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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.
Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung _ 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 TĨM TẮT Tính chất nửa liên tục tập nghiệm toán tựa cân tổng quát phụ thuộc tham số Trong báo này, thiết lập điều kiện đủ cho tập nghiệm toán tựa cân tổng quát phụ thuộc tham số có tính chất ổn định như: tính nửa liên tục tính nửa liên tục Hausdorff Từ khóa: tốn tựa cân tổng qt phụ thuộc tham số, tính nửa liên tục dưới, tính nửa liên tục Hausdorff 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 → A , K : A× Λ → A , T : A × A × Γ → B , C : A× Λ → B and F : A × B × A × M → 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 (U , V ) means U ⊆V , (u, v) wU × V means ∃u ∈ U , ∀v ∈ V and similarly for m, s , MSc., Dong Thap University 19 Số 33 năm 2012 Tạp chí KHOA HỌC ĐHSP TPHCM _ ρ1 (U , V ) means U ∩ V = ∅ and similarly for ρ Let α ∈ {w, m, s} , α ∈ {w, m, s } , ρ ∈ {ρ1 , ρ } and ρ ∈ {ρ1 , ρ } We consider the following parametric generalized quasiequilibrium problems (QEP αρ ): Find x ∈ K1 ( x , λ ) such that ( y, t )α K ( 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 ( 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 → 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 ( 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 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 , 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 = K , ρ = ρ , 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 ) 20 Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung _ (b) If T ( x, y, γ ) = {x}, Λ = Γ, A = B, X = Y , K1 = clK , K = K , ρ = ρ1 , ρ = ρ 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 = K , ρ = ρ and replace C ( x, λ ) by − int C ( x, λ ) , replace F by f be a vector function Then, (QEP α ρ ) 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 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 21 Tạp chí KHOA HỌC ĐHSP TPHCM Số 33 năm 2012 _ ( 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 ) 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 → 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 is usc and compact-valued in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is convex; 22 Nguyen Van Hung Tạp chí KHOA HỌC ĐHSP TPHCM _ (ii) in K1 ( A, Λ ) × K ( K1 ( A, Λ ), Λ) × {γ } , T is usc and compact-valued if α = s , and lsc if α = w (or α = m ); (iii) ∀t ∈ T ( K1 ( A, Λ ) × K ( K1 ( A, Λ ), Λ ), Γ), ∀µ0 ∈ M , ∀λ0 ∈ Λ , K (., λ0 ) is concave in K1 ( A, Λ ) and F (., t ,., µ0 ) is generalized C (., λ0 ) -concave in K1 ( A, Λ ) × K ( K1 ( A, Λ), Λ ) ; (iv) the set {(x, t, y, à, ) K1( A, ) ìT (K1( A, Λ), K2 (K1( A, Λ), Λ), Γ) × K ( K1 ( A, ), ) ì {à0 } × {λ0 }: ρ ( F ( x, t , y, µ ); C ( x, λ ))} is closed Then Σαρ is lower semicontinuous at (λ0 , γ , µ0 ) Proof Since α = {w, m, s} and ρ = {ρ1 , ρ } , we have in fact six cases However, the proof techniques are similar We consider only the cases α = s, ρ = ρ 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 0 at (λ0 , γ , µ0 ) , i.e., ∃x0 ∈ Σ% sρ2 (λ0 , γ , µ0 ) , ∃(λn , γ n , µn ) → (λ0 , γ , 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 m m m m m m m m m m F ( xm′ , tm , ym , µm ) ⊆/ int C ( xm′ , λm ) (2.1) As K is usc at ( x0 , λ0 ) and K ( x0 , λ0 ) is compact, one has y0 ∈ K ( x0 , λ0 ) such that ym → y0 (taking a subnet if necessary) By the lower semicontinuity of T at ( x0 , y0 , γ ) , 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 , 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 , µ0 ) Therefore, Σ% s ρ is lsc at (λ0 , γ , µ0 ) 2 Now we check that % sρ (λ , γ , µ )) Σ s ρ2 (λ0 , γ , µ0 ) ⊆ cl(Σ 0 Indeed, let x1 ∈ Σ s ρ (λ0 , γ , µ0 ) , x2 ∈ Σ% sρ (λ0 , γ , µ0 ) and xα = (1− t ) x1 + tx2 , t ∈ (0,1) 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 ), 23 Số 33 năm 2012 Tạp chí KHOA HỌC ĐHSP TPHCM _ and since K (., λ0 ) is concave, one implies that for each yα ∈ K ( xα , λ0 ) , there exist y1 ∈ K ( x1 , λ0 ) and y2 ∈ K ( 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 , µ0 ) Hence Σ s ρ (λ0 , γ , µ0 ) ⊆ cl(Σ% s ρ (λ0 , γ , µ0 )) By the lower 2 semicontinuity of Σ% s ρ at (λ0 , γ , µ0 ) , we have % sρ (λ , γ , µ )) ⊆ lim inf Σ % sρ (λ , γ , µ ) ⊆ lim inf Σ (λ , γ , µ ), Σ sρ2 (λ0 , γ , µ0 ) ⊆ cl (Σ 2 0 n n n sρ2 n n n i.e., Σ s ρ is lower semicontinuous at (λ0 , γ , µ0 ) 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 ( 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 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 = 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 ) and (iii ) 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 ( 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 ) and (iii ) of Theorem 2.1 in Anh-Khanh [1] are not fulfilled 24 Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung _ 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 ( x, λ ) = λ [0, ] and ⎧[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 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 (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 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 = and E (0) is also not convex Indeed, let x1 = −1, x2 = ∈ E (0) and t = ∈ (0,1) but tx1 + (1 − t ) x2 ∈ / E (0) 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 25 Số 33 năm 2012 Tạp chí KHOA HỌC ĐHSP TPHCM _ Example 2.8 Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.6 and let K1 ( x, λ ) = K ( x, λ ) = [λ , λ + 3] and F ( x, t , y, µ ) = F ( x, y, λ ) = x − (1 + λ ) x We show that K (., 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 x1 = 0, x2 = ∈ E (0) = [0,3] , then for all y ∈K2 ( A,0) = [0,3] , we 1 have F ( x1 , y, 0) = 0, F ( x2 , y, 0) = / , but F ( x1 + x2 , y, 0) = − ∈/ (0, +∞) 2 16 Theorem 2.9 Impose the assumption of Theorem 2.2 and the following additional conditions: (v) K 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 ( K1 ( A, Λ ), Λ ), Γ) × K ( K1 ( A, Λ ), Λ ) : ρ ( F ( x, t , y, µ0 ); C ( x, λ0 ))} is closed Then Σαρ is Hausdorff lower semicontinuous at (λ0 , γ , µ0 ) Proof We consider only for the cases: α = s, ρ = ρ We first prove that Σ s ρ (λ0 , γ , µ0 ) is closed Indeed, we let xn ∈ Σ s ρ (λ0 , γ , µ0 ) such that xn → x0 If x0 ∈/ Σ sρ (λ0 , γ , µ0 ) , 2 ∃y0 ∈ K ( x0 , λ0 ), ∃t0 ∈ T ( x0 , y0 , γ ) such that F ( x0 , t0 , y0 , µ0 ) ⊆/ C ( x0 , λ0 ) (2.2) By the lower semicontinuity of K (., λ0 ) at x0 , one has yn ∈ K ( xn , λ0 ) such that yn → y0 Since xn ∈ Σ s ρ (λ0 , γ , µ0 ) , ∀tn ∈ T ( xn , yn , γ ) such that 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 , µ0 ) is closed On the other hand, since Σsρ (λ0 , γ , µ0 ) ⊆ E(λ0 ) is compact by E (λ0 ) compact Since Σ s ρ is lower semicontinuous at (λ0 , γ , µ0 ) and Σ s ρ (λ0 , γ , µ0 ) compact Hence 2 Σ s ρ2 is Hausdorff lower semicontinuous at (λ0 , γ , 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 , Z = , Λ = M = Γ = [0,1], C ( x, λ ) = + , λ0 = , and for , K1 ( x, λ ) = K1 ( x, λ ) = {( x1 , λ x1 )} and F ( x, t , y, µ ) = + 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 ) ∈ | x2 = λ x1} and then Σαρ is not Hausdorff lower semicontinuous at (0, 0, 0) (although Σαρ is lsc at (0,0,0)) 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 bài: 08-11-2011; ngày chấp nhận đăng: 23-12-2011) 27 ... 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... 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... 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