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

Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> LOWER SEMICONTINUITY OF THE SOLUTION SETS<br /> OF PARAMETRIC GENERALIZED QUASIEQUILIBRIUM PROBLEMS<br /> <br /> NGUYEN VAN HUNG*<br /> <br /> ABSTRACT<br /> In this paper we establish sufficient conditions for the solution sets of parametric<br /> generalized quasiequilibrium problems with the stability properties such as lower<br /> semicontinuity and Hausdorff lower semicontinuity.<br /> Keyword: parametric generalized quasiequilibrium problems, lower semicontinuity,<br /> Hausdorff lower semicontinuity.<br /> TÓM TẮT<br /> Tính chất nửa liên tục dưới của các tập nghiệm<br /> của các bài toán tựa cân bằng tổng quát phụ thuộc tham số<br /> Trong bài báo này, chúng tôi thiết lập điều kiện đủ cho các tập nghiệm của các bài<br /> toán tựa cân bằng tổng quát phụ thuộc tham số có các tính chất ổn định như: tính nửa liên<br /> tục dưới và tính nửa liên tục dưới Hausdorff.<br /> Từ khóa: các bài toán tựa cân bằng tổng quát phụ thuộc tham số, tính nửa liên tục<br /> dưới, tính nửa liên tục dưới Hausdorff.<br /> <br /> 1. Introduction and Preliminaries<br /> Let X , Y , Λ, Γ, M be a Hausdorff topological spaces, let Z be a Hausdorff<br /> topological vector space, A ⊆ X and B ⊆ Y be a nonempty sets. Let K1 : A× Λ → 2 A ,<br /> 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 be<br /> multifunctions with C is a proper solid convex cone values and closed.<br /> For the sake of simplicity, we adopt the following notations. Letters w, m and s<br /> are used for a weak, middle and strong, respectively, kinds of considered problems. For<br /> ubsets U and V under consideration we adopt the notations.<br /> (u, v) w U × V means ∀u ∈ U , ∃v ∈ V ,<br /> (u, v) m U × V means ∃v ∈ V , ∀u ∈ U ,<br /> (u, v) s U × V means ∀u ∈ U , ∀v ∈ V ,<br /> ρ1 (U , V ) means U ∩V ≠ ∅ ,<br /> ρ 2 (U , V ) means U ⊆V ,<br /> (u, v) wU × V means ∃u ∈ U , ∀v ∈ V and similarly for m, s ,<br /> <br /> <br /> *<br /> MSc., Dong Thap University<br /> <br /> 19<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Số 33 năm 2012<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> ρ1 (U , V ) means U ∩ V = ∅ and similarly for ρ 2 .<br /> <br /> Let α ∈ {w, m, s} , α ∈ {w, m, s } , ρ ∈ {ρ1 , ρ 2 } and ρ ∈ {ρ1 , ρ 2 } . We consider the<br /> following parametric generalized quasiequilibrium problems.<br /> (QEP αρ ): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying<br /> ρ ( F ( x , t , y, µ ); C ( x , λ )).<br /> *<br /> We consider also the following problem (QEP αρ ) as an auxiliary problem to<br /> (QEP αρ ):<br /> *<br /> (QEP αρ ): Find x ∈ K1 ( x , λ ) such that ( y, t )α K 2 ( x , λ ) × T ( x , y, γ ) satisfying<br /> ρ ( F ( x , t , y, µ );int C ( x , λ )).<br /> For each λ ∈ Λ, γ ∈ Γ, µ ∈ M , we let E (λ ) := {x ∈ A | x ∈ K1 ( x, λ )} and let<br /> %αρ : Λ × Γ × M → 2 A be a set-valued mapping such that Σ (λ , γ , µ ) and<br /> Σαρ , Σ αρ<br /> %αρ (λ , γ , µ ) are the solution sets of (QEP ) and (QEP * ), respectively, i.e.,<br /> Σ αρ αρ<br /> <br /> 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 ))},<br /> %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 ))}.