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THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79) 2014, VOL 1 99 EXISTENCE OF SOLUTIONS FOR THE SYSTEM OF VECTOR QUASIEQUILIBRIUM PROBLEMS SỰ TỒN TẠI NGHIỆM CHO HỆ BÀI TOÁN TỰA CÂN[.]
THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79).2014, VOL 99 EXISTENCE OF SOLUTIONS FOR THE SYSTEM OF VECTOR QUASIEQUILIBRIUM PROBLEMS SỰ TỒN TẠI NGHIỆM CHO HỆ BÀI TOÁN TỰA CÂN BẰNG VÉC TƠ Nguyen Van Hung1, Le Huynh My Van2, Phan Thanh Kieu1 Dong Thap University; Email: nvhung@dthu.edu.vn, ptkieu@dthu.edu.vn Vietnam National University - HCMC, University of Information Technology; Email: vanlhm@uit.edu.vn Abstract - In this paper, we study the systems of generalized multivalued strong vector quasiequilibrium problem (in short (SQVEP)) real locally convex Hausdorff topological vector spaces This problem includes as special cases the generalized strong vector quasi-equilibrium problems, quasi-equilibrium problems, equilibrium problems and variational inequality problems Then, we establish an existence theorem for its solutions by using fixed-point theorem Moreover, we also discuss the closedness of the solution set for (SQVEP) The results presented in the paper improve and extend the main results of Long et al [Math Comput Model,47 445-451 (2008)] and Plubtieng - Sitthithakerngkietet [Fixed Point Theory Appl doi:10.1155/2011/475121 (2011)] Tóm tắt - Trong báo này, nghiên cứu hệ toán tựa cân véc tơ đa trị mạnh tổng quát (viết tắt, (SQVEP)) không gian véc tơ tơ pơ Hausdorff thực lồi địa phương Bài tốn chứa nhiều toán đặc biệt toán tựa cân véc tơ mạnh tổng quát, toán tựa cân bằng, toán cân tốn bất đẳng thức biến phân Sau đó, chúng tơi thiết lập định lý cho tồn nghiệm sử dụng định lý điểm bất động Ngồi ra, chúng tơi thảo luận tính đóng tập nghiệm cho (SQVEP) Kết báo cải thiện mở rộng kết Long tác giả [Math Comput Model,47 445 451 (2008)] Plubtieng - Sitthithakerngkietet Sitthithakerngkietet [Fixed Point Theory Appl doi:10.1155/2011/475121 (2011)] Key words - system vector quasiequilibrium problems; multivalued; fixed-point theorem; existence; closedness Từ khóa - hệ toán tựa cân véctơ; đa trị; định lý điểm bất động; tồn tại; tính đóng Introduction and Preliminaries (iii) F is said to be continuous at x0 if it is both lsc and usc at x0 F is said to be continuous at x0 A if it is Let X, Y, Z be real locally convex Hausdorff topological vector spaces A X and B Y be two nonempty compact convex subsets and C Z be a solid pointed closed convex cone Let Ki , Pi : A A → A , Ti : A A → 2B and Fi : A B A → 2Z , i = 1,2 be multifunctions We consider the following system generalized strong vector quasiequilibrium problems (in short, (SQVEP): (SQVEP): Find ( x , u ) A A and z T1 ( x , u ), v T2 ( x , u ) such that x K1 ( x , u ), u K ( x , u ) satisfying F1 ( x , z , y) C , y P1 ( x , u ), F2 (u , v , y) C , y P2 ( x , u ) We denote that ( F ) is the solution set of (SQVEP) Next, we recall some basic definitions and their some properties Definition 1.1 (See [1, 3]) Let X, Z be two topological vector spaces, A be a nonempty subset of X and F : A → 2Z be a multifunction (i) F is said to be lower semicontinuous (lsc) at x0 if F ( x0 ) U for some open set U Z implies the existence of a neighborhood