UPPER SEMICONTINUITY AND CLOSEDNESS OF THE SOLUTION SETS TO PARAMETRIC QUASIEQUILIBRIUM PROBLEMS NGUYEN VAN HUNG*, PHAN THANH KIEU** ABSTRACT In this paper we establish sufficient conditions for the s[.]
Nguyen Van Hung et al Tạp chí KHOA HỌC ĐHSP TPHCM UPPER SEMICONTINUITY AND CLOSEDNESS OF THE SOLUTION SETS TO PARAMETRIC QUASIEQUILIBRIUM PROBLEMS NGUYEN VAN HUNG*, PHAN THANH KIEU** ABSTRACT In this paper we establish sufficient conditions for the solution mappings of parametric generalized vector quasiequilibrium problems to have the stability properties such as upper semicontinuity and closedness Our results improve recent existing ones in the literature Keywords: parametric quasiequilibrium problems, upper semicontinuity, closedness TĨM TẮT Tính chất nửa liên tục tính đóng 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 đóng Kết chúng tơi cải thiện số kết tồn gần danh sách tài liệu tham khảo Từ khóa: toán tựa cân tổng quát phụ thuộc tham số, tính nửa liên tục trên, tính đóng Introduction and Preliminaries Let X ,Y , Λ, Γ, M be Hausdorff topological spaces, let Z be a Hausdorff topological vector space, A ⊆ X and B ⊆ Y be nonempty sets Let K : A ×Λ →2A , K : A ×Λ →2A , C : A ×Λ →2B and F : A × B × A × M →2Z T : A × A ×Γ →2 , multifunctions with C( x, λ ) is a proper convex cone values and closed B be Now, we adopt the following notations Letters w, m and s are used for a weak, middle and strong, respectively, kinds of considered problems For subsets U and V under consideration we adopt the notations (u, v) w U × V means ∀u ∈U , ∃ v ∈V , (u, v) m U × V (u, v) s U × V (u, v) wU × V * ** means means means ∃ v ∈V , ∀u ∈U , ∀u ∈U , ∀v ∈V , ∃ u ∈U , ∀v ∈V and similarly for m, s MSc., Dong Thap University BA., Dong Thap University Let α ∈{w, m, s} and α ∈{w, m, s } We consider the following parametric quasiequilibrium problem (in short, (QEP λγµ )): α (QEP λ γµ ): Find α x ∈ K1(x , λ ) such that ( y, t)α K2 (x , λ ) × T (x , y,γ ) statisfying F (x , t, y, µ ) − intC(x , λ ) ⊆/ For λ ∈Λ, γ ∈Γ, µ ∈ M consider the following parametric extended quasiequilibrium problem (in short, (QEEP λαγµ )): (QEEP λ γµ ): Find α x ∈ K1(x , λ ) such that ( y, t)α K2 (x , λ ) × T (x , y,γ ) statisfying F (x , t, y, µ ) ∩ − intC(x, λ )=∅ For each λ ∈Λ, γ ∈Γ, µ ∈ M , we let E(λ ) := {x ∈ A | x ∈ K1(x, λ )} Σα , Ξα : Λ × Γ × M →2 be set-valued mappings such that Σα ( λ , γ , µ) are the solution sets of (QEP λ γµ ) and (QEEP λ γµ ), respectively A α and let and Ξα ( λ , γ , µ ) α Throughout the paper we assume that Σ ( λ , γ , µ ) ≠∅ α and each ( λ , γ , µ in the ( , , à0 ) ì ì M ) neighborhoods Ξα (λ ,γ , µ ) ≠ ∅ for By the definition, the following relations are clear: Σs ⊆ Σm ⊆ Σw and Ξs ⊆ Ξm ⊆ Ξw Special cases of the problems (QEP λ γµ ) and (QEEP λγµ ) are as follows: α α (a) If T (x, y,γ ) = {t}, Λ = Γ = M , A = B, X = Y , K = K and α = m , then =K (QEP λγµ ) and (QEEP λγµ ) become to (PGQVEP) and (PEQVEP), respectively in α α Kimura-Yao [8] (PGQVEP): Find x ∈ K (x , λ ) such that and F (x , y, λ ) − int C( x , λ )), for all y ∈ K (x , λ ) ⊂/ (PEQVEP): Find x ∈ K (x , λ ) such that F (x , y, λ ) ∩− int C(x , λ ) = ∅, for all y ∈ K (x , λ ) (b) If T (x, y,γ ) = {t}, Λ = Γ, A = B, X = Y , K = clK , K = K ,α = m, C(x, λ ) and ≡C replace " − int C(x, by " ⊆ Z ‚ − int C " with C ⊆ be closed and int C ≠∅ , then ⊆/ λ )" Z λγµ (QEP α ) become to (SQEP) in Anh-Khanh [1] (SQEP): Find x ∈ K (x , λ ) such that F (x , y, λ ) ⊆ Z ‚ − int C, for all y ∈ K (x , λ ) (c) If T (x, y,γ ) = {t}, Λ = Γ = M , A = B, X = Y , K = K = K ,α =m by f α and replace F be a vector function, then (QEP λ γµ ) become to (PVQEP) in Kimura-Yao [7] (PQVEP): Find x ∈ K (x , λ ) such that f (x , y, λ ) ∈/ − int C( x , λ )), for all y ∈ K (x, λ ) The parametric generalized quasiequilibrium problems include many rather general problems as particular cases as vector minimization, variational inequalities, Nash equilibria, fixedpoint and coincidence-point problems, complementarity problems, minimax inequalities, etc Stability properties of solutions have been investigated even in models for vector quasiequilibrium problems [1, 3, 4, 7, 8, 9], variational problems [5, 6, 10, 11] and the references therein In this paper we establish sufficient conditions for the solution sets Σα , to have Ξα the stability properties such as the upper semicontinuity and closedness with respect to parameter λ , γ , µ The structure of our paper is as follows In the remaining part of this section we recall definitions for later uses Section is devoted to the upper semicontinuity and closedness of solution sets for parametric quasiequilibrium problems (QEP λγµ ) and α (QEEP λαγµ ) Now we recall some notions in [1, 2, 12] Let X and Z be as above and be a multifunction is said to be lower semicontinuous (lsc) at x0 if G G(x0 ) ∩U ≠ ∅ for some open set U ⊆ Z implies the existence of a neighborhood N of G : X →2Z x0 such that, for all x ∈ N , G(x) ∩U ≠ ∅ An equivalent formulation is that: G is lsc at x0 ∀xα →x0 , ∀z0 ∈ G(x0 ), ∃ zα ∈ G(xα ), zα →z0 G is called upper semicontinuous if (usc) at x0 if for each open set U ⊇ G(x0 ) , there is a neighborhood N of x0 such that U ⊇ G(N ) G is said to be Hausdorff upper semicontinuous (H-usc in short; Hausdorff lower semicontinuous, H-lsc, respectively) at x0 if for each neighborhood B of the origin in Z , there exists a neighborhood of x0 such that, G( x) ⊆ G( x0 ) + B,∀x ∈ N N ( G( x0 ) ⊆ G( x) + B,∀x ∈ N ) is said to be continuous at x0 if it is both lsc and usc at G x0 and to be H-continuous at x0 if it is both H-lsc and H-usc at x0 We say that G satisfies a certain property in a A ⊆ if satisfies it at all points of A subset X G Proposition 1.1 (See [1, 2, 12]) Let A and Z be as above and G : A →2Z be a multifunction (i) If is usc x0 then G is H -usc at x0 Conversely if G is H -usc x0 and if G at at G(x0 ) compact, then G is usc at x0 ; (ii If G is usc at x0 and if G(x0 ) is closed, then G is closed at (iii) If Z is compact and G is closed at x0 then G is usc at x0 ; x0 ; (iv) If G has compact values, then G is usc at x0 if and only if, for each net {xα } ⊆ which converges to x and for each net {y } ⊆ G(x ) , there y ∈ G( x0 ) α α A are and a subnet {yβ } of {yα } such that yβ →y Main results In this section, we discuss the upper semicontinuity and closedness of solution sets for parametric quasiequilibrium problems (QEP λγµα ) and (QEEP λγµα ) Theorem 2.1 Assume for problem (QEP λαγµ ) that (i) E is usc at λ0 and E(λ is compact, ) and K2 is lsc in K1( A, Λ) × {λ 0} ; (ii) in K1 ( A, Λ) × K2 (K1 ( A, Λ), Λ) × {γ 0} , is usc and compact-valued T if α = m ), and lsc if α = s ; α= w (or (iii) the set {(x, t, y, µ, λ ) ∈ K1 ( A, Λ) × T (K1 ( A, Λ), K2 (K1 ( A, Λ), Λ), Γ) × K (K1 ( A, Λ), Λ) × {à } ì { }: F (x, t, y, µ ) ⊆/ − int C(x, λ )} is closed Then Σα is both upper semicontinuous and closed at (λ , γ , µ0 ) Proof Similar arguments can be applied to three cases We present only the proof for the cases where α = w We first prove that Σw is upper semicontinuous at (λ , γ , µ0 ) Indeed, we suppose to the contrary that is not upper semicontinuous at Σw (λ , γ , µ0 ) , i.e., there is an open set U of Σw (λ 0,γ 0, µ0 ) such that for all {(λ n , γ n , µn )} convergent to {(λ ,γ , µ0 )}, there xn ∈Σw (λ n ,γ n , µn ) xn ∈/ ∀n By the upper exists , U, semicontinuity of E and compactness of E(λ ) , one can assume xn →x0 for some that x0 ∈ E(λ ) x0 ∈/ Σ w (λ , γ , µ ) , then ∃ y0 ∈ K (x0 , λ ), ∀t0 ∈ T (x0 , such that If y0 , γ ) F (x0 , t0 , y0 , µ0 ) ⊆ − int C(x0 , λ ) By the lower semicontinuity of K at (x0 , λ ) , there exists (2.1) yn ∈ K2 (xn , λ n such ) that yn →y0 Since xn ∈Σw (λ n ,γ n , µn ) , ∃ tn ∈T (xn , yn , γ such that n ) F (xn , tn , yn , µ n ) − int C(x , λ n n ⊆/ ) Since (2.2) is usc and T (x0 , y0 , γ is compact, one has a subnet tm ∈ T (xm , ym ,γ ) such that tm →t0 for some t0 ∈T (x0 , y0 , γ ) T m ) By the condition (iii) we see a contradiction between (2.1) and (2.2) Thus, x0 ∈Σw (λ 0,γ 0, µ0 ) ⊆ U , this contradicts to the fact xn ∈/ ∀n Hence, Σw is upper U, semicontinuous at (λ , γ , µ0 ) Now we prove that Σw is closed at (λ , γ , µ0 ) Indeed, we suppose that Σw is not closed at (λ , γ , µ0 ) , i.e., there is (xn , λ n , γ n , µn ) →(x0 , λ ,γ , with a net µ0 ) xn ∈Σw (λ n ,γ n , µn but x0 ∈/ Σ w (λ , γ , µ ) The further argument is the same as ) above And so we have (λ , γ , µ ) □ is closed Σw at The following example shows that the upper semicontinuity and compactness of E are essential Example 2.2 Let A = B = X = Y = □ , Λ = Γ = M = [0,1], λ = 0, C(x, λ ) = □ , F (x, t, y, λ ) = 32λ +sinx , 1K (x, λ ) = (− λ − 1, λ2], K (x, λ ) = {0} and T (x, y, λ ) = [0, 23x ] Then, we have E(0) = (− 1, 0] +2cosλ and E( λ ) = (− λ − 1, λ ], ∀λ ∈ (0,1] We show that K2 is lsc and assumption (ii) and (iii) of Theorem 2.1 are fulfilled But is neither usc Σα nor closed at λ = and Σα (0, 0, is not compact The reason is that E is not usc at 0 0) and Σα (0, 0, 0) = (− 1, is not compact In fact and E (0) 0] Σα ( λ , γ , µ ) = (− λ − 1, λ ], ∀λ ∈ (0,1] Remark 2.3 K2 is lsc in K1 ( A, Λ) × {λ 0} (which is not The assumption in Theorem 2.1 we have imposed in this Theorem 4.1 of [8] and [7]) Example 2.4 shows that the lower semicontinuity of K2 needs to be added to Theorem 4.1 of [8] and [7] Example 2.4 Let X ,Y , Λ, Γ, M , λ , C(x, λ) as in Example 2.2 and let F (x, t, y, λ ) = x + y + λ , K1(x, λ ) [0, ],T (x, y, λ ) = {t} We have = 1 K (x, λ ) = - ,0, 2 A = B = [− 1 , ], 22 if λ = 0, 0, otherwise We have E( λ ) = [0,1], ∀λ ∈[0,1] Hence E is usc at E (0) and condition (ii) and (iii) of Theorem 2.1 are easily seen to be fulfilled But Σα is not upper semicontinuous at λ = The reason is that semicontinuous In fact is compact and K2 is not lower Σ (λ ,γ , µ) = =2 if λ = 0, α otherwise The following example shows that the condition (iii) of Theorem 2.1 is essential Example 2.5 Let Λ, Γ, M ,T , λ , as in Example 2.4 and X = Y = A = B = [0,1] , C let K1 (x, λ ) = K2 (x, λ ) = [0,1] and x− y F (x, t, y, λ ) = y2 − if λ = 0, otherwise We show that assumptions (i) and (ii) of Theorem 2.1 are easily seen to be fulfilled But Σα is not usc at λ = The reason is that assumption (iii) is violated Indeed, {(x , y , λ )} → taking x = 0, t = 0, y n n = ,λ = n n n tF (x , ,=y , λ ) →0 n →∞ , as 1 >0, but (0, , 0) F (0, 0, ,1 / n) n n n n n n n 2 F (0, 0,1, 0) = − < The following example shows that all assumptions of Theorem 2.1 are fulfilled But Theorem 3.4 in Anh and Khanh [1] cannot be applied and Example 2.6 Let A, B, X ,Y , Λ, Γ, M , λ , as C K1 (x, λ ) = K2 (x, λ ) = [0, 2],T (x, y,γ ) = [0,1] 0 if F (x, t, y, λ ) = λ = 0, in = then Example2.5 and let e cos x + otherwise We show that assumptions (i), (ii) and (iii) of Theorem 2.1 are easily seen to be fulfilled Hence, Σα is usc at (0, 0, 0) But Theorem 3.4 in Anh and Khanh [1] cannot be applied The reason is that F is not lsc at (x, y, 0) Remark 2.7 (i) In Theorem 4.1 in Kimura-Yao [8] the same conclusion as Theorem 2.1 was proved in another way Its assumptions (i)-(iv) derive (i) Theorem 2.1, assumptions (v)(or (vi)) coincides with (iii) of Theorem 2.1 (ii) In Theorem 4.1 in Kimura-Yao [7] the same conclusion as Theorem 2.1 was proved in another way Its assumptions (i)-(iv) derive (i) Theorem 2.1, assumption (v) coincides with (iii) of Theorem 2.1 Theorem 2.8 Assume for problem (QEEP λγµ α ) that (i) E is usc at λ0 an d E(λ is compact, ) and K2 is lsc in K1( A, Λ) × {λ 0} ; (ii) i α = (or K ( A, Λ) × K2 (K1 ( A, Λ), Λ) × {γ 0} , is usc and compact-valued n w T if {(x, t, y, µ, λ ) ∈ K1 ( A, Λ) × T (K1 ( A, Λ), K2 (K1 ( A, Λ), Λ), Γ) × α = m ), and lsc if α = s ; (iii) the set K2 (K1 ( A, Λ), Λ) × {à0}ì { 0}: F (x, t, y, ) − int C(x, λ ) = ∅} is closed Then Ξα is both upper semicontinuous and closed at (λ , γ , µ0 ) Proof Similar arguments can be applied to three cases We present only the proof for the cases where α = m We first prove that Ξ is upper semicontinuous at (λ , γ , µ0 ) m Indeed, we suppose to the contrary that Ξ is not upper semicontinuous at (λ , γ , µ0 ) , m i.e., there is an open set V of Ξm (λ 0,γ , µ0 ) such that for all {(λ n , γ n , µn )} convergent to {(λ ,γ , µ0 )}, there xn ∈Ξm (λ n ,γ n , µn ) xn ∈/ ∀n By the upper semicontinuity exists , V, of E and compactness of E(λ ) , one can assume xn →x0 for some x0 ∈ E(λ ) If that x0 ∈/ Ξm (λ , γ , µ ) , then ∀t0 ∈T (x0 , y0 , γ ), ∃ y0 ∈ K such that (2.3) (x0 , λ ) F (x0 , t0 , y0 , µ0 ) ∩− int C(x0 , λ ) ≠ ∅ By the lower semicontinuity of K at (x0 , λ ) , there exists that yn →y0 Since xn ∈Ξm (λ n ,γ n , µn ) , ∃ tn ∈T (xn , yn , γ such that n ) yn ∈ K2 (xn , λ n ) such F (xn , tn , yn , µn ) ∩− int C(xn , λ n ) = ∅ Since (2.4) is usc and T (x0 , y0 , γ is compact, one has a subnet tm ∈ T (xm , ym ,γ ) such that tm →t0 for some t0 ∈T (x0 , y0 , γ ) T m ) By the condition (iii) we see a contradiction between (2.3) and (2.4) Thus, x0 ∈Ξm (λ 0,γ 0, µ0 ) ⊆ V , this contradicts to the xn ∈/ ∀n Hence, Ξm is upper fact V, semicontinuous at (λ , γ , µ0 ) Now we prove that Ξ m is closed at (λ , γ , µ0 ) Indeed, we suppose that Ξm is not closed at (λ , γ , µ0 ) ,i.e., there is a net (xn , λ n , γ n , µn ) →(x0 , λ ,γ , with xn ∈Ξm (λ n ,γ n , µn but x0 ∈/ Ξm (λ , γ , µ ) µ0 ) ) The further argument is the same as above And so we have Ξ m is closed at (λ , γ , µ0 ) □ Remark 2.9 Theorem 2.8 is an extension of Theorem 4.1 in [8] The Example 2.3 is also shows that the lower semicontinuity of K2 needs to be added to Theorem 4.1 of Kimura-Yao in [8] 1 10 11 12 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 Berge C (1963), Topological Spaces, Oliver and Boyd, London 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 (Received: 30/01/2012; Revised: 21/10/2012; Accepted: 28/10/2012) ... the upper semicontinuity and closedness of solution sets for parametric quasiequilibrium problems (QEP λγµ ) and α (QEEP λαγµ ) Now we recall some notions in [1, 2, 12] Let X and Z be as above and. .. is not The assumption in Theorem 2.1 we have imposed in this Theorem 4.1 of [8] and [7]) Example 2.4 shows that the lower semicontinuity of K2 needs to be added to Theorem 4.1 of [8] and [7]... variational problems [5, 6, 10, 11] and the references therein In this paper we establish sufficient conditions for the solution sets Σα , to have Ξα the stability properties such as the upper semicontinuity