Nonlinear Equations - UFRJ

84 [CH. 7: NEWTON ITERATION
As long as there is no ambiguity, we drop the subscripts of the
norm.
Definition 7.1 (Smale’s γ invariant). Let f : D ⊆ E → F be an
analytic mapping between Banach spaces, and x ∈ E. When Df(x)
is invertible, define
γ(f, x 0 ) = sup
k≥2
Otherwise, set γ(f, x 0 ) = ∞.
( ‖Df(x0 ) −1 D k f(x 0 )‖
k!
) 1
k−1
.
In the one variable setting, this can be compared to the radius of
convergence ρ of f ′ (x)/f ′ (x 0 ), that satisfies
More generally,
ρ −1 = lim sup
k≥2
( ‖f ′ (x 0 ) −1 f (k) (x 0 )‖
k!
) 1
k−1
.
Proposition 7.2. Let f : D ⊆ E → F be a C ∞ map between Banach
spaces, and x 0 ∈ D such that γ(f, x 0 ) < ∞. Then f is analytic in x 0
if and only if, γ(f, x 0 ) is finite. The series
f(x 0 ) + Df(x 0 )(x − x 0 ) + ∑ k≥2
1
k! Dk f(x 0 )(x − x 0 ) k (7.2)
is uniformly convergent for x ∈ B(x 0 , ρ) for any ρ < 1/γ(f, x 0 )).
Proposition 7.2, if. The series
Df(x 0 ) −1 f(x 0 ) + (x − x 0 ) + ∑ k≥2
1
k! Df(x 0) −1 D k f(x 0 )(x − x 0 ) k
is uniformly convergent in B(x 0 , ρ) where
(
ρ −1 ‖Df(x0 ) −1 D k f(x 0 )‖
< lim sup
k≥2
k!
≤
lim sup γ(f, x 0 ) k−1
k
k≥2
= lim
k→∞ γ(f, x 0) k−1
k
= γ(f, x 0 )
) 1 k