Tạp chí KHOA HỌC ĐHSP TPHCM

Nguyen Van Hung

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LOWER SEMICONTINUITY OF THE SOLUTION SETS

OF PARAMETRIC GENERALIZED QUASIEQUILIBRIUM PROBLEMS

NGUYEN VAN HUNG*

ABSTRACT

In this paper we establish sufficient conditions for the solution sets of parametric

generalized quasiequilibrium problems with the stability properties such as lower

semicontinuity and Hausdorff lower semicontinuity.

Keyword: parametric generalized quasiequilibrium problems, lower semicontinuity,

Hausdorff lower semicontinuity.

TÓM TẮT

Tính chất nửa liên tục dưới 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 dưới và tính nửa liên tục dưới Hausdorff.

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

dưới, tính nửa liên tục dưới Hausdorff.

1.

Introduction and Preliminaries

Let X , Y , Λ, Γ, M be a Hausdorff topological spaces, let Z be a Hausdorff

topological vector space, A ⊆ X and B ⊆ Y be a nonempty sets. Let K1 : A× Λ → 2 A ,

K 2 : A× Λ → 2 A , T : A × A × Γ → 2 B , C : A× Λ → 2 B and F : A × B × A × M → 2 Z

multifunctions with C is a proper solid convex cone values and closed.

be

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

ubsets U and V under consideration we adopt the notations.

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

(u, v) m U × V

*

means

∃v ∈ V , ∀u ∈ U ,

(u, v) s U × V

means

∀u ∈ U , ∀v ∈ V ,

ρ1 (U , V )

means

U ∩V ≠ ∅ ,

ρ 2 (U , V )

means

U ⊆V ,

(u, v) wU × V

means

∃u ∈ U , ∀v ∈ V and similarly for m, s ,

MSc., Dong Thap University

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ρ1 (U , V )

means

U ∩ V = ∅ and similarly for ρ 2 .

Let α ∈ {w, m, s} , α ∈ {w, m, s } , ρ ∈ {ρ1 , ρ 2 } and ρ ∈ {ρ1 , ρ 2 } . We consider the

following parametric generalized quasiequilibrium problems.

(QEP αρ ): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying

ρ ( F ( x , t , y, µ ); C ( x , λ )).

*

We consider also the following problem (QEP αρ

) as an auxiliary problem to

(QEP αρ ):

*

): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying

(QEP αρ

ρ ( F ( x , t , y, µ );int C ( x , λ )).

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

%αρ : Λ × Γ × M → 2 A be a set-valued mapping such that Σ (λ , γ , µ ) and

Σαρ , Σ

αρ

%αρ (λ , γ , µ ) are the solution sets of (QEP ) and (QEP * ), respectively, i.e.,

Σ

αρ

αρ

Σαρ (λ , γ , µ ) = {x ∈ E (λ ) | ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) : ρ ( F ( x , t , y, µ ); C ( x , λ ))},

%αρ (λ , γ , µ ) = {x ∈ E (λ ) | ( y, t )α K ( x , λ ) × T ( x , y, γ ) : ρ ( F ( x , t , y, µ );int C ( x , λ ))}.

Σ

2

Clearly Σ%αρ (λ , γ , µ ) ⊆ Σαρ (λ , γ , µ ) . Throughout the paper we assume that

%αρ (λ , γ , µ ) ≠ ∅ for each (λ , γ , µ ) in the neighborhood of

Σαρ (λ , γ , µ ) ≠ ∅ and Σ

(λ0 , γ 0 , µ0 ) ∈ Λ × Γ × M .

By the definition, the following relations are clear:

% sρ ⊆ Σ

% mρ ⊆ Σ

% wρ .

Σ ⊆Σ ⊆Σ

and Σ

sρ

mρ

wρ

The parametric generalized quasiequilibrium problems is more general than many

following problems.

(a) If T ( x, y, γ ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 , ρ = ρ1 and

replace C ( x, λ ) by − int C ( x, λ ) . Then, (QEP α ρ2 ) and (QEP α ρ1 ) becomes to (PGQVEP)

and (PEQVEP), respectively, in Kimura-Yao [7].

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

F ( x , y, λ ) ⊂/ − intC ( x , λ )), for all y ∈ K ( x, λ ).

and

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

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

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(b) If T ( x, y, γ ) = {x}, Λ = Γ, A = B, X = Y , K1 = clK , K 2 = K , ρ = ρ1 , ρ = ρ 2 and

replace C ( x, λ ) by Z \ − int C with C ⊆ Z be closed and int C ≠ ∅ . Then, (QEP αρ1 ) and

(QEP αρ2 ) becomes to (QEP) and (SQEP), respectively, in Anh - Khanh [1].

(QEP): Find x ∈ clK ( x , λ ) such that

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

and

(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 ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 and replace

C ( x, λ ) by − int C ( x, λ ) , replace F by f be a vector function. Then, (QEP α ρ )

2

becomes to (PVQEP) in Kimura-Yao [6].

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

f ( x , y, λ ) ∈

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

Note that generalized quasiequilibrium problems encompass many optimizationrelated models like vector minimization, variation inequalities, Nash equilibrium, fixed

point and coincidence-point problems, complementary problems, minimum

inequalities, etc. Stability properties of solutions have been investigated even in models

for vector quasiequilibrium problems [1, 2, 3, 6, 7, 8], variation problems [4, 5, 9, 10]

and the references therein.

