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. u(x) | <. f(t) f(u(x)) <. , u(x) v(x) < f(t),.. v(x) = f(u(x)) f(t). (.) . < X , x 2 X " > 0 y 2 X, j j y ; xj j < " y x.

.

. u(x) , u(x) X.

, <, y >> x ) y x:

.

4 , . .

(p w), Rn R+, Rn | p, R+ |+ + w. (p w) . (p w) , , , ( , ). (p w) .

, p 2 Rn , + u(x), X, w . , Bp w(X) ,.. , p:

Bp w(X) = fx j x 2 X x p wg:

p 2 Rn w + (p w), u(x) Bp w(X), x 2 Bp w (X), . Rn R+ X Rn. + + .

, (p w) 0,.. > 0 :

( p w) = (p w):

, , . , , .

, w p,.. w(p). (p) = (p w(p)), . Rn X Rn.

+ + w(p) , , w(p) . . 2.10: , > 0 : w( p) = w(p). Bp w(p) p, ^ X (p). 1 w(p) ^ , X (p) 0,..

^ ^ B p w( p) = Bp w(p) X ( p) = X (p) ( p) = (p),.. (p) 0. w(p) . , .

, Rn. + Bp w(Rn ) + , p,.. pi, . , X = Rn, u(x) + , w > 0, p =(0 p2 : : : pn) 2 Rn nf0g Bp w (X) + u(x) Bp w(X), . . 2.10.

, X: xk 2 X xk j k !1, xk. , , . .

, Bp w(X) , . 2.11 ( . 2.11: . 2.12: ).

, , . x 2 (p w).

(p) (p w). , . , , : x x0 2 X y 2 X, [x x0],.. (1 ; )x + x0 2 (0 1), u(y) > (1 ; )u(x) + u(x0), . 2.12.

5 .

A) (UMP): w > 0 p >> 0.

, (p w).

B) (EMP): p >> 0 w, u > u(0). h(p u), (p u) x 2 X, , .

.

. , .

, X Rn, u(x) + :

@u @u @u > 0 lim = 1 lim =0 i =1 : : : n x !0 x !1 @xi @xi i @xi i @2u U (x) = (x) 8x 2 X:

@xi@xj p , K | . u(x) , (p K ) , p K x (p K ) (p K ) :

u(x) ! max hp xi = K x (). , , x (p K ) 0, x (p K ) . :

;

L(x ) = u(x) ; hp xi ; K :

, (3.1) hp x i ; K =@u (3.2) (x ) ; pi =0 i =1 : : : n:

@xi , (3.2) | , ( p ).

, pn, . pn. :

@x (3.3) hp i = ;x n @pn @x @ def (3.4) U ; p =(0 : : : 0 ) = :

@pn @pn U ( ), (3.4) @x =@pn (3.3), @ =@pn:

; @ x n + pU = ; :

; @pn pU p ;

; ; ; pU p , M ; [M ](i) i- . , U = ; [U ](n). :

@ ; = x n + p[U ](n):

@pn ; z [U ](n) (.. z | n- ; U ). @ = x n + hz pi:

@pn @x =@pn:

@x ; 1 ; (3.5) = x n U p + hp ziU p + z :

@pn (3.5) .

.

, , x (p K ), p , K . (3.1) (3.2) K, :

@x (3.6) hp i =@K @x @ (3.7) U ; p =0:

@K @K (3.7) @x =@K (3.6), @ =@K :

@ = = ;

; @K pU p @x ; (3.8) = ; U p @K @x @x ; = ; x n + hp ziU p + z :

@pn @K (3.5). p,..

, K , . , , K p,.. ;

K (p), : u x p K (p) = const.

;

x p K p, K K (p). x pi,.. @x =@pi + @x =@K @K=@pi pi (@x =@pi)comp. i = n.

(3.1) pn. :

@x @K (3.9) x n + p ; =0:

@pn comp @pn p u(x ) , , (@x =@pn)comp u = const, u,.. @u=@x.

