Lower semicontinuity of the solution sets of parametric generalized quasiequilibrium problems

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.

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⊆ be a nonempty sets Let Y K A1: × Λ →2A,

2: 2A

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

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

( , ) 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,

ρ1( , )U V means U V∩ ≠ ∅ ,

ρ2( , )U V means 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}, α∈{ , , }w m s , ρ∈{ , }ρ ρ1 2 and ρ∈{ , }ρ ρ1 2 We consider the following parametric generalized quasiequilibrium problems

(QEPαρ): Find x K x∈ 1( , )λ such that ( , )y t K xα 2( , )λ ×T x y( , , )γ satisfying

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

We consider also the following problem (QEP*

αρ) as an auxiliary problem to (QEPαρ):

(QEP*

αρ): Find x K x∈ 1( , )λ such that ( , )y t K xα 2( , )λ ×T x y( , , )γ satisfying

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

For each λ∈ Λ ∈Γ ∈,γ ,µ M , we let E( ) : {λ = ∈x A x K x| ∈ 1( , )}λ and let

%

αρ

Σ Σ Λ×Γ× → be a set-valued mapping such that Σαρ( , , )λ γ µ and

% ( , , )αρ λ γ µ

Σ are the solution sets of (QEPαρ) and (QEP*

αρ), respectively, i.e., 2

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

%αρ( , , ) {λ γ µ x E( ) | ( , )λ y tαK x2( , )λ T x y( , , ) : ( ( , , , );int ( , ))}.γ ρ F x t y µ C x λ

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

αρ λ γ µ

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

0 0 0

( , ,λ γ µ )∈ Λ ×Γ×M

By the definition, the following relations are clear:

sρ mρ wρ and ρ ρ w ρ

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 K, 1 =K2 =K,ρ ρ ρ ρ= 2, = 1 and replace ( , )C x λ by int ( , )− C x λ Then, (QEP

2

α ρ ) and (QEP

1

α ρ ) becomes to (PGQVEP) and (PEQVEP), respectively, in Kimura-Yao [7]

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

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

F x y λ ⊂ −/ C x λ y K x∈ λ

and

(PEQVEP): 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 Λ = Γ,A B X Y K= , = , 1 =clK K, 2 =K,ρ ρ ρ ρ= 1, = and replace ( , )C x λ by Z\ int− C with C⊆Z be closed and int C≠ ∅ Then, (QEP

1

αρ ) and (QEPαρ2) becomes to (QEP) and (SQEP), respectively, in Anh - Khanh [1]

(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

(c) If T x y( , , ) { },γ = x Λ = Γ =M A B X, = , =Y K, 1=K2 =K,ρ ρ= 2 and replace ( , )

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

2

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

(PQVEP): Find x K x∈ ( , )λ such that

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

f x y λ ∈ −/ C x λ y K x∈ λ

Note that generalized quasiequilibrium problems encompass many optimization-related 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 Σ to have αρ 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 (QEPαρ)

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 for

some open set U implies the existence of a neighborhood of

0 ( )

G x ∩ ≠ ∅U Z

An equivalent formulation is that: is lsc at , ( )

x N G x∈ ∩ ≠ ∅U G x0 if ∀ → , xα x0

is called upper semicontinuous (usc) at

0 ( ),0 ( ), 0

z G x zα G xα zα z

each open set U ⊇G x( )0 , there is a neighborhood N of x0 such that is said to be Hausdorff upper semicontinuous (H-usc in short; Hausdorff lower semicontinuous, H-lsc, respectively) at

( )

U ⊇G N Q

0

x if for each neighborhood B of the origin in

Z, there exists a neighborhood N of x such that, Q x( )⊆Q x( )+ ∀ ∈B x N,

(Q x( )0 ⊆Q x( )+ ∀ ∈B x N, ) is said to be continuous at G x0 if it is both lsc and usc at 0

x and to be H-continuous at x0 if it is both H-lsc and H-usc at x0 is called closed

at

G

0

x if for each net {( , )} graph : {( , )x zα α ⊆ G = x z z G x∣ ∈ ( )},( , )x zα α →( , )x z0 0 , must z0

belong to The closeness is closely related to the upper (and Hausdorff upper) semicontinuity We say that G satisfies a certain property in a subset if G

satisfies it at every points of

0 ( )

