Untitled 126 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No K2 2017 Abstract— The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequal[.]
126 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 The GAP function of a parametric mixed strong vector quasivariational inequality problem Le Xuan Dai, Nguyen Van Hung, Phan Thanh Kieu Abstract— The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequality problems, fixed point problems, coincidence point problems, complementary problems etc There are many authors who have been studied the gap functions for vector variational inequality problem This problem plays an important role in many fields of applied mathematics, especially theory of optimization In this paper, we study a parametric gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) in Hausdorff topological vector spaces (SQVIP) Find x K ( x, ) and z T ( x, ) such that < z, y x > f ( y , x, ) Rn , y K ( x, ), n where we denote the nonnegative of R by Rn = {t = (t1, t2 ,, tn )T Rn | ti 0, i = 1,2,, n} Moreover, we also discuss the lower semicontinuity, upper semicontinuity and the continuity for the parametric gap function for this problem To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces Hence the results presented in this paper (Theorem 1.3 and Theorem 1.4) are new and different in comparison with some main results in the literature Manuscript Received on July 13th, 2016 Manuscript Revised December 06th, 2016 This research is funded by Ho Chi Minh City University of Technology - VNU-HCM under grant number T-KHUD-2016107 Le Xuan Dai is with Department of Applied Mathematics, Ho Chi Minh City University of Technology - VNU-HCM, Vietnam National University - Ho Chi Minh City, Vietnam, Email: ytkadai@hcmut.edu.vn Nguyen Van Hung is with Department of Mathematics, Dong Thap University, Cao Lanh City, Vietnam, Email: nvhung@dthu.edu.vn Phan Thanh Kieu is with Department of Mathematics, Dong Thap University, Cao Lanh City, Vietnam, Email: ptkieu@dthu.edu.vn Index Terms—Vector quasivariational inequality problem; parametric gap function; lower semicontinuity; upper semicontinuity, continuity INTRODUCTION L et X and be Hausdorff topological vector spaces Let L( X , R n ) be the space of all linear continuous operators from X to R n K : X X , n T : X 2L ( X , R ) are set-valued mappings and let f : X R be continuous single-valued mappings For consider the following parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) (SQVIP) Find x K ( x, ) and z T ( x, ) such that n < z, y x > f ( y , x, ) Rn , y K ( x, ), where we denote the nonnegative of R n by Rn = {t = (t1, t2 ,, tn )T Rn | ti 0, i = 1,2,, n} Here the symbol also denote T denotes the transpose We intRn = {t = (t1, t ,, tn )T Rn | ti > 0, i = 1,2,, n} For each we let E ( ) := {x X | x K ( x, )} and : X be set-valued mapping such that ( ) is the solution set of (SQVIP) Throughout the paper, we always assume that ( ) for each in the neighborhood The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational nequality problems, fixed point problems, coincidence point problems, complementarity problems etc, There are many authors have been studied the gap functions for vector variational inequality problem, see ([2]-[6], [8]-[10]) and the references therein The structure of our paper is as follows In Section of this article, we introduce the model vector quasivariational inequality problem and recall definitions for later uses In Section 2, we establish the lower semicontinuity, the upper semicontinuity and the continuity for the gap 127 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 function of parametric mixed strong vector quasivariational inequality problem A Preliminaries Next, we recall some basic definitions and their some properties Definition 1.1 (See [1], [7]) Let X and Z be Hausdorff topological vector spaces and F : X Z 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 An equivalent formulation is that: F is lsc at x0 if x x0 , z0 F ( x0 ), z F ( x ), z z0 F is said to be lower semicontinuous in X if it is lower semicontinuous at each x0 X 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 such that U F (N ) F is said to be upper semicontinuous in X if it is upper semicontinuous at each x0 X 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 if it