In this paper we establish sufficient conditions for the solution sets Σαρ to have

the stability properties such as the lower semicontinuity and the Hausdorff lower

semicontinuity with respect to parameter λ , γ , µ under relaxed assumptions about

generalized convexity of the map F .

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 lower semicontinuity and the

Hausdorff lower semicontinuity of solution sets of problems (QEP αρ ).

Now we recall some notions. Let X and Z be as above and G : X → 2Z 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 ) . Q 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, Q( x) ⊆ Q( x0 ) + B, ∀x ∈ N

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( Q( x0 ) ⊆ Q( 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 . G is called closed

at x0 if for each net {( xα , zα )} ⊆ graphG := {( x, z )∣ z ∈ G ( x)}, ( xα , zα ) → ( x0 , z0 ) , z0 must

belong to G ( x0 ) . The closeness is closely related to the upper (and Hausdorff upper)

semicontinuity. We say that G satisfies a certain property in a subset A ⊆ X if G

satisfies it at every points of A . If A = X we omit ``in X " in the statement.

Let A and Z be as above and G : A → 2Z be a 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 ( x0 ) compact, then G usc at x0 ;

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

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

{xα } ⊆ A which converges to x0 and for each net { yα } ⊆ G ( xα ) , there are y ∈ G ( x) and

a subnet { yβ } of { yα } such that yβ → y.

Definition. (See [1], [11]) Let X and Z be as above. Suppose that A is a nonempty

convex set of X and that G : X → 2Z be a multifunction.

(i) G is said to be convex in A if for each x1 , x2 ∈ A and t ∈ [0,1]

G (tx1 + (1 − t ) x2 ) ⊃ tG ( x1 ) + (1 − t )G ( x2 )

(ii) G is said to be concave A if for each x1 , x2 ∈ A and t ∈ [0,1]

G (tx1 + (1 − t ) x2 ) ⊂ tG ( x1 ) + (1 − t )G ( x2 )

2.

Main results

In this section, we discuss the lower semicontinuity and the Hausdorff lower

semicontinuity of solution sets for parametric generalized quasiequilibrium problems

(QEP αρ ).

Definition 2.1

Let A and Z be as above and C : A → 2 Z with a proper solid convex cone values.

Suppose G : A → 2Z . We say that G is generalized C -concave in A if for each

x1 , x2 ∈ A , ρ (G ( x1 ), C ( x1 )) and ρ (G ( x2 ),int C ( x2 )) imply

ρ (G (tx1 + (1 − t ) x2 ),int C (tx1 + (1 − t ) x2 )), for all t ∈ (0,1).

Theorem 2.2

Assume for problem (QEP αρ ) that

(i) E is lsc at λ0 , K 2 is usc and compact-valued in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is

convex;

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(ii) in K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ) × {γ 0 } , T is usc and compact-valued if α = s ,

and lsc if α = w (or α = m );

(iii) ∀t ∈ T ( K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ ), Γ), ∀µ0 ∈ M , ∀λ0 ∈ Λ , K 2 (., λ0 ) is concave

in

K1 ( A, Λ )

and

F (., t ,., µ0 )

is

generalized

C (., λ0 ) -concave

in

K1 ( A, Λ ) × K 2 ( K1 ( A, Λ), Λ ) ;

(iv) the set {(x, t, y, µ, λ) ∈ K1( A, Λ) ×T (K1( A, Λ), K2 (K1( A, Λ), Λ), Γ) × K 2 ( K1 ( A, Λ), Λ ) ×

{µ0 } × {λ0 }: ρ ( F ( x, t , y, µ ); C ( x, λ ))} is closed.

Then Σαρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .

Proof.

Since α = {w, m, s} and ρ = {ρ1 , ρ 2 } , we have in fact six cases. However, the

proof techniques are similar. We consider only the cases α = s, ρ = ρ 2 . We prove that

% s ρ is lower semicontinuous at (λ , γ , µ ) . Suppose to the contrary that Σ

% s ρ is not lsc

Σ

2

2

0

0

0

at (λ0 , γ 0 , µ0 ) , i.e., ∃x0 ∈ Σ% sρ2 (λ0 , γ 0 , µ0 ) , ∃(λn , γ n , µn ) → (λ0 , γ 0 , µ0 ) , ∀xn ∈ Σ% sρ2 (λn , γ n , µn ),

xn →

/ x0 . Since E is lsc at λ0 , there is a net xn′ ∈ E (λn ) , xn′ → x0 . By the above

contradiction assumption, there must be a subnet xm′ of xn′ such that, ∀m ,

% sρ (λ , γ , µ ) , i.e., ∃y ∈ K ( x′ , λ ) , ∃t ∈ T ( x′ , y , γ ) such that

x′ ∈

/ Σ

m

2

m

m

m

m

2

m

m

m

m

m

m

F ( xm′ , tm , ym , µm ) ⊆/ int C ( xm′ , λm ).

(2.1)

As K 2 is usc at ( x0 , λ0 ) and K 2 ( x0 , λ0 ) is compact, one has y0 ∈ K 2 ( x0 , λ0 ) such

that ym → y0 (taking a subnet if necessary). By the lower semicontinuity of T at

( x0 , y0 , γ 0 ) ,

one has tm ∈ T ( xm , ym , γ m ) such that tm → t0 .

