L (i) Lu uc (ii) Lu is h ct is m ng d n c t qu m t H L IC cg il ic nt tc d ng ng d ih c t ki n th c h ng d c bi Gi ih u th ng d c hi h ng th i c nh ng ki n th n u khoa h c c bi ct om n u ki n t t nh t , chia s t c c hi M tc g ng thi mong nh t h c H Thang Long University Libraty M CL C C S T N T I NGHI M C NG 1.1 t qu b tr 1.2 S t n t i nghi m c i gi thi t gi u 14 1.3 S t n t i nghi m c i gi thi t t u 19 1.4 C ng h p t 23 M H U HI U HI U HENIG C A 27 27 2.2 30 2.3 S t n t i nghi m 34 a t p nghi m 41 K T LU N 46 U THAM KH O 47 M U c nhi bao g m nhi ng h c bi bi ng th c mb ng s t n t i nghi u ki n t nh nghi m, thu Nhi u k t qu v s t n t i nghi m c n c Bianchi, Hadjisavv t qu v s t n t i nghi m h u hi u y u c v thi t u ho c t tl pm ts k t qu v s t n t i nghi m h u hi u, nghi m h u hi u Henig c b a t p nghi m h u hi u Henig c a b th c bi ng c nhi u c s t n t i nghi m c a Lu t qu u v s t n t i nghi m a t p nghi m c c a Bianchi, Hadjisavv Lu m ph n m t lu u tham kh o t n t i nghi m c ng vec t qu c Schaible [3] v s t n t i nghi m h u hi u y u c u ho c t ng vec im u ki n b c Thang Long University Libraty m h u hi u hi u Henig c b ng vec m nghi m h u hi u Henig c vec ng t qu v t qu v t n t i nghi m h u hi t p nghi m h u hi u y u c a b a t p nghi m h u hi ng th c bi t qu Stampacchia a X Gong [7] t qu v s t n t i nghi m h u hi u y u c a u ho c t u ki n b t qu a M Bianchi, N Schaible [3] 1.1 t qu b tr Cho X g gian vec Y gian vec C C sinh x y Do int C Y, int C Y y x C , Y x y y x int C, x y y x C, x x, y F(y, x) x, y F(x, y) > F(y, x) K, F F(x, y) F: K x K K 0, F(y, x) < F x y , x, y K, F(x, y) F(y, x) < f:K Y, Y }, L( ) {x K: f (x) K Y f:K Y U ( ) {x K: f (x) } ong K C L( ) {x K: f (x) - int C} = f 1[( L( ) {x K: f (x) - ( int C )} = f 1[( f:K (t ) Y int C )c ] int C )c ] - x, y K f ( x t ( y x)) , t [0, 1] Ta c C : V Y f:K C- f (x*) Y f ( x) V C, f U x U C1.3 H x* x* X cho K K f: K K CY x K - 10 Thang Long University Libraty T: A L (X, Y) (i) (Tx - Ty, x - y) (ii) Cho f 0, x, y A C*\{0} T - f ((Tx, x - y)) + f ((Ty, y - x)) 0, A C*\{0}, Cho T: A x, y A Cho f , L(X, Y) - hemi (t) := (T(ty + (l - t) x), y - x), t C*\{0} f (t) := f((T(ty + (1- t) x), y - x)), t - [0, 1] x, y [0, 1] i T C*\{0}, T ( f ) ( x0 ) q(x) v A A x0 q( x0 ) Y, A x0 X cho V + C, U( x0 ) 35 A, f [q(x) U( x0 ) V - C, q C A] C- C A CCho f C*\ f q: A x q R A f A A N q x0 C x0 T A q q1 q1, q2 C A N q2 C f q: A A A -q C- C f C*\ R A [3] Cho A C- X X x1, x2 A q(t x1 + (1- t) x2) A, t q: A Y [0, 1], tq( x1 ) + (1- t) q( x2 ) (Fan) f, cho A A0 A0 (x) n { x1, , x n A0 E( xi ) i x0 {E(x): x A0 } A0 cho E( x0 ) 36 Thang Long