N of x0 such that, for all x N , F ( x) U said to be lower semicontinuous in continuous at each x0 A (iv) F is said to be closed if Graph(F)={(x, y) : x A, y F(x)} is a closed subset in A Y F is said to be closed in A if it is closed at each x0 A Definition 1.2 (See [1, 2]) Let X, Z be two topological vector spaces, A be a nonempty subset of X and F : A → 2Z be a multifunction and C Z is a nonempty closed convex cone (i) F is called upper C -continuous at x0 A , if for any neighborhood U of the origin in Z , there is a neighbourhood V of x0 such that F ( x) F ( x0 ) + U + C , x V (ii) F is called lower C -continuous at x0 A , if for any neighborhood U of the origin in Z , there is a neighborhood V of x0 such that F ( x0 ) F ( x) + U − C , x V Definition 1.3 (See [3]) Let X and Z be two topological vector spaces and A is a nonempty convex subset of X A set-valued mapping F : A → 2Z is said to be properly C -quasiconvex if for any x, y A and t [0,1] , we have A if it is lower semicontinuous at each x0 A either F ( x) F (tx + (1 − t ) y ) + C (ii) F is said to be upper semicontinuous (usc) at x0 if for each open set U F ( x0 ) , there is a neighborhood N of x0 or F ( y ) F (tx + (1 − t ) y) + C such that U F ( N ) F is said to be upper semicontinuous in A if it is upper semicontinuous at each x0 A Lemma 1.1 (See [3]) Let X, Z be two topological vector spaces, A be a nonempty convex subset of X and F : A → 2Z be a multifunction 100 Nguyen Van Hung, Le Huynh My Van, Phan Thanh Kieu (i) If F is upper semicontinuous at x0 A with closed values, then F is closed at x0 A ; Then ( F ) Moreover ( F ) is closed Proof For all ( x, z, u, v) A B A B , define be set- (ii) If F is closed at x0 A and Z is compact, then F is upper semicontinuous at x0 A valued mappings: , : A B A → A by (iii) If F has compact values, then F is usc at x0 if and only if, for each net {x } A which converges to x0 and for each net { y } F ( x ) , there are y F ( x0 ) and a ( x, v, u ) = {b K ( x, u ) : F (b, v, y) C , y P2 ( x, u )} subnet { y } of { y } such that y → y Lemma 1.1 (See [1,5]) Let A be a nonempty compact subset of a locally convex Hausdorff vector topological space Z If M : A → A is upper semicontinuous and for any x A, M ( x) is nonempty, convex and closed, then there exists an x* A such that x* M ( x* ) Main Results Definition 2.1 Let X, Z be two real locally convex Hausdorff topological vector spaces and A be a nonempty compact convex subset of X , and C Z is a nonempty closed convex cone Suppose F : A → 2Z be a multifunction F is said to be strongly C -quasiconvex in A if x1 , x2 A, [0,1] , F ( x1 ) C and F ( x2 ) C Then, it follows that F ( x1 + (1 − ) x2 ) C Remark 2.1 In the Definition 2.1, if we let X = A = Z = , C = R− , and let F : → be a singlevalued mapping Then, we have • x1 , x2 A, t [0,1] , if F ( x1 ) , F ( x2 ) , then F ((1 − t ) x1 + tx2 )) This means that F is modified 0-level quasiconvex, since the classical quasiconvexity says that x1 , x2 A, [0,1] , F ((1 − ) x1 + x2 )) max{F ( x1 ), F ( x2 )} Now, we let X and Z be two topological vector spaces, A be a nonempty convex subset of X F : A → 2Z , Z and C Z be a solid pointed closed convex cone, we will use the following level-sets types: L F := {x A : F ( x) + C}; L F := {x A : F ( x) + C} ( x, z , u ) = {a K1 ( x, u ) : F ( a, z , y ) C , y P1 ( x, u )}, and Step Show that ( x, z, u ) and ( x, v, u ) are nonempty Indeed, for all ( x, z, u) A B A and ( x, v, u) A B A , for each {i = 1, 2} , Ki ( x, u ), Pi ( x, u ) are nonempty Thus, by assumption (iii), we have ( x, z, u ) and ( x, v, u ) are nonempty Step Show that ( x, z, u ) and ( x, v, u ) are convex subsets of A Let a1 , a2 ( x, z , u ) and [0,1] and put a = a1 + (1 − )a2 Since a1 , a2 K1 ( x, u ) and K1 ( x, u ) is a convex set, we have a1 , a2 ( x, z , u ) , we have a K1 ( x, u ) Thus, for F1 (a1 , z, y) C, y P1 ( x, u ), F1 (a2 , z, y) C, y P1 ( x, u ) By (iv), F_1(., z, y) is strongly C -quasiconvex F1 ( a1 + (1 − )a2 , z, y) C , [0,1], i.e., a ( x, z, u ) Therefore, ( x, z, u ) is a convex subset of A Similarly, we have ( x, v, u ) is a convex subset of A Step ( x, z, u ) and ( x, v, u ) are closed subsets of A {an } ( x, z , u ) an → a0 Then, Let with an K1 ( x, u ) Since K1 ( x, u ) is a closed subset of A , it follow that a0 K1 ( x, u ) By the lower semicontinuity of P1 , we have y0 P1 ( x, u ) and any net {( xn , un )} → ( x, u ) , there exists a net { yn } such that yn P1 ( xn , un ) and yn → y0 As an ( x, z, u ) , we have F1 (an , z, yn ) C (2.1) By assumption (v), (2.1) yields that F1 (a0 , z , y0 ) C , Theorem 2.1 For each {i = 1, 2} , assume for the problem (SQVEP) that i.e., a0 ( x, z, u ) Therefore, ( x, z, u ) is closed Similarly, we also have ( x, v, u ) is closed (i) K i is upper semicontinuous, K i ( x, u ) is nonempty closed convex, for all ( x, u ) A A and Pi is lower semicontinuous and nonempty; Step Now, we need to show that ( x, z, u ) and ( x, v, u ) are upper semicontinuous (ii) Ti is upper semicontinuous and compact-valued and Ti ( x, u ) is nonempty convex, for all ( x, u ) A A ; (iii) for all ( x, z ) A B , Fi ( x, z, Pi ( x, u )) C ; (iv) for all ( z , y ) B A , Fi (., z, y) is strongly C quasiconvex; (v) for all z B , L Fi (., z ,.) is closed First, we show that ( x, z, u ) is upper semicontinuous Indeed, since A is a compact set and ( x, z, u ) A Hence ( x, z, u ) is compact Now we need to show that is a closed mapping Indeed, Let a net {( xn , zn , un ) : n I } A B A such that ( xn , zn , un ) → ( x, z, u ) A B A , and let an ( xn , zn , un ) such that an → a0 THE UNIVERSITY OF DANANG, JOURNAL OF SCIENCE AND TECHNOLOGY, NO 6(79).2014, VOL Now we need to show that a0 ( x, z, u ) Since an K1 ( xn , un ) and K is upper semicontinuous, we have a0 K1 ( x, u ) Suppose to the contrary a0 ( x, z , u ) Then, y0 P1 ( x, u ) such that (2.2) By the lower semicontinuity of P1 , there is a net yn P1 ( xn , un ) such that yn → y0 Since an ( xn , zn , un ), we have F1 (an , zn , yn ) C (2.3) By the condition (v) we have F1 (a0 , z, y0 ) C (2.4) This is the contradiction between (2.2) and (2.4) Thus, a0 ( x, z, u ) Hence, is upper semicontinuous also have a net {( xn , un ), n I } ( F ) : As ( xn , un ) ( F ) , zn T1 ( xn , un ), T2 ( xn , un ) , there exist xn K1 ( xn , un ), un K ( xn , un ) such that F1 (a0 , z, y0 ) C Similarly, we semicontinuous Indeed, let ( xn , un ) → ( x0 , u0 ) 101 ( x, v, u ) is upper , : A B A : ( x, z, u ) = ( ( x, z, u ), T1 ( x, u)), ( x, z, u) A B A and ( x, v, u ) = (( x, v, u ), T2 ( x, u )), ( x, v, u ) A B A Then , are upper semicontinuous ( x, z, u ) A B A , ( x, v, u ) A B A , and ( x, z , u ) and ( x, v, u ) be two nonempty closed convex subsets of A B A Define the set-valued mapping H : ( A B) ( A B) → 2( A B )( A B ) by H (( x, z ), (u, v)) = (( x, z, u ), ( x, v, u )), for all (( x, z ), (u, v)) ( A B) ( A B) Then H is also upper semicontinuous and (( x, z ), (u, v)) ( A B) ( A B) , H (( x, z ), (u, v)) is a nonempty closed convex subset of ( A B) ( A B) By Lemma 1.1 there exists a point (( xˆ, zˆ)(uˆ, vˆ)) ( A B) ( A B) such that (( xˆ , zˆ )(uˆ , vˆ)) H (( xˆ, zˆ)(uˆ, vˆ)) , that is ( xˆ, uˆ ) ( xˆ, zˆ, uˆ ), (uˆ, vˆ) ( xˆ, uˆ, vˆ) which implies that xˆ ( xˆ, zˆ, uˆ ), uˆ T1 ( xˆ, uˆ ) and uˆ ( xˆ, uˆ, vˆ), vˆ T2 ( xˆ, uˆ ) Hence, there exist ( xˆ, uˆ ) A A, zˆ T1 ( xˆ, uˆ ), vˆ T2 ( xˆ, uˆ ) such that xˆ K1 ( xˆ, uˆ ), uˆ K ( xˆ , uˆ ) satisfying F1 ( xˆ, zˆ, y ) C , y P1 ( xˆ, uˆ ), and F2 (uˆ, vˆ, y) C , y P2 ( xˆ, uˆ ), i.e., (SQVEP) has a solution Step Now we prove that ( F ) is closed F2 (un , , y ) C , y P2 ( xn , un ) Since K1 , K are upper semicontinuous and closedK1 , K valued Thus, are closed Hence, x0 K1 ( x0 , u0 ), u0 K ( x0 , u0 ) Since T1 , T2 are upper semicontinuous and T1 ( x0 , u0 ), T2 ( x0 , u0 ) are compact There exist z T1 ( x0 , u0 ) and v T2 ( x0 , u0 ) such that zn → z , → v (taking subnets if necessary), we have z T ( x0 ) such that zn → z By the condition (v), we have F1 ( x0 , z, y ) C , y P1 ( x0 , u0 ), and F2 (u0 , v, y ) C , y P2 ( x0 , u0 ) Step Now we need to the solutions set ( F ) Define the set-valued mappings → AB by F1 ( xn , zn , y ) C , y P1 ( xn , un ) and This means that ( x0 , u0 ) ( F ) Thus ( F ) is a closed set Remark 2.2 (a) If K1 ( x , u ) = P1 ( x , u ) = S1 ( x ) , K ( x , u ) = P2 ( x , u ) = S ( x ) , T1 ( x , u ) = T1 ( x ) , T2 ( x , u ) = T2 ( x ) , then (SQVEP) become to (SGSVQEPs) in [4] (b) If K1 ( x , u ) = P1 ( x , u ) = K ( x , u ) = P2 ( x , u ) = S ( x ), T1 ( x , u ) = T2 ( x , u ) = T ( x ) , then (SQVEP) become to (SGSVQEP) in [2] Remark 2.3 In this special case as Remark 2.2, Theorem 2.1 reduces to Theorem 3.1 in [4] and Theorem 3.1 in [2] However, our Theorem 2.1 is stronger than Theorem 3.1 in [4] and Theorem 3.1 in [2] Moreover, we omit the assumption F is lower (-C)-continuous in Theorem 3.1 in [4] and Theorem 3.1 in [2] The following example shows that in this the special case, all the assumptions of Theorem 2.1 may be satisfied, but Theorem 3.1 in [4] and Theorem 3.1 in [2] are not fulfilled Example 2.1 Let X = Y = Z = R , A = B = [−1,1], C = [0, +) and let K1 ( x) = K2 ( x) = [0,1], F :[−1,1] → 2R [0, ] and T1 ( x, u ) = T2 ( x, u ) = [0, ] and if x0 = u0 = , otherwise F1 ( x, z, y ) = F2 (u, v, y ) = 1 [ , ] F ( x) = [ ,1] if x0 = , otherwise We show that all assumptions of Theorem 2.1 are satisfied So by this Theorem the considered problem has 102 Nguyen Van Hung, Le Huynh My Van, Phan Thanh Kieu solutions However, F is not lower ( −C ) -continuous at 1 Indeed, we let a neighborhood U = [− , ] of the 8 1 origin in Z , then for any neighborhood V = [ − , + ] 3 1 of x0 = , where 0, choose x* V We have, 3 x0 = 1 F ( x0 ) = F ( ) = [ , ] F ( x* ) + U + C 3 1 = [ ,1] + [− , ] + + 8 = [ , ]+ +, 8 Also, Theorem 3.1 in [4] and