Since ( xm′ , tm , ym , λm , γ m , µm ) → ( x0 , t0 , y0 , λ0 , γ 0 , µ0 ) and by condition (iv) and (2.1)

yields that

F ( x0 , t0 , y0 , µ0 ) ⊆/ int C ( x0 , λ0 ) ,

which is impossible since x0 ∈ Σ% sρ (λ0 , γ 0 , µ0 ) . Therefore, Σ% s ρ is lsc at (λ0 , γ 0 , µ0 ) .

2

2

Now we check that

% sρ (λ , γ , µ )).

Σ s ρ2 (λ0 , γ 0 , µ0 ) ⊆ cl(Σ

2

0

0

0

Indeed, let x1 ∈ Σ s ρ (λ0 , γ 0 , µ0 ) , x2 ∈ Σ% sρ (λ0 , γ 0 , µ0 ) and xα = (1− t ) x1 + tx2 , t ∈ (0,1) .

2

2

By the convexity of E , we have xα ∈ E (λ0 ) . By the generalized C (., λ0 ) -concavity of

F (., t , y, µ0 ) , we have

F ( xα , t , y, µ0 ) ⊆ int C ( xα , λ0 ),

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and since K 2 (., λ0 ) is concave, one implies that for each yα ∈ K 2 ( xα , λ0 ) , there exist

y1 ∈ K 2 ( x1 , λ0 ) and y2 ∈ K 2 ( x2 , λ0 ) such that yα = ty1 + (1 − t ) y2 . By the generalized

C (., λ0 ) -concavity of F (., t ,., µ0 ) , we have

F ( xα , t , yα , µ0 ) ⊆ int C ( xα , λ0 ),

i.e., xα ∈ Σ% s ρ (λ0 , γ 0 , µ0 ) . Hence Σ s ρ (λ0 , γ 0 , µ0 ) ⊆ cl(Σ% s ρ (λ0 , γ 0 , µ0 )) . By the lower

2

2

2

semicontinuity of Σ% s ρ at (λ0 , γ 0 , µ0 ) , we have

2

% sρ (λ , γ , µ )) ⊆ lim inf Σ

% sρ (λ , γ , µ ) ⊆ lim inf Σ (λ , γ , µ ),

Σ sρ2 (λ0 , γ 0 , µ0 ) ⊆ cl (Σ

2

2

0

0

0

n

n

n

sρ2

n

n

n

i.e., Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .

2

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

Example 2.3

Let A = B = X = Y = Z = , Λ = Γ = M = [0,1], λ0 = 0, C ( x, λ ) = [0, +∞ ) and let

F ( x, t , y, λ ) = 2λ , T ( x, y, λ ) = {x}, K 2 ( x, λ ) = [0,1]

and

⎧[-1,1]

K1 ( x, λ ) = ⎨

⎩[-1-λ , 0]

if λ = 0,

otherwise.

We have E (0) = [−1,1] , E (λ ) = [−λ − 1, 0], ∀λ ∈ (0,1] . Hence K 2 is usc and the

condition (ii), (iii) and (iv) of Theorem 2.2 is 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 Σαρ (0, 0, 0) = [−1,1] and Σαρ (λ , γ , µ ) = [−λ − 1, 0], ∀λ ∈ (0,1] .

The following example shows that in this the special case, assumption (iv) of

Theorem 2.2 may be satisfied even in cases, but both assumption (ii 1 ) and (iii 1 ) of

Theorem 2.1 in Anh-Khanh [1] are not fulfilled.

Example 2.4

Let A, B, X , Y , Z , T , Λ, Γ, M , λ0 , C as in Example 2.3, and let K1 ( x, λ ) =

K 2 ( x, λ ) = [0,1] and

⎧[-4,0]

K1 ( x, λ ) = ⎨

⎩[-1-λ , 0]

if λ = 0,

otherwise.

We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and

Σαρ (λ , γ , µ ) = [0,1], ∀λ ∈ [0,1] . But both assumption (ii 1 ) and (iii 1 ) of Theorem 2.1

in Anh-Khanh [1] are not fulfilled.

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

Theorem2.2 may be satisfied even in cases, but Theorem 2.1 and Theorem 2.3 in AnhKhanh [1] are not fulfilled.

Example 2.5

Let A, B, X , Y , T , Λ, Γ, M , λ0 , C as in Example 2.4, and let K1 ( x, λ ) = K 2 ( x, λ ) =

λ

[0, ] and

2

⎧[0,1]

K1 ( x, λ ) = ⎨

⎩[2, 4]

if λ = 0,

otherwise.

We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and

λ

Σαρ (λ , γ , µ )) = [0, ], ∀λ ∈ [0,1] . Theorem 2.1 and Theorem 2.3 in Anh-Khanh [1] are

2

not fulfilled. The reason is that F is neither usc nor lsc at ( x, y, 0) .

Remark 2.6

In special cases, as in Section 1 (a) and (c). Then, Theorem 2.2 reduces to

Theorem 5.1 in Kimura-Yao [7, 6]. However, the proof of the theorem 5.1 is in a

different way. Its assumption (i) - (v) of Theorem 5.1 coincides with (i) of Theorem 2.2

and assumption (vi), (vii) coincides with (iii), (iv) of Theorem 2.2 Theorem 2.2 slightly

improves Theorem 5.1 in Kimura-Yao [7, 6], since no convexity of the values of E is

imposed.

