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=1 |a ij | < 1 A : R n → R n ! " % ' A = (a ij ) ! x = (x 1 , x 2 , . . . , x n , . . . ) ! Ax = ∞ i =1 a 1i x i , ∞ i =1 a 2i x i , . . . , ∞ i =1 a ni x i , . . . < - 5 sup 1≤j<∞ ∞ i =1 |a ij | < 1 A : 1 → 1 " ! - 5 sup 1≤i<∞ ∞ j =1 |a ij | < 1 A : m → m " ! - !5 ∞ j =1 ∞ i =1 |a ij | 2 < 1 A : 2 → 2 " ! % . * A : C[0, 1] → C[0, 1], Ax(t) = λ x(t α ), α ≥ 0 λ ! c % / f g |f(0)| < 1, |f(1)| < 1 +" x (t) = f(t) · x(t α ) + g(t) α > 0, α = 1 C [0, 1] ! % A : L 2 [0, 1] → L 2 [0, 1] , (Ax)(t) = λ · x(t α ), 0 < α ≤ 1 λ & ! c % K (t, s) α < 1 λ A : C[0, 1] → C[0, 1] (Ax)(t) = λ · $ 1 0 K (t, s) |t − s| α x (s) ds ! c &9 "0 ,+ 0 0 7 * K ⊂ X & ! & ! & " + K * X ! & ! & + X = & & " ! (: * K & {x n } " K ! " + K * M & & [M] & & M * x ∈ M ! a ∈ A + ρ (x, a) ≤ ε + A & M & ! ε * ε > 0 ! M & ! ε + M ! & ! " & & &"" (X, ρ) M C [a, b] F * C > 0 + φ ∈ F ! x ∈ [a, b] ! | φ(x) | ≤ C + F * ε > 0 ! δ > 0 + |x 1 − x 2 | < δ ! x 1 , x 2 ∈ [a, b] ! φ ∈ F ! | φ (x 1 ) − φ (x 2 ) | < ε + F & "" $! & M ⊂ C [a, b] [ R n C n & & & & "" R n ( C n ) K &-5 R n ( C n ) K & X A B & & A ∪B, A∩ B & & A B & & ! #8#" " & ! \ A B & ! A ε B ε ! ε + , A ∪ B & ! A ε ∪ B ε & ! ε ' A ∪ B & ! #8#" A ∪ B & & ! #8`" A ∩B ⊂ A & & & C [a, b] F = y (s) = $ b a K (s, t) x (t) dt , x ∈ B[0, 1] (17.1) & ' B [0, 1] & " C [a, b] 3 x (t) ≡ 0 5 # & K (s, t) " [a, b] × [a, b] ! "# * F ! " + K (s, t) " [a, b] × [a, b] ! ! ( C > 0 + ! s, t ∈ [a, b] ! |K(s, t)| ≤ C x ∈ B[0, 1] max a ≤t≤b | x (t) | ≤ 1 ! " [ F ! - | y (s) | = ## ## $ b a K (s, t) x (t) dt ## ## ≤ $ b a |K (s, t) | · | x (t) | dt ≤ C · 1 · (b − a) . ' F ! " [ F + - | y (s 1 ) − y (s 2 ) | = ## # + b a K (s 1 , t ) x (t) dt − + b a K (s 2 , t ) x (t) dt ## # ≤ ≤ + b a |K (s 1 , t ) − K (s 2 , t ) | · | x (t) | dt ≤ ε · 1 · (b − a) . 2 |s 1 − s 2 | < δ ! ! s 1 , s 2 ∈ [a, b] x ∈ B[0, 1] ! \ F + 2 * 3#8#5 F & & & C [0, 1] Φ = x α (t) = 2 α t 1 + α 2 t 2 ; α ∈ (0, ∞) (17.2) & ! "# * 3#8%5 Φ ! + (1 − α t) 2 = 1−2 α t+α 2 t 2 ≥ 0 | x α (t) | ≤ 1 ! \ Φ ! 1 + ! (: * ε > 0 δ > 0 ! x α ∈ Φ t 1 , t 2 ∈ [0, 1] + | t 1 − t 2 | < δ + | x α (t 1 ) − x α (t 2 ) | ≥ ε + Φ [ ε = 1/2 δ > 0 " * α > 1 δ t 1 = 1 α , t 2 = 0 | t 1 − t 2 | = 1 α < δ | x α (t 1 ) − x α (t 2 ) | = 2 α · 1 α 1 + α 2 · 1 α 2 = 1 > ε \ Φ + 2 3#8%5 Φ & & & X & & ! "# X = (−∞, ∞) M = 0, 2, 2 −1 , . . . , 2 −n , . . . M & > \ M & & M & > % ! ( ! " & #8`" M & & () * +, * *+- + # - &$ X A & & , B (B ⊂ A) & & &% X A & & 2 B (B ⊂ A) & & & && X A & & , " B (B ⊂ A) & & &' X A B & & A ∪ B, A ∩ B & & &. = & &/ = & & & = & & & X & {F α } α ∈ I & & * F α & ! α ∈I F α & = & & & & X, Y Y & f : X → Y , X f (X) &$ X & f : X → X ! ρ (f(x), f(y)) ≥ ρ(x, y), ∀ x, y ∈ X + f &% X & f : X → X ρ (f(x), f(y)) < ρ(x, y), x = y , f (x) = x ! && X & f : X → X , f (X) = X &' R 5 Z 5 M n = k n : k ∈ Z & R ! c &. Z 2 & R 2 c & / 1 A = {1 ≤ x ≤ 5, 0 ≤ y ≤ 4} & ! ! ε = √ 2 d ε = √ 2 ! & & f : X → Y M ⊂ X & ! Download 1.57 Mb. Do'stlaringiz bilan baham: 
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