G x

A⊆ X

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 x( )0 compact, then G usc at x0;

(ii) If G is H-lsc at x0 then G is lsc The converse is true if G x( )0 is compact; (iii) If has compact values, then G G is usc at x0 if and only if, for each net { }xα ⊆ which converges to A x0 and for each net{ }yα ⊆G x( )α , there are and

a subnet {

( )

y G x∈ }

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 x x1, 2∈ and A t∈[0,1]

( (1 ) ) ( ) (1 ) (

G tx + −t x ⊃tG x + −t G x2)

(ii) G is said to be concave A if for each x x1, 2∈ and A t∈[0,1]

( (1 ) ) ( ) (1 ) (

G tx + −t x ⊂tG x + −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 (QEPαρ)

Definition 2.1

Let A and Z be as above and : C A→2Z with a proper solid convex cone values Suppose G A: →2Z We say that is generalized G C -concave in if for each A

1, 2

x x ∈ , A ρ( ( ), ( ))G x C x1 1 and ρ( ( ),int ( ))G x2 C x2 imply

( (G tx (1 t x) ),int (C tx (1 t x) )), for all t (0,1)

Theorem 2.2

Assume for problem (QEPαρ) that

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

( , ) ( ( , ), ) { }

K A K K A

(ii) in 1 Λ × 2 1 Λ Λ × γ0 , is usc and compact-valued if T α =s ,

and lsc if α =w (or α =m );

(iii) ∀ ∈t T K A( ( , )1 Λ ×K K A2( ( , ), ), ),1 Λ Λ Γ ∀ ∈µ0 M,∀ ∈ Λ , λ0 K2(., )λ0 is concave

in K A1( , )Λ and F t(., ,.,µ0) is generalized C(., )λ0 -concave in

;

1( , ) 2( ( , ), )1

K A Λ ×K K A Λ Λ

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

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

Then Σ is lower semicontinuous at αρ ( , ,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 ρ= 2 We prove that

%sρ2

Σ is lower semicontinuous at ( , ,λ γ µ0 0 0) Suppose to the contrary that %Σ is not lsc sρ 2

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

0

n

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

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

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

such that ( , ,m m m, m) int ( ,m m)

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

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

0 0 0

( , , )x y γ ,

one has t m∈T x y( ,m m,γm) such that t m→t0

Since ( , ,x t y m′ m m,λ γ µm, m, m)→ ( , , , , ,x t y0 0 0 λ γ µ0 0 0)and by condition (iv) and (2.1)

yields that

( , , , ) int ( , )

F x t y µ ⊆/ C x λ ,

which is impossible since x0∈ Σ %sρ2( , ,λ γ µ0 0 0) Therefore, %Σ is lsc at sρ 2 ( , ,λ γ µ0 0 0)

Now we check that

% 2

2( , ,0 0 0) cl( s ( , ,0 0 0))

Indeed, let x1∈Σsρ2( , ,λ γ µ0 0 0), x2∈ Σ%sρ2( ,λ γ µ0 0, 0) and xα = −(1 )t x tx t1+ 2, ∈(0,1)

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

0

(., , , )

F t y µ , we have

( , , , ) int ( , ),

F x t y µ ⊆ C x λ

and since K2(., )λ0 is concave, one implies that for each yα∈K x2( , )α λ0 , there exist

1 2( , )1 0

y ∈K x λ and y2∈K x2( , )2 λ0 such that yα =ty1+ −(1 t y) 2 By the generalized 0

(., )

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

( , , , ) int ( , ),

F x t yα α µ ⊆ C xα λ

i.e., xα∈ Σ%sρ 2( , ,λ γ µ0 0 0) Hence Σsρ2( , ,λ γ µ0 0 0)⊆cl(Σ%sρ 2( , ,λ γ µ0 0 0)) By the lower

semicontinuity of %Σ at sρ 2 ( , ,λ γ µ0 0 0), we have

2 ( , , 0 0 0 ) ( s ( , , 0 0 0 )) lim inf s ( , , ) lim i nf 2 ( , , ),

sρ λ γ µ cl ρ λ γ µ ρ λ γ µn n n sρ λ γ µn n n

i.e., Σ is lower semicontinuous at sρ2 ( , ,λ γ µ0 0 0)