The following example shows that the convexity and lower semicontinuity of K

is essential.

Example 2.7

Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.5 and let

⎧⎪{−1, 0,1}

K1 ( x, λ ) = ⎨

⎪⎩{0,1}

if λ = 0,

otherwise.

Then, we shows that K 2 is usc and has compact-valued K1 ( X , A) × {λ0 } and assumption

(ii), (iii) and (iv) of Theorem 2.2 are fulfilled. But Σαρ (λ , γ , µ )) is not lsc at (0, 0, 0) .

The reason is that E is not lsc at λ0 = 0 and E (0) is also not convex. Indeed, let

1

x1 = −1, x2 = 0 ∈ E (0) and t = ∈ (0,1) but tx1 + (1 − t ) x2 ∈

/ E (0) .

2

In fact, Σαρ (0, 0, 0) = {−1, 0,1} and Σαρ (λ , γ , µ ) = {0,1}, ∀λ ∈ (0,1] .

The following example shows that the concavity of F (., t., µ0 ) is essential.

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Example 2.8

Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.6 and let K1 ( x, λ ) = K 2 ( x, λ )

= [λ , λ + 3] and F ( x, t , y, µ ) = F ( x, y, λ ) = x 2 − (1 + λ ) x . We show that K 2 (., λ0 ) is

concave and the assumptions (i), (ii), (iv) of Theorem 2.2. are satisfied. But Σαρ is not

lsc at (0, 0, 0) . The reason is that the concavity of F is violated. Indeed, taking

3

x1 = 0, x2 =

∈ E (0) = [0,3] ,

then

for

all

y ∈K2 ( A,0) = [0,3] ,

we

2

1

1

3

have F ( x1 , y, 0) = 0, F ( x2 , y, 0) = 3 / 4 , but F ( x1 + x2 , y, 0) = − ∈/ (0, +∞) .

2

2

16

Theorem 2.9

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

(v) K 2 is lsc in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is compact;

(vi) the set {( x, t , y ) ∈ K1 ( A, Λ) × T ( K1 ( A, Λ ), K 2 ( K1 ( A, Λ ), Λ ), Γ) × K 2 ( K1 ( A, Λ ), Λ ) :

ρ ( F ( x, t , y, µ0 ); C ( x, λ0 ))} is closed.

Then Σαρ is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) .

Proof.

We consider only for the cases: α = s, ρ = ρ 2 . We first prove that Σ s ρ (λ0 , γ 0 , µ0 )

2

is closed. Indeed, we let xn ∈ Σ s ρ (λ0 , γ 0 , µ0 ) such that xn → x0 . If x0 ∈/ Σ sρ (λ0 , γ 0 , µ0 ) ,

2

2

∃y0 ∈ K 2 ( x0 , λ0 ), ∃t0 ∈ T ( x0 , y0 , γ 0 ) such that

F ( x0 , t0 , y0 , µ0 ) ⊆/ C ( x0 , λ0 ) .

(2.2)

By the lower semicontinuity of K 2 (., λ0 ) at x0 , one has yn ∈ K 2 ( xn , λ0 ) such that

yn → y0 . Since xn ∈ Σ s ρ (λ0 , γ 0 , µ0 ) , ∀tn ∈ T ( xn , yn , γ 0 ) such that

2

F ( xn , tn , yn , µ0 ) ⊆ C ( xn , λ0 ) .

(2.3)

By the condition (vi), we see a contradiction between ( 2.2) and (2.3). Therefore,

Σ s ρ (λ0 , γ 0 , µ0 ) is closed.

2

On the other hand, since Σsρ (λ0 , γ 0 , µ0 ) ⊆ E(λ0 ) is compact by E (λ0 ) compact.

2

Since Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) and Σ s ρ (λ0 , γ 0 , µ0 ) compact. Hence

2

2

Σ s ρ2 is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) . So we complete the proof.

The following example shows that the assumed compactness in (v) is essential.

Example 2.10

Let X = Y = A = B =

x = ( x − 1, x2 ) ∈

26

2

2

, Z = , Λ = M = Γ = [0,1], C ( x, λ ) =

+

, λ0 = 0 ,

and

for

, K1 ( x, λ ) = K1 ( x, λ ) = {( x1 , λ x1 )} and F ( x, t , y, µ ) = 1 + λ . We shows

Nguyen Van Hung

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that the assumptions of Theorem 2.8 are satisfied, but the compactness of E (λ0 ) is not

satisfied. Direct computations give Σαρ (λ , γ , µ ) = {( x1 , x2 ) ∈

2

| x2 = λ x1} and then Σαρ

is not Hausdorff lower semicontinuous at (0, 0, 0) (although Σαρ is lsc at (0,0,0)).

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

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

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

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

Optimization., 55, pp. 221-230.

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parametric multivalued vector quasivariational inequalities and applications", J.

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

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and approximate solution sets to parametric multivalued quasivariational

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

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

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

pp. 2233-2268.

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

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

443.

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

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

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

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Mathematical Systems, Springer-Verlag Berlin Heidelberg.