<br /> Σ 2<br /> <br /> Clearly Σ%αρ (λ , γ , µ ) ⊆ Σαρ (λ , γ , µ ) . Throughout the paper we assume that<br /> Σαρ (λ , γ , µ ) ≠ ∅ and Σ %αρ (λ , γ , µ ) ≠ ∅ for each (λ , γ , µ ) in the neighborhood of<br /> (λ0 , γ 0 , µ0 ) ∈ Λ × Γ × M .<br /> By the definition, the following relations are clear:<br /> Σ ⊆Σ ⊆Σ and Σ% sρ ⊆ Σ<br /> % mρ ⊆ Σ% wρ .<br /> sρ mρ wρ<br /> <br /> The parametric generalized quasiequilibrium problems is more general than many<br /> following problems.<br /> (a) If T ( x, y, γ ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 , ρ = ρ1 and<br /> replace C ( x, λ ) by − int C ( x, λ ) . Then, (QEP α ρ2 ) and (QEP α ρ1 ) becomes to (PGQVEP)<br /> and (PEQVEP), respectively, in Kimura-Yao [7].<br /> (PGQVEP): Find x ∈ K ( x , λ ) such that<br /> F ( x , y, λ ) ⊂/ − intC ( x , λ )), for all y ∈ K ( x, λ ).<br /> and<br /> (PEQVEP): Find x ∈ K ( x , λ ) such that<br /> F ( x , y, λ ) ∩ (− int C ( x , λ )) = ∅, for all y ∈ K ( x, λ ).<br /> <br /> <br /> <br /> <br /> 20<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> (b) If T ( x, y, γ ) = {x}, Λ = Γ, A = B, X = Y , K1 = clK , K 2 = K , ρ = ρ1 , ρ = ρ 2 and<br /> replace C ( x, λ ) by Z \ − int C with C ⊆ Z be closed and int C ≠ ∅ . Then, (QEP αρ1 ) and<br /> (QEP αρ2 ) becomes to (QEP) and (SQEP), respectively, in Anh - Khanh [1].<br /> (QEP): Find x ∈ clK ( x , λ ) such that<br /> F ( x , y, λ ) ∩ ( Z \ − int C ) ≠ ∅, for all y ∈ K ( x, λ ).<br /> and<br /> (SQEP): Find x ∈ K ( x , λ ) such that<br /> F ( x , y, λ ) ⊆ Z \ − int C , for all y ∈ K ( x, λ ).<br /> (c) If T ( x, y, γ ) = {x}, Λ = Γ = M , A = B, X = Y , K1 = K 2 = K , ρ = ρ 2 and replace<br /> C ( x, λ ) by − int C ( x, λ ) , replace F by f be a vector function. Then, (QEP α ρ )<br /> 2<br /> <br /> becomes to (PVQEP) in Kimura-Yao [6].<br /> (PQVEP): Find x ∈ K ( x , λ ) such that<br /> f ( x , y, λ ) ∈<br /> / − int C ( x , λ )), for all y ∈ K ( x, λ ).<br /> Note that generalized quasiequilibrium problems encompass many optimization-<br /> related models like vector minimization, variation inequalities, Nash equilibrium, fixed<br /> point and coincidence-point problems, complementary problems, minimum<br /> inequalities, etc. Stability properties of solutions have been investigated even in models<br /> for vector quasiequilibrium problems [1, 2, 3, 6, 7, 8], variation problems [4, 5, 9, 10]<br /> and the references therein.<br /> In this paper we establish sufficient conditions for the solution sets Σαρ to have<br /> the stability properties such as the lower semicontinuity and the Hausdorff lower<br /> semicontinuity with respect to parameter λ , γ , µ under relaxed assumptions about<br /> generalized convexity of the map F .<br /> The structure of our paper is as follows. In the remaining part of this section, we<br /> recall definitions for later uses. Section 2 is devoted to the lower semicontinuity and the<br /> Hausdorff lower semicontinuity of solution sets of problems (QEP αρ ).