, x, x u = const, p @u=@x u = const (. (3.2)). (3.9) . , @K (3.10) x n = :

@pn (3.2) pn. @x @ (3.11) U = p + @pn comp @pn, , hp (@x =@pn)compi = 0, , (@x =@pn)comp (3.11) p, @ =@pn:

@ ; = pU = hz pi:

@pn (3.11), :

@x ; 1 ;= hz piU p + U = @pn comp ;

; 1 ; hz piU p + z = hz piU p + z :

(3.5), , x pn. , .

3.1 , :

@x @x @x (3.12) = ; x n:

@pn @pn comp @K . (3.12) .

. (@x =@pn)comp, , :

(n) ; 1 ; 1 ; (@x =@pn)comp = U p0pU + U p0 p, , p, -. , ;

;1 ;1 ; (@x =@pi)comp = hz piU p0 + z = U p0hp zi + U = ;

; 1 ; 1 ; 1 ; 1 ; 1 ; U p0pU + U = U p0pU + U = (n) ; 1 ;1 ; U p0pU + U :

; 1 ; 1 ; . H = U p0pU + U .

H . .

1) H . , U | ; , U | . , p0p | n n , (i j)- pipj, . ;

T ; 1 ; 1 ; 1 ; 1 ; 1 ; U p0pU = (U )T (p0p)T (U )T = U p0pU ; 1 ;, , U p0pU . , .

2) : pH = Hp0 = 0. , pH = 0 ( H ).

, ; 1 ; 1 ; 1 ; 1 ; 1 ;pH = pU p0pU + pU = ; (pU p0)pU + pU =; pU p .

3) H ,.. v 2 Rn vH v0 0. , vH v0 = , v p .

v | Rn. , vH v0 0.

; ;U ( ), w | Rn, v , p.

; , v = p + w, 2 R, wU p0 = 0. :

vH v0 =( p + w)H ( p0 + w0) = pH p0 + pH w0 + wH p0 + wH w0 = ; 1 ; 1 ; wH w0 = w( U p0pU + U )w0 = ;1 ; 1 ; 1 ; (wU p0)pU w0 + wU w0 = wU w0 , w = 0.

.

.

3.1 , :

@x n < 0:

@pn comp , .

. , (@x =@pn)comp = [H ](n), (@x n=@pn)comp = hnn, hnn | H. ei , hnn = enHen. p 0, p en , , (3) , hnn < 0, .

.

n- , @x n=@K > 0,.. .

, , .

3.2 .

. 3.6 p. .

3.3 .

. , x n . .

. i j , (@x j=@pi)comp > 0,.. i- ( i) j . (@x j =@pi)comp < 0, i j .

. , | .

3.4 i j, i .

. , , i = n. 3.1, (@x n=@pn)comp < 0. , , h(@x =@pn)comp pi = 0, , p 0, , j, (@x j=@pn)comp > 0. .

. j i, @x j=@pi > 0. , x (p K ) , ,.. @x j=@pi i = j. @x j=@pi > 0 i = j, 6 .

P n 3.1 u(x) = x i i, > 0, 0 < < 1, x (p K ) | i i i i= . , x (p K ) .

1 . , ( , ) . . , .

. , . , : \ " .

. , , (): , .

2 . M = fx1 : : : xmg | , S = fy1 : : : yng | , . yk :

k k xi xim :

, ,.. , 1 k : : : <:

1 k <= f( : : : ):

a b:

a b (a b) a = b (a b ) b a (a b) a < b (a b):

, a , a a : : :,.. a .

. .

. 10 1 : : : 10, a, b c. .

1 2 : : : a b : : : c b a : : : b c c : : : a . . - 2 3 a b c b a b c c a . .

2.1 . , , | .

. n = 10 m = 4 .

:

- 2 3 4 a a b c b c a a c d c d d b d b , a 5 , b |4 , c |1 d |0 . a.

.

- 2 3 a a b b c a c d c d b d a b , c d | . , . , , .

: a b.

2.2 . , , , . , .

, , , . .

2.3 , n m . S Si, . p , i- ki : j Sij = ki. , ki > 0 k1 + kp = n.

, xi . i (xi yj) - . yj xi ij , 0 , | 1 ,.., | (m ; 1) . , xi k- yj, = k ; 1.

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