(Ngày Tòa soạn nhận được bài: 08-11-2011; ngày chấp nhận đăng: 23-12-2011)

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LOWER SEMICONTINUITY OF THE SOLUTION SETS

OF PARAMETRIC GENERALIZED QUASIEQUILIBRIUM PROBLEMS

NGUYEN VAN HUNG*

ABSTRACT

In this paper we establish sufficient conditions for the solution sets of parametric

generalized quasiequilibrium problems with the stability properties such as lower

semicontinuity and Hausdorff lower semicontinuity.

Keyword: parametric generalized quasiequilibrium problems, lower semicontinuity,

Hausdorff lower semicontinuity.

TÓM TẮT

Tính chất nửa liên tục dưới 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 dưới và tính nửa liên tục dưới Hausdorff.

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

dưới, tính nửa liên tục dưới Hausdorff.

1.

Introduction and Preliminaries

Let X , Y , Λ, Γ, M be a Hausdorff topological spaces, let Z be a Hausdorff

topological vector space, A ⊆ X and B ⊆ Y be a nonempty sets. Let K1 : A× Λ → 2 A ,

K 2 : A× Λ → 2 A , T : A × A × Γ → 2 B , C : A× Λ → 2 B and F : A × B × A × M → 2 Z

multifunctions with C is a proper solid convex cone values and closed.

be

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

ubsets U and V under consideration we adopt the notations.

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

(u, v) m U × V

*

means

∃v ∈ V , ∀u ∈ U ,

(u, v) s U × V

means

∀u ∈ U , ∀v ∈ V ,

ρ1 (U , V )

means

U ∩V ≠ ∅ ,

ρ 2 (U , V )

means

U ⊆V ,

(u, v) wU × V

means

∃u ∈ U , ∀v ∈ V and similarly for m, s ,

MSc., Dong Thap University

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ρ1 (U , V )

means

U ∩ V = ∅ and similarly for ρ 2 .

Let α ∈ {w, m, s} , α ∈ {w, m, s } , ρ ∈ {ρ1 , ρ 2 } and ρ ∈ {ρ1 , ρ 2 } . We consider the

following parametric generalized quasiequilibrium problems.

(QEP αρ ): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying

ρ ( F ( x , t , y, µ ); C ( x , λ )).

*

We consider also the following problem (QEP αρ

) as an auxiliary problem to

(QEP αρ ):

*

): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying

(QEP αρ

ρ ( F ( x , t , y, µ );int C ( x , λ )).

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

%αρ : Λ × Γ × M → 2 A be a set-valued mapping such that Σ (λ , γ , µ ) and

Σαρ , Σ

αρ

%αρ (λ , γ , µ ) are the solution sets of (QEP ) and (QEP * ), respectively, i.e.,

Σ

αρ

αρ

Σαρ (λ , γ , µ ) = {x ∈ E (λ ) | ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) : ρ ( F ( x , t , y, µ ); C ( x , λ ))},

%αρ (λ , γ , µ ) = {x ∈ E (λ ) | ( y, t )α K ( x , λ ) × T ( x , y, γ ) : ρ ( F ( x , t , y, µ );int C ( x , λ ))}.

Σ

2

Clearly Σ%αρ (λ , γ , µ ) ⊆ Σαρ (λ , γ , µ ) . Throughout the paper we assume that

%αρ (λ , γ , µ ) ≠ ∅ for each (λ , γ , µ ) in the neighborhood of

Σαρ (λ , γ , µ ) ≠ ∅ and Σ

(λ0 , γ 0 , µ0 ) ∈ Λ × Γ × M .

By the definition, the following relations are clear:

% sρ ⊆ Σ

% mρ ⊆ Σ

% wρ .

Σ ⊆Σ ⊆Σ

and Σ

sρ

mρ

wρ

The parametric generalized quasiequilibrium problems is more general than many

following problems.

(a) If T ( x, y, γ ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 , ρ = ρ1 and

replace C ( x, λ ) by − int C ( x, λ ) . Then, (QEP α ρ2 ) and (QEP α ρ1 ) becomes to (PGQVEP)

and (PEQVEP), respectively, in Kimura-Yao [7].

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

F ( x , y, λ ) ⊂/ − intC ( x , λ )), for all y ∈ K ( x, λ ).

and

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

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

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(b) If T ( x, y, γ ) = {x}, Λ = Γ, A = B, X = Y , K1 = clK , K 2 = K , ρ = ρ1 , ρ = ρ 2 and

replace C ( x, λ ) by Z \ − int C with C ⊆ Z be closed and int C ≠ ∅ . Then, (QEP αρ1 ) and

(QEP αρ2 ) becomes to (QEP) and (SQEP), respectively, in Anh - Khanh [1].

(QEP): Find x ∈ clK ( x , λ ) such that

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

and

(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 ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 and replace

C ( x, λ ) by − int C ( x, λ ) , replace F by f be a vector function. Then, (QEP α ρ )

2

becomes to (PVQEP) in Kimura-Yao [6].

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

f ( x , y, λ ) ∈

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

Note that generalized quasiequilibrium problems encompass many optimizationrelated models like vector minimization, variation inequalities, Nash equilibrium, fixed

point and coincidence-point problems, complementary problems, minimum

inequalities, etc. Stability properties of solutions have been investigated even in models

for vector quasiequilibrium problems [1, 2, 3, 6, 7, 8], variation problems [4, 5, 9, 10]

and the references therein.