University Libraty 2A E: A n co { x1, , xn } E( xi ), x1, , xn } i A, X, f co (D C D L(X, Y) f(X, X*)], [ V(A, F) , Vf (A, F) (x, y) = (Tx, y - x) + q(y) - q(x), x, y x , A A f(F(x, y)) = f((Tx, y - x)) + f(q(y)) - f(q(x y E, G: A E(y) = {x A: f((Tx, x - y)) + f(q(x A: f((Ty, y - x)) + f(q(y 2A f(q(y))}, G(y) = {x A f(q(x))} 2.4 [10] A {E(y): y A} = {E(y): y {G(y): y A} A} n 37 E Vf (A, F) A - convexlike, [0, 1], (i) Cho x1, x2 F( x3 , y) (ii) : x3 tF( x1 , y) + (1 - t) F( x2 , y), Cho y1, y2 F(x, y3 ) A; y3 A, tF(x, y1 ) + (1 - t) F(x, y2 ), A , X, y C f A A, - , f(F(x, y)) F(x, x) 0, V(A, F) G: A G(y) = {x A: f(F(x, y)) , 0}, 2A A , y G(y {x : x0 f(F( x , y)) y A I G(y) cho { x x } x0 G(y) I 38 Thang Long University Libraty G(y { G(y): y A T A} n Do A ta i G(yi ) , y1, , yn A n B = { y1, , yn } x i A G(yi ) yi B cho x G( yi ) f(F(x, yi )) < i cho f(F(x, yi )) < - i x f(F(x, y x A, A yj > cho B cho f(F(x, y j )) + < g: A (2.5) Rn g(x) = ( f(F(x, y1 )) - , - f(F(x, y2 )) - , , - f(F(x, yn )) - ), x 39 A 2.5), - g(x) int Rn+ C F(x, y Do f x A (2.6) - t [0, 1], x1, x2 A A x3 g( x3 ) tg( x1 ) + (1 - t) g( x2 ) [8], g(A) + Rn+ Theo g(A) + int Rn+ t1, t2 , , tn n i ti cho n t ti ( ( f ( F ( x, yi ))) x ) A, n t x ti f ( F ( x, yi )) Theo y A (2.7) A cho n F ( x, y) f t x ti f (F ( x, yi )) A C , n f (F ( x, y)) t x ti f ( F ( x, yi )) A (2.8) 40 Thang Long University Libraty (2.7) - (2.8), ta suy x f(F(x, y)) A x = y, f(F(y, y)) - F(y, y) , f(F(y, y)) {G(y): y A} Ta suy x {G(y): y A} f(F(x, y)) , v x Vf (A, F) y A V(A, F) - 41 ng, L (X, Y) - hemi n , q: A [ (X, X*)], q(A) f theo q(x), C C } ,y (X, X*), , {Vf (A, F): (x, y) = (Tx, y - x) + q(y) - A 2A H: C H(f) = Vf (A, F), f Do C Y C 2.3.1, C f C H(f) H(f f C H(f) Khi x1, x2 i = 1, 2, f(F( xi , y)) = f((T xi , y - xi ) + q(y) - q( xi )) E, G: A y A 2A E(y) = {x A: f((Tx, x - y)) + f(q(x)) f(q(y))}, G(y) = {x A: f((Ty, y - x)) + f(q(y)) f(q(x))} x1, x2 Vf (A, F) x1, x2 E ( y) : y A 42 Thang Long University Libraty {E(y): y A} = {G(y): y A} i =1, 2, f((Ty, y - xi )) + f(q(y)) t [0, 1], f(q( xi y C l q f((Ty: y - (t x1 + (1- t) x2 )) + f(q(y)) A f C f(q(t x1 + (1- t) x2 y , t x1 + (1 - t) x2 t x1 + (1 - t) x2 {G(y): y A} = {E(y): y A} H(f), H(f H(f) ta C H Cho ( fn , xn ) graph (H) = {(f, x) ( fn , xn ) ( f0 , x0 ) xn H ( fn ) C C A Vf ( A, F ) , n 43 A: x H(f)}, A A; fn ((Txn , xn y)) Do fn (q( y)) y A fn (q( xn )) y A fn (q( xn )) T fn ((Ty, y xn )) fn C , fn (q( y)) Cho y A Do Ty Ty, y - x0 L(X, Y fn ((Ty, y- xn )) lim fn (q( xn )) n {(Ty, y - xn x0 xn f0 (Ty, y- x0 )) fn fn (q( y)) (2.9) f0 f0 (q( y)) f0 (q( x0 )) f0 ((Ty, y- x0 )) 2.10) f0 ((Ty, y x0 )) Do T v f0 ((Tx0 , x0 y)) f0 (q( x0 )) f0 (q( y)) y A D f0 (q( x0 )) f0 (q( y)) q (2.10) f0 C C- f0 (q( x0 )) y f0 (q( y)) A b y A, 44 Thang Long University Libraty y f0[(Tx0 , y- x0 ) q(y) q( x0 )] = f0 ( F ( x0 , y)) A x0 Vf ( A, F ) H ( f0 ) C Theo H(f) {H(f): f C } = {Vf (A, F): f [11], C } (X, X*) VH , w (A, F (X, X*) Theo x F(x, A) + C A VH(A, F) = {Vf(A, F): f C } Vf (A, F C H: C H(f) = Vf(A, F), f 2A C H(f) [11], VH(A, F C Do (X, X*) int C 45 Vw (A, F) = {Vf (A, F): f C*\{0}} ng theo (X, X* Vw(A, F f C f C*\{0} K T LU N Lu t qu v s t n t i nghi a t p nghi m c - n t qu v t n t i nghi m h u hi u y u c Bianchi Hadjisavvas - m: a Schaible [3]; t qu v t n t i nghi m h u hi u c a Gong [7]; - t qu v h u hi u y u c a b a t p nghi m h u hi ng th c bi p nghi m Stampacchia c a Gong [7] S t n t i nghi m p nghi m c c nhi tri n 46 Thang Long University Libraty [1] Aubin, J P., and Ekeland, I (1984), Applied Nonlinear Analysis, John Wiley, New York, NY [2] Bianchi, M., and Schaible, S (1996), Generalized monotone bifunctions and equilibrium problems, Journal of Optimization Theory and Applications, vol 90, pp 31-43 [3] Bianchi, M., Hadjisavvas, N., and Schaible, S (1997), Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, vol 92, pp 527 542 [4] Borwein, J M., and Zhuang, D (1993), Superefficiency in vector optimization, Transactions of the American Mathematical Society, vol 338, pp 105 122 [5] Chen, G Y (1992), Existence of solutions for a vector variational inequality: An extension of the Hartman Stampacchia theorem, Journal of Optimization Theory and Applications, vol 74, pp 445 456 [6] F a n , K (1961), A Generalization of Tychonoff's fixed - point theorem, Mathematische Annalen, vol 142, pp 305 47 310 [7] Gong, X H (2001), Efficiency and Henig efficiency for vector equilibrium problems, J Optim Theory Appl., vol 108, 139 154 [8] Jahn, J (1986), Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt am Main, Germany [9] Jeyakumar, V., Oettli, W., and Natividad, M (1993), A Solvability theorem for a class of quasiconvex mappings with applications to Optimization, Journal of Mathematical Analysis and Applications, vol 179, pp 537 546 [10] Lassonde, M (1983), On the use of KKM multifunctions in fixed point theory and related topics, Journal of Mathematical Analysis and Applications, vol 97, pp 151 201 [11] Warburton, A R (1983), Quasiconcave vector maximization: connectedness of the sets of Pareto optimal and weak Pareto optimal Alternatives, Journal of Optimization Theory and Applications, vol 40, pp 537 557 48 Thang Long University Libraty 49