<br /> Now we recall some notions. Let X and Z be as above and G : X → 2Z be a<br /> multifunction. G is said to be lower semicontinuous (lsc) at x0 if G ( x0 ) ∩ U ≠ ∅ for<br /> some open set U ⊆ Z implies the existence of a neighborhood N of x0 such that, for all<br /> x ∈ N , G ( x) ∩ U ≠ ∅ . An equivalent formulation is that: G is lsc at x0 if ∀xα → x0 ,<br /> ∀z0 ∈ G ( x0 ), ∃zα ∈ G ( xα ), zα → z0 . G is called upper semicontinuous (usc) at x0 if for<br /> each open set U ⊇ G ( x0 ) , there is a neighborhood N of x0 such that U ⊇ G ( N ) . Q is<br /> said to be Hausdorff upper semicontinuous (H-usc in short; Hausdorff lower<br /> semicontinuous, H-lsc, respectively) at x0 if for each neighborhood B of the origin in<br /> Z , there exists a neighborhood N of x0 such that, Q( x) ⊆ Q( x0 ) + B, ∀x ∈ N<br /> <br /> 21<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Số 33 năm 2012<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> ( Q( x0 ) ⊆ Q( x) + B, ∀x ∈ N ). G is said to be continuous at x0 if it is both lsc and usc at<br /> x0 and to be H-continuous at x0 if it is both H-lsc and H-usc at x0 . G is called closed<br /> at x0 if for each net {( xα , zα )} ⊆ graphG := {( x, z )∣ z ∈ G ( x)}, ( xα , zα ) → ( x0 , z0 ) , z0 must<br /> belong to G ( x0 ) . The closeness is closely related to the upper (and Hausdorff upper)<br /> semicontinuity. We say that G satisfies a certain property in a subset A ⊆ X if G<br /> satisfies it at every points of A . If A = X we omit ``in X " in the statement.<br /> Let A and Z be as above and G : A → 2Z be a multifunction.<br /> (i) If G is usc at x0 then G is H -usc at x0 . Conversely if G is H -usc at x0 and<br /> if G ( x0 ) compact, then G usc at x0 ;<br /> (ii) If G is H-lsc at x0 then G is lsc. The converse is true if G ( x0 ) is compact;<br /> (iii) If G has compact values, then G is usc at x0 if and only if, for each net<br /> {xα } ⊆ A which converges to x0 and for each net { yα } ⊆ G ( xα ) , there are y ∈ G ( x) and<br /> a subnet { yβ } of { yα } such that yβ → y.<br /> Definition. (See [1], [11]) Let X and Z be as above. Suppose that A is a nonempty<br /> convex set of X and that G : X → 2Z be a multifunction.<br /> (i) G is said to be convex in A if for each x1 , x2 ∈ A and t ∈ [0,1]<br /> G (tx1 + (1 − t ) x2 ) ⊃ tG ( x1 ) + (1 − t )G ( x2 )<br /> (ii) G is said to be concave A if for each x1 , x2 ∈ A and t ∈ [0,1]<br /> G (tx1 + (1 − t ) x2 ) ⊂ tG ( x1 ) + (1 − t )G ( x2 )<br /> 2. Main results<br /> In this section, we discuss the lower semicontinuity and the Hausdorff lower<br /> semicontinuity of solution sets for parametric generalized quasiequilibrium problems<br /> (QEP αρ ).<br /> Definition 2.1<br /> Let A and Z be as above and C : A → 2 Z with a proper solid convex cone values.<br /> Suppose G : A → 2Z . We say that G is generalized C -concave in A if for each<br /> x1 , x2 ∈ A , ρ (G ( x1 ), C ( x1 )) and ρ (G ( x2 ),int C ( x2 )) imply<br /> ρ (G (tx1 + (1 − t ) x2 ),int C (tx1 + (1 − t ) x2 )), for all t ∈ (0,1).