In this paper we establish sufficient conditions for the solution sets Σαρ to have

the stability properties such as the lower semicontinuity and the Hausdorff lower

semicontinuity with respect to parameter λ , γ , µ under relaxed assumptions about

generalized convexity of the map F .

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 lower semicontinuity and the

Hausdorff lower semicontinuity of solution sets of problems (QEP αρ ).

Now we recall some notions. Let X and Z be as above and G : X → 2Z 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 ) . Q 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, Q( x) ⊆ Q( x0 ) + B, ∀x ∈ N

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( Q( x0 ) ⊆ Q( 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 . G is called closed

at x0 if for each net {( xα , zα )} ⊆ graphG := {( x, z )∣ z ∈ G ( x)}, ( xα , zα ) → ( x0 , z0 ) , z0 must

belong to G ( x0 ) . The closeness is closely related to the upper (and Hausdorff upper)

semicontinuity. We say that G satisfies a certain property in a subset A ⊆ X if G

satisfies it at every points of A . If A = X we omit ``in X " in the statement.

Let A and Z be as above and G : A → 2Z be a 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 ( x0 ) compact, then G usc at x0 ;

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

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

{xα } ⊆ A which converges to x0 and for each net { yα } ⊆ G ( xα ) , there are y ∈ G ( x) and

a subnet { yβ } of { yα } such that yβ → y.

Definition. (See [1], [11]) Let X and Z be as above. Suppose that A is a nonempty

convex set of X and that G : X → 2Z be a multifunction.

(i) G is said to be convex in A if for each x1 , x2 ∈ A and t ∈ [0,1]

G (tx1 + (1 − t ) x2 ) ⊃ tG ( x1 ) + (1 − t )G ( x2 )

(ii) G is said to be concave A if for each x1 , x2 ∈ A and t ∈ [0,1]

G (tx1 + (1 − t ) x2 ) ⊂ tG ( x1 ) + (1 − t )G ( x2 )

2.

Main results

In this section, we discuss the lower semicontinuity and the Hausdorff lower

semicontinuity of solution sets for parametric generalized quasiequilibrium problems

(QEP αρ ).

Definition 2.1

Let A and Z be as above and C : A → 2 Z with a proper solid convex cone values.

Suppose G : A → 2Z . We say that G is generalized C -concave in A if for each

x1 , x2 ∈ A , ρ (G ( x1 ), C ( x1 )) and ρ (G ( x2 ),int C ( x2 )) imply

ρ (G (tx1 + (1 − t ) x2 ),int C (tx1 + (1 − t ) x2 )), for all t ∈ (0,1).

Theorem 2.2

Assume for problem (QEP αρ ) that

(i) E is lsc at λ0 , K 2 is usc and compact-valued in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is

convex;

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(ii) in K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ) × {γ 0 } , T is usc and compact-valued if α = s ,

and lsc if α = w (or α = m );

(iii) ∀t ∈ T ( K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ ), Γ), ∀µ0 ∈ M , ∀λ0 ∈ Λ , K 2 (., λ0 ) is concave

in

K1 ( A, Λ )

and

F (., t ,., µ0 )

is

generalized

C (., λ0 ) -concave

in

K1 ( A, Λ ) × K 2 ( K1 ( A, Λ), Λ ) ;

(iv) the set {(x, t, y, µ, λ) ∈ K1( A, Λ) ×T (K1( A, Λ), K2 (K1( A, Λ), Λ), Γ) × K 2 ( K1 ( A, Λ), Λ ) ×

{µ0 } × {λ0 }: ρ ( F ( x, t , y, µ ); C ( x, λ ))} is closed.

Then Σαρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .

Proof.

Since α = {w, m, s} and ρ = {ρ1 , ρ 2 } , we have in fact six cases. However, the

proof techniques are similar. We consider only the cases α = s, ρ = ρ 2 . We prove that

% s ρ is lower semicontinuous at (λ , γ , µ ) . Suppose to the contrary that Σ

% s ρ is not lsc

Σ

2

2

0

0

0

at (λ0 , γ 0 , µ0 ) , i.e., ∃x0 ∈ Σ% sρ2 (λ0 , γ 0 , µ0 ) , ∃(λn , γ n , µn ) → (λ0 , γ 0 , µ0 ) , ∀xn ∈ Σ% sρ2 (λn , γ n , µn ),

xn →

/ x0 . Since E is lsc at λ0 , there is a net xn′ ∈ E (λn ) , xn′ → x0 . By the above

contradiction assumption, there must be a subnet xm′ of xn′ such that, ∀m ,

% sρ (λ , γ , µ ) , i.e., ∃y ∈ K ( x′ , λ ) , ∃t ∈ T ( x′ , y , γ ) such that

x′ ∈

/ Σ

m

2

m

m

m

m

2

m

m

m

m

m

m

F ( xm′ , tm , ym , µm ) ⊆/ int C ( xm′ , λm ).

(2.1)

As K 2 is usc at ( x0 , λ0 ) and K 2 ( x0 , λ0 ) is compact, one has y0 ∈ K 2 ( x0 , λ0 ) such

that ym → y0 (taking a subnet if necessary). By the lower semicontinuity of T at

( x0 , y0 , γ 0 ) ,

one has tm ∈ T ( xm , ym , γ m ) such that tm → t0 .

