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k = m) $$$ S , S × [0, 1] S × S R 3 + $$% ,! R 2 + 1 R 2 + R 3 + " $$& 1 + R n + $$' L 2 [0, 1] L 1 [0, 1] 5 5 + c %9 @2 0 " - * f : (X, ρ) → (Y, d) ! L > 0 + x 1 , x 2 ∈ X ! d (f (x 1 ) , f (x 2 )) ≤ L ρ (x 1 , x 2 ) + f 1 * f : X → X /& /& L < 1 f * A : X → X ! x ∈ X + Ax = x + x A %"" $< & % x = 1 3 cos x − 2 ! = ! >>># & ! "# & R− f : R → R f (x) = 1 3 cos x − 2 ρ (f (x 1 ) , f (x 2 )) ≤ 1 3 ρ (x 1 , x 2 ) ( f ! 2 ! #7#" ! , ! x ≈ −2, 194749. % C [−a, a] f i : C[−a, a] → C[−a, a] (i = 1, 2) " f 1 (x(t)) = x(−t) f 2 (x(t)) = −x(−t) 2 & ! "# C [−a, a] + & C + [−a, a] & C − [−a, a] , x + ∈ C + [−a, a] ! f 1 (x + ) = x + " x − ∈ C − [−a, a] f 2 (x − ) = x − \ ! + f 1 " ! f 2 % R f : R → R ! x, y ∈ R, x = y ! ρ (f(x), f(y)) < ρ(x, y) f (x) = x ! ! "# '! x ∈ R f (x) = √ x 2 + 1 + 1 √ x 2 + 1 ' ! ! x, y ∈ R, x = y ρ (f(x), f(y)) < ρ(x, y) f (x) = x ! " ! x ≤ 0 f (x) = x ! 2 ! x > 0 - f (x) − x = " x 2 + 1 − x + 1 √ x 2 + 1 = 1 √ x 2 + 1 + x + 1 √ x 2 + 1 > 0. \ ! x ∈ R f (x) > x [ ρ (f(x), f(y)) < ρ(x, y) ⇐⇒ |f(x) − f(y)| < |x − y| ! f (x) = x √ x 2 + 1 − x ( √ x 2 + 1) 3 = x √ x 2 + 1 (1 − 1 x 2 + 1 ) = x 3 (x 2 + 1) 3/2 ' ! ! x ∈ R f (x) < 1 2 ! |f(x) − f(y)| = |f (ξ)| |x − y| < |x − y|, x = y () * +, * *+- + # - % * f : R → R |f (x)| ≤ q < 1 x = f(x) ! %$ * f : [a, b] → [a, b] x = f(x) " ! + %% * 0 ≤ a ≤ 1 x n +1 = x n − 1 2 (x 2 n − a), x 0 = 0 ! { x n } " √ a " %& \ ! c %' S = {z : |z| = 1} f : S → S ! + c %. X f : X → X ! , f %/ X f i : X → X (i = 1, 2) * f 1 ! f 1 ◦ f 2 = f 2 ◦ f 1 f 2 + % R 2 ; % X f : X → X f " 3 + 5 f 2 = f ◦ f, f n = f n −1 ◦ f * n ! f n ! f : X → X " % K (t, s) [a, b] × [a, b] , " K (t, s) ! ( ! t, s ∈ [a, b] ! |K(t, s)| ≤ M , (Ax)(t) = $ b a K (t, s)x(s)ds C [a, b] " ! x, y ∈ C[a, b] ! ρ (Ax, Ay) ≤ M(b − a)ρ(x, y) + % K (t, s) ! ! $ b a $ b a |K(t, s)| 2 dtds ≤ M , L 2 [a, b] x (t) = λ $ b a K (t, s)x(s) ds + y(t), y ∈ L 2 [a, b] λ & ! ! 3 ! 5 ! %$ K (t, s) a ≤ s ≤ t ≤ b ! ! 3 ! 5 , (Ax)(t) = $ t a K (t, s)x(s)ds A : C[a, b] → C[a, b] ! " n + A n : C[a, b] → C[a, b] " ! %% K (t, s) a ≤ s ≤ t ≤ b ! ! y (t) [a, b] x (t) = λ $ t a K (t, s)x(s)ds + y(t) λ ∈ R ! ! %& (Ax)(t) = t + 0 x 2 (s)ds A : C[0, a] → C[0, a] ! a > 0 ! ! %' a > 0 x (t) = 1 + $ t 0 x 2 (s)ds C [0, a] ! c %. R n S n −1 = x = (x 1 , x 2 , . . . , x n ) : n i =1 x i = 1, x i ≥ 0 & * P = (p ij ), i, j = 1, n " p ij ≥ 0 n i =1 p ij = 1, j = 1, 2, . . . , n " P x → P x S n −1 & " S n −1 & ρ (x, y) = n i =1 |x i − y i |, x, y ∈ S n −1 * P " 3 p i 1 > 0, p i 2 > 0, . . . , p in > 0 5 P x = x S n −1 & ! %/ K (t, s) [0, 1] × [0, 1] max 0≤t≤1 $ 1 0 |K(t, s)| ds = M * 4M|λ| < 1 + x (t) = 1 + λ $ 1 0 K (t, s)x(s)ds C [0, 1] ! " % = 5 5x − 3 sin x = 7, 5 3x + e −|x| = 10, !5 x = ln 3 √ 1 + x 2 − 3 ! >># & % f x (t) − 1 2 sin x(t) + f(t) = 0 ! x ∈ C[0, 1] + % , x (t) = e −x(t) + sin t C [0, 1] ! + % X A : X → X ρ (Ax, Ay) ≤ q ρ(x, y) 0 ≤ q < 1 ! x 0 ∈ X ! Ax = x " ! B [x 0 , ρ (x 0 , Ax 0 ) 1 − q ] %$ X B [x 0 , r ] ⊂ X & f : B[x 0 , r ] → X * f B [x 0 , r ] ( ρ (f(x), f(y)) < q · ρ(x, y), 0 ≤ q < 1, x, y ∈ B[x 0 , r ] ρ (f(x), x 0 ) ≤ (1 − q) · r + f (x) = x B [x 0 , r ] ! %% X f : X → X ρ (f(x), f(y)) ≥ α · ρ(x, y), α > 1, x, y ∈ X , f (x) = x ! c %& R n ρ (x, y) = n i =1 |x i − y i | A = (a ij ), i, j = 1, n max 1≤j≤n n Download 1.57 Mb. Do'stlaringiz bilan baham: |
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