<br /> Theorem 2.2<br /> Assume for problem (QEP αρ ) that<br /> (i) E is lsc at λ0 , K 2 is usc and compact-valued in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is<br /> convex;<br /> <br /> <br /> 22<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> (ii) in K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ) × {γ 0 } , T is usc and compact-valued if α = s ,<br /> and lsc if α = w (or α = m );<br /> (iii) ∀t ∈ T ( K1 ( A, Λ ) × K 2 ( K1 ( A, Λ ), Λ ), Γ), ∀µ0 ∈ M , ∀λ0 ∈ Λ , K 2 (., λ0 ) is concave<br /> in K1 ( A, Λ ) and F (., t ,., µ0 ) is generalized C (., λ0 ) -concave in<br /> K1 ( A, Λ ) × K 2 ( K1 ( A, Λ), Λ ) ;<br /> (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 ) ×<br /> {µ0 } × {λ0 }: ρ ( F ( x, t , y, µ ); C ( x, λ ))} is closed.<br /> Then Σαρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .<br /> Proof.<br /> Since α = {w, m, s} and ρ = {ρ1 , ρ 2 } , we have in fact six cases. However, the<br /> proof techniques are similar. We consider only the cases α = s, ρ = ρ 2 . We prove that<br /> % s ρ is lower semicontinuous at (λ , γ , µ ) . Suppose to the contrary that Σ<br /> Σ % s ρ is not lsc<br /> 2 0 0 0 2<br /> <br /> <br /> 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 ),<br /> xn →/ x0 . Since E is lsc at λ0 , there is a net xn′ ∈ E (λn ) , xn′ → x0 . By the above<br /> contradiction assumption, there must be a subnet xm′ of xn′ such that, ∀m ,<br /> x′ ∈/ Σ% sρ (λ , γ , µ ) , i.e., ∃y ∈ K ( x′ , λ ) , ∃t ∈ T ( x′ , y , γ ) such that<br /> m 2 m m m m 2 m m m m m m<br /> <br /> F ( xm′ , tm , ym , µm ) ⊆/ int C ( xm′ , λm ). (2.1)<br /> As K 2 is usc at ( x0 , λ0 ) and K 2 ( x0 , λ0 ) is compact, one has y0 ∈ K 2 ( x0 , λ0 ) such<br /> that ym → y0 (taking a subnet if necessary). By the lower semicontinuity of T at<br /> ( x0 , y0 , γ 0 ) ,<br /> one has tm ∈ T ( xm , ym , γ m ) such that tm → t0 .<br /> Since ( xm′ , tm , ym , λm , γ m , µm ) → ( x0 , t0 , y0 , λ0 , γ 0 , µ0 ) and by condition (iv) and (2.1)<br /> yields that<br /> F ( x0 , t0 , y0 , µ0 ) ⊆/ int C ( x0 , λ0 ) ,<br /> which is impossible since x0 ∈ Σ% sρ (λ0 , γ 0 , µ0 ) . Therefore, Σ% s ρ is lsc at (λ0 , γ 0 , µ0 ) .<br /> 2 2<br /> <br /> <br /> Now we check that<br /> % sρ (λ , γ , µ )).<br /> Σ s ρ2 (λ0 , γ 0 , µ0 ) ⊆ cl(Σ 2 0 0 0<br /> <br /> <br /> 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) .<br /> 2<br /> 2<br /> <br /> <br /> By the convexity of E , we have xα ∈ E (λ0 ) . By the generalized C (., λ0 ) -concavity of<br /> F (., t , y, µ0 ) , we have<br /> F ( xα , t , y, µ0 ) ⊆ int C ( xα , λ0 ),<br /> <br /> <br /> 23<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Số 33 năm 2012<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> and since K 2 (., λ0 ) is concave, one implies that for each yα ∈ K 2 ( xα , λ0 ) , there exist<br /> y1 ∈ K 2 ( x1 , λ0 ) and y2 ∈ K 2 ( x2 , λ0 ) such that yα = ty1 + (1 − t ) y2 . By the generalized<br /> C (., λ0 ) -concavity of F (., t ,., µ0 ) , we have<br /> F ( xα , t , yα , µ0 ) ⊆ int C ( xα , λ0 ),<br /> 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<br /> 2<br /> 2<br /> 2<br /> <br /> <br /> <br /> semicontinuity of Σ% s ρ at (λ0 , γ 0 , µ0 ) , we have<br /> 2<br /> <br /> <br /> % sρ (λ , γ , µ )) ⊆ lim inf Σ<br /> Σ sρ2 (λ0 , γ 0 , µ0 ) ⊆ cl (Σ % sρ (λ , γ , µ ) ⊆ lim inf Σ (λ , γ , µ ),<br /> 2 0 0 0 2 n n n sρ2 n n n<br /> <br /> <br /> i.e., Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) .<br /> 2<br /> <br /> <br /> The following example shows that the lower semicontinuity of E is essential.