Since ( xm′ , tm , ym , λm , γ m , µm ) → ( x0 , t0 , y0 , λ0 , γ 0 , µ0 ) and by condition (iv) and (2.1)

yields that

F ( x0 , t0 , y0 , µ0 ) ⊆/ int C ( x0 , λ0 ) ,

which is impossible since x0 ∈ Σ% sρ (λ0 , γ 0 , µ0 ) . Therefore, Σ% s ρ is lsc at (λ0 , γ 0 , µ0 ) .

2

2

Now we check that

% sρ (λ , γ , µ )).

Σ s ρ2 (λ0 , γ 0 , µ0 ) ⊆ cl(Σ

2

0

0

0

Indeed, let x1 ∈ Σ s ρ (λ0 , γ 0 , µ0 ) , x2 ∈ Σ% sρ (λ0 , γ 0 , µ0 ) and xα = (1− t ) x1 + tx2 , t ∈ (0,1) .

2

2

By the convexity of E , we have xα ∈ E (λ0 ) . By the generalized C (., λ0 ) -concavity of

F (., t , y, µ0 ) , we have

F ( xα , t , y, µ0 ) ⊆ int C ( xα , λ0 ),

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and since K 2 (., λ0 ) is concave, one implies that for each yα ∈ K 2 ( xα , λ0 ) , there exist

y1 ∈ K 2 ( x1 , λ0 ) and y2 ∈ K 2 ( x2 , λ0 ) such that yα = ty1 + (1 − t ) y2 . By the generalized

C (., λ0 ) -concavity of F (., t ,., µ0 ) , we have

F ( xα , t , yα , µ0 ) ⊆ int C ( xα , λ0 ),

i.e., xα ∈ Σ% s ρ (λ0 , γ 0 , µ0 ) . Hence Σ s ρ (λ0 , γ 0 , µ0 ) ⊆ cl(Σ% s ρ (λ0 , γ 0 , µ0 )) . By the lower

2

2

2

semicontinuity of Σ% s ρ at (λ0 , γ 0 , µ0 ) , we have

2

% sρ (λ , γ , µ )) ⊆ lim inf Σ

% sρ (λ , γ , µ ) ⊆ lim inf Σ (λ , γ , µ ),

Σ sρ2 (λ0 , γ 0 , µ0 ) ⊆ cl (Σ

2

2

0

0

0

n

n

n

sρ2

n

n

n

i.e., Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .

2

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

Example 2.3

Let A = B = X = Y = Z = , Λ = Γ = M = [0,1], λ0 = 0, C ( x, λ ) = [0, +∞ ) and let

F ( x, t , y, λ ) = 2λ , T ( x, y, λ ) = {x}, K 2 ( x, λ ) = [0,1]

and

⎧[-1,1]

K1 ( x, λ ) = ⎨

⎩[-1-λ , 0]

if λ = 0,

otherwise.

We have E (0) = [−1,1] , E (λ ) = [−λ − 1, 0], ∀λ ∈ (0,1] . Hence K 2 is usc and the

condition (ii), (iii) and (iv) of Theorem 2.2 is 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 Σαρ (0, 0, 0) = [−1,1] and Σαρ (λ , γ , µ ) = [−λ − 1, 0], ∀λ ∈ (0,1] .

The following example shows that in this the special case, assumption (iv) of

Theorem 2.2 may be satisfied even in cases, but both assumption (ii 1 ) and (iii 1 ) of

Theorem 2.1 in Anh-Khanh [1] are not fulfilled.

Example 2.4

Let A, B, X , Y , Z , T , Λ, Γ, M , λ0 , C as in Example 2.3, and let K1 ( x, λ ) =

K 2 ( x, λ ) = [0,1] and

⎧[-4,0]

K1 ( x, λ ) = ⎨

⎩[-1-λ , 0]

if λ = 0,

otherwise.

We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and

Σαρ (λ , γ , µ ) = [0,1], ∀λ ∈ [0,1] . But both assumption (ii 1 ) and (iii 1 ) of Theorem 2.1

in Anh-Khanh [1] are not fulfilled.

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

Theorem2.2 may be satisfied even in cases, but Theorem 2.1 and Theorem 2.3 in AnhKhanh [1] are not fulfilled.

Example 2.5

Let A, B, X , Y , T , Λ, Γ, M , λ0 , C as in Example 2.4, and let K1 ( x, λ ) = K 2 ( x, λ ) =

λ

[0, ] and

2

⎧[0,1]

K1 ( x, λ ) = ⎨

⎩[2, 4]

if λ = 0,

otherwise.

We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and

λ

Σαρ (λ , γ , µ )) = [0, ], ∀λ ∈ [0,1] . Theorem 2.1 and Theorem 2.3 in Anh-Khanh [1] are

2

not fulfilled. The reason is that F is neither usc nor lsc at ( x, y, 0) .

Remark 2.6

In special cases, as in Section 1 (a) and (c). Then, Theorem 2.2 reduces to

Theorem 5.1 in Kimura-Yao [7, 6]. However, the proof of the theorem 5.1 is in a

different way. Its assumption (i) - (v) of Theorem 5.1 coincides with (i) of Theorem 2.2

and assumption (vi), (vii) coincides with (iii), (iv) of Theorem 2.2 Theorem 2.2 slightly

improves Theorem 5.1 in Kimura-Yao [7, 6], since no convexity of the values of E is

imposed.