is continuous at each x0 X iv) F is said to be closed at x0 X if and only if xn x0 , yn y0 such that yn F ( xn ), we have y0 F ( x0 ) F is said to be closed in X if it is closed at each x0 X Lemma 1.1 (See [1], [7]) If F has compact values, then F is usc at x0 if and only if, for each net { x } X which converges to x0 and for each net { y } F ( x ), there are y F (x) and a subnet { y } of { y } such that y y B Main Results In this section, we introduce the parametric gap functions for parametric mixed strong vector quasivariational inequality problem, then we study some properties of this gap function Definition 1.2 A function h : X R is said to be a parametric gap function of (SQVIP) if it satisfies the following properties [i)] i) h( x, ) for all x E ( ) i) h( x0 , ) = if and only if x0 ( ) Now we suppose that K ( x, ) and T ( x, ) are compact sets for any ( x, ) X We define function h : X R as follows h ( x, ) = max zT ( x, ) yK ( x , ) (< z, x y > f ( y , x, ))i (1) where (< z , x y > f ( y , x, )) i is the i th component of < z , x y > f ( y, x, ), i = 1,2,, n Since K ( x, ) and T ( x, ) are compact sets, h( x, ) is well-defined In the following, we will always assume that f ( x, x, ) = for all x E ( ) Theorem 1.2 The function h( x, ) defined by (1) is a parametric gap function for the (SQVIP) Proof We h1 : X L( X , R n ) R n h1 ( x, z ) = define as follows a function max (< z, x y > f ( y , x, ))i , max yK ( x , ) 1i n where x E ( ), z T ( x, ) i) It is easy to see that h1( x, z ) Suppose to the contrary that there exists x0 E ( ) and z T ( x0 , ) such that h1( x0 , z ) < 0, then > h1( x0 , z0 ) = max (< z0 , x y > f ( y , x0 , ))i max yK ( x0 , ) 1i n max (< z0 , x y > f ( y, x0 , ))i , y K ( x0 , ), 1i n which is impossible when h1 ( x, z ) = max y = x0 Hence, max (< z, x y > f ( y , x, ))i 0, yK ( x , ) 1 i n where x E ( ), z T ( x, ) Thus, since arbitrary, we have h ( x, ) = max zT ( x , ) yK ( x , ) z T ( x, ) is (< z, x y > f ( y , x, ))i ii) By definition, h( x0 , ) = if and only if there exists z T ( x0 , ) such that h1( x0 , z ) = 0, i.e., max for max (< z0 , x0 y > f ( y, x0 , ))i = 0, yK ( x0 , ) 1i n x E ( ) if and only if, for any y K ( x0 , ), max (< z0 , x0 y > f ( y , x0 , ))i 0, 1i n namely, there is an index (< z0 , x0 y > f ( y, x0 , ))i 0, to such that which is equivalent i0 n, < z0 , x0 y > f ( y, x0 , ) Rn , y K ( x0 , ), that is, x0 ( ) Remark 1.1 As far as we know, there have not been any works on parametric gap functions for mixed strong vector quasiequilibrium problems, and hence our the parametric gap functions is new and cannot compare with the existing ones in the literature SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 128 Example 1.1 Let X = R, n = 2, = [0,1], 1 K ( x, ) = [0,1], T ( x, ) = ,3 x x 2 = and Now we consider the problem (QVIP), finding x K ( x, ) and z T ( x, ) such that f ( y , x, ) = = h0 ( x , z , ) = 1 < z, y x > f ( y , x, ) = ( y x ), (3 x x )( y x) 2 R2 It follows from a direct computation for all [0,1] Now we show that h(.,.) is a parametric gap function of (SQVIP) Indeed, taking e = (1,1) intRn , we have ( ) = {0} h( x , ) = = max yK ( x, ) max (< z, x y > f ( y, x, ))i max zT ( x , ) yK ( x, ) 1i n if x = 0, ((3 x x4 )( x y)) = x x , if x (0,1] Hence, h(.,.) is a parametric gap function of (SQVIP) The following Theorem 1.3 gives sufficient condition for the parametric gap function h(.,.) is continuous in X Theorem 1.3 Consider (SQVIP) If the following conditions hold: i) K (.,.) is continuous with compact values in = Proof First, we prove that h(.,.) semicontinuous in X Indeed, we let suppose that {( x , )} X h ( x , ) a , and ( x , ) ( x0 , ) as follows that h( x , ) = = 1i n Since T (.,.) is upper semicontinuous with compact values in X , there exists z T ( x0 , ) such that z z (taking a subnet {z } of {z } if necessary) 1i n limit in (2), we have max (< z , x y > f ( y, x, ))i E ( ) yK ( x , ) 1i n are continuous, we have is continuous, and since K (.,.) is continuous with compact values in X Thus, by Proposition 19 in Section of Chapter [1] we can deduce that h0 ( x, z , ) is continuous By the compactness of T ( x , ), there exists z T ( x , ) Since g and such that h ( x , ) = (3) max (< z0 , x0 y0 > f ( y0 , x0 , ))i a 1i n Since (3) that y K ( x0 , ) is arbitrary, it follows from