<br /> Example 2.3<br /> Let A = B = X = Y = Z = , Λ = Γ = M = [0,1], λ0 = 0, C ( x, λ ) = [0, +∞ ) and let<br /> F ( x, t , y, λ ) = 2λ , T ( x, y, λ ) = {x}, K 2 ( x, λ ) = [0,1]<br /> and<br /> ⎧[-1,1] if λ = 0,<br /> K1 ( x, λ ) = ⎨<br /> ⎩[-1-λ , 0] otherwise.<br /> We have E (0) = [−1,1] , E (λ ) = [−λ − 1, 0], ∀λ ∈ (0,1] . Hence K 2 is usc and the<br /> condition (ii), (iii) and (iv) of Theorem 2.2 is easily seen to be fulfilled. But Σαρ is not<br /> upper semicontinuous at λ0 = 0 . The reason is that E is not lower semicontinuous. In<br /> fact Σαρ (0, 0, 0) = [−1,1] and Σαρ (λ , γ , µ ) = [−λ − 1, 0], ∀λ ∈ (0,1] .<br /> The following example shows that in this the special case, assumption (iv) of<br /> Theorem 2.2 may be satisfied even in cases, but both assumption (ii 1 ) and (iii 1 ) of<br /> Theorem 2.1 in Anh-Khanh [1] are not fulfilled.<br /> Example 2.4<br /> Let A, B, X , Y , Z , T , Λ, Γ, M , λ0 , C as in Example 2.3, and let K1 ( x, λ ) =<br /> K 2 ( x, λ ) = [0,1] and<br /> ⎧[-4,0] if λ = 0,<br /> K1 ( x, λ ) = ⎨<br /> ⎩[-1-λ , 0] otherwise.<br /> We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and<br /> Σαρ (λ , γ , µ ) = [0,1], ∀λ ∈ [0,1] . But both assumption (ii 1 ) and (iii 1 ) of Theorem 2.1<br /> in Anh-Khanh [1] are not fulfilled.<br /> <br /> <br /> <br /> <br /> 24<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> The following example shows that in this the special case, assumption of<br /> Theorem2.2 may be satisfied even in cases, but Theorem 2.1 and Theorem 2.3 in Anh-<br /> Khanh [1] are not fulfilled.<br /> Example 2.5<br /> 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 ) =<br /> λ<br /> [0, ] and<br /> 2<br /> ⎧[0,1] if λ = 0,<br /> K1 ( x, λ ) = ⎨<br /> ⎩[2, 4] otherwise.<br /> We shows that the assumptions (i), (ii) and (iii) of Theorem 2.2 satisfied and<br /> λ<br /> Σαρ (λ , γ , µ )) = [0, ], ∀λ ∈ [0,1] . Theorem 2.1 and Theorem 2.3 in Anh-Khanh [1] are<br /> 2<br /> not fulfilled. The reason is that F is neither usc nor lsc at ( x, y, 0) .<br /> Remark 2.6<br /> In special cases, as in Section 1 (a) and (c). Then, Theorem 2.2 reduces to<br /> Theorem 5.1 in Kimura-Yao [7, 6]. However, the proof of the theorem 5.1 is in a<br /> different way. Its assumption (i) - (v) of Theorem 5.1 coincides with (i) of Theorem 2.2<br /> and assumption (vi), (vii) coincides with (iii), (iv) of Theorem 2.2 Theorem 2.2 slightly<br /> improves Theorem 5.1 in Kimura-Yao [7, 6], since no convexity of the values of E is<br /> imposed.<br /> The following example shows that the convexity and lower semicontinuity of K<br /> is essential.<br /> Example 2.7<br /> Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.5 and let<br /> ⎧⎪{−1, 0,1} if λ = 0,<br /> K1 ( x, λ ) = ⎨<br /> ⎪⎩{0,1} otherwise.<br /> Then, we shows that K 2 is usc and has compact-valued K1 ( X , A) × {λ0 } and assumption<br /> (ii), (iii) and (iv) of Theorem 2.2 are fulfilled. But Σαρ (λ , γ , µ )) is not lsc at (0, 0, 0) .<br /> The reason is that E is not lsc at λ0 = 0 and E (0) is also not convex. Indeed, let<br /> 1<br /> x1 = −1, x2 = 0 ∈ E (0) and t = ∈ (0,1) but tx1 + (1 − t ) x2 ∈ / E (0) .<br /> 2<br /> In fact, Σαρ (0, 0, 0) = {−1, 0,1} and Σαρ (λ , γ , µ ) = {0,1}, ∀λ ∈ (0,1] .