The following example shows that the convexity and lower semicontinuity of K

is essential.

Example 2.7

Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.5 and let

⎧⎪{−1, 0,1}

K1 ( x, λ ) = ⎨

⎪⎩{0,1}

if λ = 0,

otherwise.

Then, we shows that K 2 is usc and has compact-valued K1 ( X , A) × {λ0 } and assumption

(ii), (iii) and (iv) of Theorem 2.2 are fulfilled. But Σαρ (λ , γ , µ )) is not lsc at (0, 0, 0) .

The reason is that E is not lsc at λ0 = 0 and E (0) is also not convex. Indeed, let

1

x1 = −1, x2 = 0 ∈ E (0) and t = ∈ (0,1) but tx1 + (1 − t ) x2 ∈

/ E (0) .

2

In fact, Σαρ (0, 0, 0) = {−1, 0,1} and Σαρ (λ , γ , µ ) = {0,1}, ∀λ ∈ (0,1] .

The following example shows that the concavity of F (., t., µ0 ) is essential.

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Example 2.8

Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.6 and let K1 ( x, λ ) = K 2 ( x, λ )

= [λ , λ + 3] and F ( x, t , y, µ ) = F ( x, y, λ ) = x 2 − (1 + λ ) x . We show that K 2 (., λ0 ) is

concave and the assumptions (i), (ii), (iv) of Theorem 2.2. are satisfied. But Σαρ is not

lsc at (0, 0, 0) . The reason is that the concavity of F is violated. Indeed, taking

3

x1 = 0, x2 =

∈ E (0) = [0,3] ,

then

for

all

y ∈K2 ( A,0) = [0,3] ,

we

2

1

1

3

have F ( x1 , y, 0) = 0, F ( x2 , y, 0) = 3 / 4 , but F ( x1 + x2 , y, 0) = − ∈/ (0, +∞) .

2

2

16

Theorem 2.9

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

(v) K 2 is lsc in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is compact;

(vi) the set {( x, t , y ) ∈ K1 ( A, Λ) × T ( K1 ( A, Λ ), K 2 ( K1 ( A, Λ ), Λ ), Γ) × K 2 ( K1 ( A, Λ ), Λ ) :

ρ ( F ( x, t , y, µ0 ); C ( x, λ0 ))} is closed.

Then Σαρ is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) .

Proof.

We consider only for the cases: α = s, ρ = ρ 2 . We first prove that Σ s ρ (λ0 , γ 0 , µ0 )

2

is closed. Indeed, we let xn ∈ Σ s ρ (λ0 , γ 0 , µ0 ) such that xn → x0 . If x0 ∈/ Σ sρ (λ0 , γ 0 , µ0 ) ,

2

2

∃y0 ∈ K 2 ( x0 , λ0 ), ∃t0 ∈ T ( x0 , y0 , γ 0 ) such that

F ( x0 , t0 , y0 , µ0 ) ⊆/ C ( x0 , λ0 ) .

(2.2)

By the lower semicontinuity of K 2 (., λ0 ) at x0 , one has yn ∈ K 2 ( xn , λ0 ) such that

yn → y0 . Since xn ∈ Σ s ρ (λ0 , γ 0 , µ0 ) , ∀tn ∈ T ( xn , yn , γ 0 ) such that

2

F ( xn , tn , yn , µ0 ) ⊆ C ( xn , λ0 ) .

(2.3)

By the condition (vi), we see a contradiction between ( 2.2) and (2.3). Therefore,

Σ s ρ (λ0 , γ 0 , µ0 ) is closed.

2

On the other hand, since Σsρ (λ0 , γ 0 , µ0 ) ⊆ E(λ0 ) is compact by E (λ0 ) compact.

2

Since Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) and Σ s ρ (λ0 , γ 0 , µ0 ) compact. Hence

2

2

Σ s ρ2 is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) . So we complete the proof.

The following example shows that the assumed compactness in (v) is essential.

Example 2.10

Let X = Y = A = B =

x = ( x − 1, x2 ) ∈

26

2

2

, Z = , Λ = M = Γ = [0,1], C ( x, λ ) =

+

, λ0 = 0 ,

and

for

, K1 ( x, λ ) = K1 ( x, λ ) = {( x1 , λ x1 )} and F ( x, t , y, µ ) = 1 + λ . We shows

Nguyen Van Hung

Tạp chí KHOA HỌC ĐHSP TPHCM

_____________________________________________________________________________________________________________

that the assumptions of Theorem 2.8 are satisfied, but the compactness of E (λ0 ) is not

satisfied. Direct computations give Σαρ (λ , γ , µ ) = {( x1 , x2 ) ∈

2

| x2 = λ x1} and then Σαρ

is not Hausdorff lower semicontinuous at (0, 0, 0) (although Σαρ is lsc at (0,0,0)).

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

REFERENCES

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

multivalued vector quasiequilibrium problems", J. Math. Anal. Appl., 294, pp. 699711.

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

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

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

Optimization., 55, pp. 221-230.

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.

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.

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

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

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.

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

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

443.

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

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

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.

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

Mathematical Systems, Springer-Verlag Berlin Heidelberg.

(Ngày Tòa soạn nhận được bài: 08-11-2011; ngày chấp nhận đăng: 23-12-2011)

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