Theorem 3.1 in [2] does not work The following example shows that the assumption (v) of Theorem 2.1 is strictly weaker than the assumption upper C-continuous in [2,4] Example 2.2 Let X = Y = Z = R , A = B = [−1,1], C = [0, +) and let K1 ( x) = K ( x) = [0,1], T1 ( x, u ) = T2 ( x, u ) = {1}, F :[−1,1] → 2R and F1 ( x, z, y ) = F2 (u, v, y ) = [1, ] F ( x) = [ , ] if x0 = , otherwise We show that all assumptions of Theorem 2.1 are satisfied So (SQVEP) has solution However, F is not upper C -continuous at x0 = Indeed, we let a 1 neighborhood U = [− , ] of the origin in Z , then for any 3 1 neighborhood V = [ − , + ] of x0 = , where 0, 3 choose We see that, x* V 1 F ( x* ) = [ , ] F ( x0 ) + U + C 3 1 = [1, ] + [− , ] + + 3 11 = [ , ]+ +, Also, Theorem 3.1 in [4] and Theorem 3.1 in [2] does not work The following example shows that our strongly C quasiconvexity is strictly weaker than the C quasiconvexity in [2,4] Example 2.3 Let X = Y = Z = R , A = B = [0,1], C = [0, +) and let K1 ( x) = K ( x) = [0,1], T1 ( x, u ) = T2 ( x, u ) = [1, 2], F :[0,1] → 2R and F1 ( x, z, y ) = F2 (u, v, y ) = [ ,2] if x0 = , 2 F ( x) = 1 [ , ] otherwise We show that all assumptions of Theorem 2.1 are satisfied However, F is not properly C -quasiconvex Indeed, we let = and x1 = 0, x2 = Then, 1 F ( x1 ) = F (0) = [ , ] F ( x1 + (1 − ) x2 ) + C 3 = F ( ) + + = [ , 2] + + , 2 1 F ( x2 ) = F (1) = [ , ] F ( x1 + (1 − ) x2 ) + C 3 = F ( ) + + = [ , 2] + + 2 Thus, it gives case where Theorem 2.1 can be applied but Theorem 3.1 in [4] and Theorem 3.1 in [2] does not work Acknowledgment This research is the output of the project “On existence of solution maps for system multivalued quasi-equilibrium problems and its application” under grant number D2014-06 which belongs to University of Information Technology-Vietnam National University HoChiMinh City REFERENCES [1] Aubin J P., Ekeland I, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984 [2] Long X J., Huang N J., Teo K.L., Existence and stability of solutions for generalized strong vector quasi-equilibrium problems, Math Computer Model 47, 445-451 (2008) [3] Luc D T, Theory of Vector Optimization: Lecture Notes in Economics and Mathematical Systems, Springer-Verlag Berlin Heidelberg, 1989 [4] Plubtieng S., Sitthithakerngkiet K, On the existence result for system of generalized strong vector quasi-equilibrium problems, Fixed Point Theory Appl Article ID 475121, doi:10.1155/2011/475121 (2011) [5] Yu J, Essential weak efficient solution in multiobjective optimization problems, Nonlinear Anal TMA., 70 (2009), 15281535 (The Board of Editors received the paper on 01/01/2014, its review was completed on 23/02/2014) ... Huang N J., Teo K.L., Existence and stability of solutions for generalized strong vector quasi- equilibrium problems, Math Computer Model 47, 445-451 (2008) [3] Luc D T, Theory of Vector Optimization:... and Mathematical Systems, Springer-Verlag Berlin Heidelberg, 1989 [4] Plubtieng S., Sitthithakerngkiet K, On the existence result for system of generalized strong vector quasi- equilibrium problems, ... where Theorem 2.1 can be applied but Theorem 3.1 in [4] and Theorem 3.1 in [2] does not work Acknowledgment This research is the output of the project “On existence of solution maps for system