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

Example 2.3

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

2

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

F x t y λ = λ T x y λ = x K x λ =

and

1

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

[-1- ,0] er

K x

oth wise

λ λ

λ

=

⎧

= ⎨

⎩

We have E(0) [ 1,1]= − , E( ) [λ = − −λ 1,0],∀ ∈λ (0,1] Hence is usc and the condition (ii), (iii) and (iv) of Theorem 2.2 is easily seen to be fulfilled But

2

K

αρ

Σ is not upper semicontinuous at λ0 = The reason is that E is not lower semicontinuous In 0 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 ) and (iii1) of Theorem 2.1 in Anh-Khanh [1] are not fulfilled

1

Example 2.4

Let A B X Y Z T, , , , , , , ,Λ Γ M, ,λ0 C as in Example 2.3, and let K x1( , )λ =

2( , ) [0,1]

K x λ = and

1

[-4,0] if 0, ( , )

[-1- ,0] er

K x

oth wise

λ λ

λ

=

⎧

= ⎨

⎩

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

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

Σ = ∀ ∈ But both assumption (ii1) and (iii ) of Theorem 2.1

in Anh-Khanh [1] are not fulfilled

1

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 Anh-Khanh [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 K x1( , )λ =K x2( , )λ = [0, ]

2

λ

and

1

( , )

K x

oth wise

λ

⎩

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

αρ λ γ µ

2

∀ ∈ Theorem 2.1 and Theorem 2.3 in Anh-Khanh [1] are not fulfilled The reason is that is neither usc nor lsc at F ( , ,0)x y

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

is essential

K

Example 2.7

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

{ }

1

1,0,1 if 0, ( , )

K x

oth wise

λ

⎪⎩

Then, we shows that K2 is usc and has compact-valued K X A1( , ) { }× λ0 and assumption (ii), (iii) and (iv) of Theorem 2.2 are fulfilled But Σαρ( , , ))λ γ µ is not lsc at (0 The reason is that

,0,0)

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

1 1, 2 0 (0)

2

t= ∈ but tx1+ −(1 )t x2∈/E(0)

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

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

Example 2.8

Let A X Y Z C, , , , , ,Λ M, ,Γ λ0 as in Example 2.6 and let K x1( , )λ =K x2( , )λ

[ ,λ λ 3]

= + and F x t y( , , , )µ =F x y( , , )λ =x2− +(1 λ)x We show that K2(., )λ0 is

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

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

1 0, 2

x = x = 3 (0) [0,3]

2∈E = , then for all y K A∈ 2( ,0) [0,3]= , we haveF x y( , ,0) 0, ( , ,0) 3 / 41 = F x y2 = , but (1 1 1 2, ,0) 3 (0,

F x + x y = − ∈/ +∞)

Theorem 2.9

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

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

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

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

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

Proof

We consider only for the cases: α =s,ρ ρ= 2 We first prove that

2( , ,0 0 0)

sρ λ γ µ

Σ

is closed Indeed, we let x n∈Σsρ2( , ,λ γ µ0 0 0) such that x n → If x0

2

0 s ( , ,0 0 )

x ∈Σ/ ρ λ γ µ0 ,

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

y K x λ t T x y γ

0 0 0 0 0 0

( , , , ) ( , )

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

Since

0

n

y → y

2( , ,0 0 0)

n s

x ∈Σρ λ γ µ , ∀ ∈t n T x y( , , )n n γ0 such that 0

( , , ,n n n ) ( , )n

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

2( , ,0 0 0)

sρ λ γ µ

On the other hand, since Σsρ2( , , )λ γ µ0 0 0 ⊆E( )λ0 is compact by E( )λ0 compact

Since

2

sρ

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

2( , ,0 0 0)

sρ λ γ µ

2

sρ

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

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

Example 2.10

0

X = = = =Y A B Z = Λ =M = Γ = C x λ = + λ = 0

1 }

2

( 1, ) , ( , ) ( , ) {( , )

x= x− x ∈ K x λ =K x λ = x λx and ( , , , ) 1F x t y µ = + We shows λ

that the assumptions of Theorem 2.8 are satisfied, but the compactness of E( )λ0 is not satisfied Direct computations give 2

is not Hausdorff lower semicontinuous at (0,0,0) (although Σ is lsc at (0,0,0)) αρ

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