assumption (iii) is violated

Indeed, taking

x = 0, t = 0,

{(x , y , λ )} → 1 and F (x , t , y , = λ ) 1 = 1 > 0 , but

n n (0, , 0)

F (0, 0,1, 0) = − 1 < 0

4 The following example shows that all assumptions of Theorem 2.1 are fulfilled But Theorem 3.4 in Anh and Khanh [1] cannot be applied

Example 2.6.

Let A, B, X ,Y , Λ, Γ, M , λ0 ,

K1 (x, λ) = K2 (x, λ) = [0, 2],T (x, y,γ ) = [0,1]

λ = 0,

 2





 

y



ecos2 x+ 2 otherwise.

We show that assumptions (i), (ii) and (iii) of Theorem 2.1 are easily seen to be fulfilled Hence, Σα is usc at (0, 0, 0) But Theorem 3.4 in Anh and Khanh [1] cannot

be applied The reason is that F is not lsc at (x, y, 0)



Remark 2.7.

(i) In Theorem 4.1 in Kimura-Yao [8] the same conclusion as Theorem 2.1 was proved in another way Its assumptions (i)-(iv) derive (i) Theorem 2.1, assumptions (v)(or (vi)) coincides with (iii) of Theorem 2.1

(ii) In Theorem 4.1 in Kimura-Yao [7] the same conclusion as Theorem 2.1 was proved in

another way Its assumptions (i)-(iv) derive (i) Theorem 2.1, assumption (v) coincides with (iii) of Theorem 2.1

Theorem 2.8.

(i) E is usc at

d

E(λ0

and

K2 is lsc in

K1( A, Λ) ×{λ0} ;

(ii) i

n K1 ( A, Λ) × K2 (K1 ( A, Λ), Λ) ×{γ 0} ,

T

is usc and compact-valued if

α=

α= m ), and lsc if α = s ;

{(x, t, y, µ, λ) ∈ K1 ( A, Λ) ×T (K1 ( A, Λ), K2 (K1 ( A, Λ), Λ), Γ) ×

K2 (K1 ( A, Λ), Λ) ×{µ0}×{λ0}: F (x, t, y, µ) ∩− int C(x, λ) = ∅} is closed.

Then Ξα is both upper semicontinuous and closed at (λ0 ,γ 0 , µ0 )

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

the cases where α= m We first prove that Ξ

m

is upper semicontinuous at (λ0 ,γ 0 , µ0 ) Indeed, we suppose to the contrary that Ξ

m

is not upper semicontinuous at (λ0 ,γ 0 , µ0 ) ,

i.e., there is an open set V of Ξm (λ0,γ 0, µ0) such that for all {(λn ,γ n , µn )} convergent to

{(λ0 ,γ 0 , µ0 )}, there

exists x n ∈Ξm (λn ,γ n , µn )

,

x n ∈/

V ,

∀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

x0 ∈/ Ξm (λ0,γ 0, µ0) , then ∀t0 ∈T (x0 , y0 ,γ 0 ), ∃y0 ∈ K2

(x0 , λ0 )

F (x0 , t0 , y0 , µ0 ) ∩− int C(x0 , λ0 ) ≠ ∅

such that (2.3)

By the lower semicontinuity of K2 at (x0 , λ0 ) , there

exists

y n ∈ K2 (x n , λn

)

such

that y n → y0 Since x n ∈Ξm (λn ,γ n , µn ) , ∃t n ∈T (x n , y n ,γ

n )

such that α

F (x n , t n , y n , µn ) ∩− int C(x n , λn ) =

Since

T is usc and T (x0 , y0 ,γ 0

)

is compact, one has a subnet t m ∈ T (x m , y m ,γ m )

such that t m → t0 for some t0 ∈T (x0 , y0 ,γ 0 )

By the condition (iii) we see a contradiction between (2.3) and (2.4) Thus,

x0 ∈Ξm (λ0,γ 0, µ0) ⊆ V , this contradicts to the

fact

x n ∈/

V ,

∀n Hence, Ξm is upper semicontinuous at (λ0 ,γ 0 , µ0 ) Now we prove that Ξm is closed at (λ0 ,γ 0 , µ0 ) Indeed,

we suppose that Ξm is not closed at (λ0 ,γ 0 , µ0 ) ,i.e., there is a net

(x n , λn ,γ n , µn ) → (x0 , λ0 ,γ 0 ,

x n ∈Ξm (λn ,γ n , µn

)

but x0 ∈/ Ξm (λ0 ,γ 0 , µ0 ) The

further argument is the same as above And so we have Ξm

Remark 2.9.

is closed at (λ0 ,γ 0 , µ0 ) □

Theorem 2.8 is an extension of Theorem 4.1 in [8] The Example 2.3 is also shows that the lower semicontinuity of

Kimura-Yao in [8]

K2 needs to be added to Theorem 4.1 of

1 Anh L Q., Khanh P Q (2004), "Semicontinuity of the solution sets of parametric

multivalued vector quasiequilibrium problems", J Math Anal Appl., 294, pp

699-711

2 Berge C (1963), Topological Spaces, Oliver and Boyd, London.

3 Bianchi M., Pini R (2003), "A note on stability for parametric equilibrium

problems" Oper Res Lett., 31, pp 445-450.

4 Bianchi M., Pini R (2006), "Sensitivity for parametric vector equilibria",

Optimization., 55, pp 221-230.

5 Khanh P Q., Luu L M (2005), "Upper semicontinuity of the solution set of

parametric multivalued vector quasivariational inequalities and applications", J.

Glob.Optim., 32, pp 551-568.

6 Khanh P Q., Luu L M (2007), "Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational

inequalities", J Optim Theory Appl., 133, pp 329-339.

7 Kimura K., Yao J C (2008), "Sensitivity analysis of solution mappings of

parametric vector quasiequilibrium problems", J Glob Optim., 41 pp 187-202.

8 Kimura K., Yao J C (2008), "Sensitivity analysis of solution mappings of

parametric generalized quasi vector equilibrium problems", Taiwanese J Math., 9,

pp 2233-2268

9 Kimura K., Yao J C (2008), "Semicontinuity of Solution Mappings of parametric

Generalized Vector Equilibrium Problems", J Optim Theory Appl., 138, pp 429–

443

10 Lalitha C S., Bhatia Guneet (2011), "Stability of parametric quasivariational

inequality of the Minty type", J Optim Theory Appl., 148, pp 281-300.

11 Li S J., Chen G Y., Teo K L (2002), "On the stability of generalized vector

quasivariational inequality problems", J Optim Theory Appl., 113, pp 283-295.

12 Luc D T (1989), Theory of Vector Optimization: Lecture Notes in Economics and

Mathematical Systems, Springer-Verlag Berlin Heidelberg