关于反函数定理

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√ 2 , ∀x ∈ B x0 , 1 − 2
pn x = xn−1 , E
=41 =2 1998 3
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ACTA MATHEMATICA SINICA
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Vol.41, No. 2 March, 1998
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√ 1 + α0 + (1 + α0 )2 − 8α0 1 2 < r2 (α0 ) = < . 2 4 2 √ 1 α0 < 2 (7 − 3 5) = 0.1458 · · · ] - iNl 1−
f (x0 )−1 f (i) (x0 ) ≤ i!γ i−1 , f (x0 )−1 f (k+1) (x) ≤ i = 2, · · · , k; (k + 1)!γ k (1 − γ x − x0 )k+2
h = A8P( ) 4T 4v^?Z77 B(C"` 7 K'Zcr77 2 !8}U7C ) |7m 4 m bq / 7 h 2 (Wi7].4C/) lZ%X B x0 , 1 2 p f P > { (∗) 7K 4 7 4 | ] I b √ q α0 , 0 < α0 < 3 − 2 2, P (C"
L7 &3r l E o F vp7{Y7 Banach * IR* p7? x of| ρ, E B (x, ρ) s a x ρ 7%X *L XM B (x, ρ) s E L(E, F) s) E 5 F 7)447 x7 B <E L(En, F) s) En 5 F 747 n- )4 x7 l f CJZ E 7E^YV D JZ F pZk7l| {b x ? l f Z D Frechet IJ) VE f : D → L(E, F) S4| I x ∈ D, f (x) : E → F v47)4 x 3 f * n () o E f (n) : D → L(En, F) S n (4| −1 , e f (x)−1 *Z Nl|C/H=*ZB^CJR a f (x) 7E^%X B 7Nl| fx >{
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xn = x1 − f (x0 )−1
1 0
(1 − τ )f (x0 + τ (xn−1 − x0 ))(xn−1 − x0 )2 dτ.
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R
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xn − xn−1 = −
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f (x0 )−1 f (x0 + σ (xn−2+τ − x0 ))(xn−2+τ − x0 )(xn−1 − xn−2 )dσdτ,
MR(1991) O177.91 58C15, 46E15 On the Inverse Function Theorem
Wang Xinghua Han Danfu (Department of Mathematics, Hangzhou University, Hangzhou 310028, China)
1 0 1 0
f (x0 )−1 f (x0 + τ (x − x0 ))(x − x0 )dx
1 2 x − x0 dτ = − 1 < 1. (1 − τ x − x0 )3 (1 − x − x0 )2
−1 fx 0
EO Banach C/ f (x) )M h 1 ( k4o) OC/7hA J&$ pF*Z 2 ) 1 l f Z x0 √
xn − x0 ≤ x1 − x0 + xn−1 − x0 < α0 + (r1 − α0 ) = r1 .
a\7 {xn } ⊂ B (x0 , r1 ) Dzh h7 =k {E A {xn } w5 2\ A8 {xn } Zp|* vw57 c2 {tn} v f = h, y = s, x = t oI t0 = 0 o s = −α0 1 O s = −1 − 2t + 1− t , 0 ≤ t < 1 CJ
c2 xn−2+τ = xn−2 + τ (xn−1 − xn−2 ). EfI<P
xn − xn−1 ≤
1 0 0 1
2 xn−2+τ − x0 · xn−1 − xn−2 dσdτ (1 − σ xn−2+τ − x0 )3
426
1 0 1 0 1 0 1
}
≤ =
0
9
9
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#
2tn−2+τ (tn−1 − tn−2 ) dσdτ (1 − σtn−2+τ )3 h (σtn−2+τ )tn−2+τ (tn−1 − tn−2 )dσdτ = tn − tn−1 .
Abstract In the hypthesis that f is analytic, Smale gave an estimation of the size of the radius of the ball in which the inverse function exists, by using the criterion α. In this paper we not only make this estimation more precise, but also weaken the hypothesis to the second continuous differentiable. Keywords Banach space, The inverse function theorem, α-criterion 1991 MR Subject Classification 58C15, 46E15 Chinese Library Classification O177.91
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B f (x0 ), α0 γ f (x0 )−1 ⊂ f B x0 , r1 (α0 ) 1 γ ,
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2
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xn − x1 ≤
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2
=
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dG>h g
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7B^" w {tn } $7 c2 h : t → s
1 + α0 −
(1 + α0 )2 − 8α0 > 4
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0
√ 2 ⊂ f B x0 , 1 − 2
.
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.
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