h0 ( x0 , z0 , ) = = max max (< z0 , x0 y > f ( y, x0 , ))i a yK ( x0 , ) 1i n and so, for any = z T ( x0 , ), max we have h ( x0 , ) = max (< z, x0 y > f ( y, x0 , ))i a zT ( x0 , ) yK ( x0 , ) 1i n is lower and satisfying It We define the function h0 : X L( X , R n ) R by (< z , x y > f ( y , x, )) i Since is continuous Taking the This proves that, for a R, the level set {( x, ) X | h( x, ) a} is closed Hence, h(.,.) is lower semicontinuous in X Theorem 1.4 Consider (SQVIP) If the following conditions hold: [i)] K (.,.) is continuous with compact values in max (< z, x y > f ( y, x , ))i a max max as max (< z, x y > f ( y, x, ))i a R zT ( x , ) yK ( x , ) 1i n h0 ( x, z, ) = max (< z, x y > f ( y , x , ))i a max yK ( x , ) 1i n Since K (.,.) is lower semicontinuous in X , for any y0 K ( x0 , ), there exists y K ( x , ) such that y y0 For y K ( x , ), we have (2) max (< z, x y > f ( y , x , ))i a X ; ii) T (.,.) is upper semicontinuous with compact values in X Then h(.,.) is lower semicontinuous in X max (< z, x y > f ( y, x , ))i max zT ( x , ) yK ( x , ) 1i n f X ; T (.,.) is continuous with compact values in X Then h(.,.) is continuous in X Proof Now, we need to prove that h(.,.) is upper semicontinuous in X Indeed, let a R and suppose that satisfying {( x , )} X h ( x , ) a , for all and ( x , ) ( x0 , ) as , then h( x , ) = = max max (< z, x y > f ( y , x , ))i a zT ( x , ) yK ( x , ) 1i n and so 129 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 20, SỐ K2-2017 max max (< z, x y > f ( y, x , ))i a , z T ( x , ) yK ( x , ) 1i n (4) REFERENCES [1] J P Aubin and I Ekeland, Applied Nonlinear Analysis, 1em plus 0.5em minus Since T (.,.) is lower semicontinuous with 0.4em John Wiley and Sons, New York, 1984 compact values in X 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[9] S K Mishra and S Y Wang and K K Lai, zT ( x0 , ) yK ( x0 , ) 1i n Gap function for set-valued vector variationallike, 1em plus 0.5em minus 0.4em J Optim This proves that, for a R, the level set Theory Appl., 138, pp 77–84, 2008 {( x, ) X | h( x, ) a} is closed Hence, h(.,.) is [10] X Q Yang and J C Yao, Gap Functions and upper semincontinuous in X Existence of Solutions to Set-Valued Vector Variational Inequalities, 1em plus 0.5em minus 0.4em J Optim Theory Appl 115, pp CONCLUSION 407–417, 2002 To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces Hence our results, Theorem 1.3 and Theorem 1.4 are new SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 20, No.K2- 2017 130 Hàm GAP cho toán bất đẳng thức tựa biến phân véc tơ tham số hỗn hợp mạnh Lê Xuân Đại, Nguyễn Văn Hưng, Phan Thanh Kiều Tóm tắt - Bài tốn bất đẳng thức tựa biến phân véc tơ tham số hỗn hợp mạnh bao gồm nhiều vấn đề toán bất đẳng thức biến phân, toán điểm bất động, toán điểm trùng lặp, toán bù nhau, v.v Có nhiều tác giả nghiên cứu tìm hàm gap cho toán bất đẳng thức biến phân véc tơ Bài tốn đóng vai trị quan trọng nhiều lĩnh vực toán ứng dụng, đặc biệt lý thuyết tối ưu Trong báo này, nghiên cứu hàm gap tham số với hỗ trợ hàm phi tuyến vơ hướng cho tốn bất đẳng thức tựa biến phân véc tơ tham số hỗn hợp mạnh (viết tắt (SQVIP)) không gian tô pô véc tơ Hausdorff (SQVIP) Tìm x K ( x, ) z T ( x, ) cho < z, y x > f ( y , x, ) Rn , y K ( x, ), với Rn = {t = (t1, t2 ,, tn )T Rn | ti 0, i = 1,2,, n} Ngồi ra, chúng tơi thảo luận tính nửa liên tục dưới, nửa liên tục tính liên tục hàm gap tham số cho tốn Theo hiểu biết mình, chúng tơi cho tới chưa có báo nghiên cứu tính nửa liên tục dưới, tính liên tục hàm gap mà không cần trợ giúp hàm phi tuyến vô hướng toán bất đẳng thức tựa biến phân véc tơ tham số hỗn hợp mạnh không gian tô pô véc tơ Hausdorff Do kết trình bày báo (Định lý 1.3 Định lý 1.4) khác biệt so với số kết tài liệu tham khảo Từ khóa - Bài toán bất đẳng thức tựa biến phân véctơ; hàm gap tham số; tính nửa liên tục dưới; tính nửa liên tục trên, tính liên tục ... devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem. .. we introduce the parametric gap functions for parametric mixed strong vector quasivariational inequality problem, then we study some properties of this gap function Definition 1.2 A function h... that is, x0 ( ) Remark 1.1 As far as we know, there have not been any works on parametric gap functions for mixed strong vector quasiequilibrium problems, and hence our the parametric gap