<br /> The following example shows that the concavity of F (., t., µ0 ) is essential.<br /> <br /> <br /> <br /> <br /> 25<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Số 33 năm 2012<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> Example 2.8<br /> Let A, X , Y , Z , C , Λ, M , Γ, λ0 as in Example 2.6 and let K1 ( x, λ ) = K 2 ( x, λ )<br /> = [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<br /> concave and the assumptions (i), (ii), (iv) of Theorem 2.2. are satisfied. But Σαρ is not<br /> lsc at (0, 0, 0) . The reason is that the concavity of F is violated. Indeed, taking<br /> 3<br /> x1 = 0, x2 = ∈ E (0) = [0,3] , then for all y ∈K2 ( A,0) = [0,3] , we<br /> 2<br /> 1 1 3<br /> have F ( x1 , y, 0) = 0, F ( x2 , y, 0) = 3 / 4 , but F ( x1 + x2 , y, 0) = − ∈/ (0, +∞) .<br /> 2 2 16<br /> Theorem 2.9<br /> Impose the assumption of Theorem 2.2 and the following additional conditions:<br /> (v) K 2 is lsc in K1 ( A, Λ ) × {λ0 } and E (λ0 ) is compact;<br /> (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 ) :<br /> ρ ( F ( x, t , y, µ0 ); C ( x, λ0 ))} is closed.<br /> Then Σαρ is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) .<br /> Proof.<br /> We consider only for the cases: α = s, ρ = ρ 2 . We first prove that Σ s ρ (λ0 , γ 0 , µ0 ) 2<br /> <br /> 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 ) ,<br /> 2 2<br /> <br /> <br /> ∃y0 ∈ K 2 ( x0 , λ0 ), ∃t0 ∈ T ( x0 , y0 , γ 0 ) such that<br /> F ( x0 , t0 , y0 , µ0 ) ⊆/ C ( x0 , λ0 ) . (2.2)<br /> By the lower semicontinuity of K 2 (., λ0 ) at x0 , one has yn ∈ K 2 ( xn , λ0 ) such that<br /> yn → y0 . Since xn ∈ Σ s ρ (λ0 , γ 0 , µ0 ) , ∀tn ∈ T ( xn , yn , γ 0 ) such that<br /> 2<br /> <br /> <br /> F ( xn , tn , yn , µ0 ) ⊆ C ( xn , λ0 ) . (2.3)<br /> By the condition (vi), we see a contradiction between ( 2.2) and (2.3). Therefore,<br /> Σ s ρ (λ0 , γ 0 , µ0 ) is closed.<br /> 2<br /> <br /> <br /> On the other hand, since Σsρ (λ0 , γ 0 , µ0 ) ⊆ E(λ0 ) is compact by E (λ0 ) compact.<br /> 2<br /> <br /> Since Σ s ρ is lower semicontinuous at (λ0 , γ 0 , µ0 ) and Σ s ρ (λ0 , γ 0 , µ0 ) compact. Hence<br /> 2 2<br /> <br /> Σ s ρ2 is Hausdorff lower semicontinuous at (λ0 , γ 0 , µ0 ) . So we complete the proof.<br /> The following example shows that the assumed compactness in (v) is essential.<br /> Example 2.10<br /> Let X = Y = A = B = 2<br /> , Z = , Λ = M = Γ = [0,1], C ( x, λ ) = + , λ0 = 0 , and for<br /> x = ( x − 1, x2 ) ∈ 2<br /> , K1 ( x, λ ) = K1 ( x, λ ) = {( x1 , λ x1 )} and F ( x, t , y, µ ) = 1 + λ . We shows<br /> <br /> <br /> 26<br /> Tạp chí KHOA HỌC ĐHSP TPHCM Nguyen Van Hung<br /> _____________________________________________________________________________________________________________<br /> <br /> <br /> <br /> <br /> that the assumptions of Theorem 2.8 are satisfied, but the compactness of E (λ0 ) is not<br /> satisfied. Direct computations give Σαρ (λ , γ , µ ) = {( x1 , x2 ) ∈ 2<br /> | x2 = λ x1} and then Σαρ<br /> is not Hausdorff lower semicontinuous at (0, 0, 0) (although Σαρ is lsc at (0,0,0)).<br /> <br /> REFERENCES<br /> 1. Anh L. Q., Khanh P. Q. 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