Nonlinear Equations - UFRJ

[SEC. 7.2: THE γ-THEOREMS 91
When u 0 < 5−√ 17
4
, (7.3) implies that the sequence u i is decreasing,
and by induction
u i = γ|t i |.
Moreover,
( ) 2 ( ) 2 ( ) 2
u i+1 ui u 0
=
u 0 u 0 ψ(u i ) ≤ ui u 0
u 0 ψ(u 0 ) < ui
.
u 0
By induction,
( ) 2
u i −1
i u0
≤
.
u 0 ψ(u 0 )
This also implies that lim t i = 0.
When furthermore u 0 ≤ (3 − √ 7)/2, u 0 /ψ(u 0 ) ≤ 1/2 by (7.4)
hence u i /u 0 ≤ 2 −2i +1 . For the converse, if u 0 > (3 − √ 7)/2, then
|t 1 |
|t 0 | = u 0
ψ(u 0 ) > 1 2 .
Before proceeding to the proof of Theorem 7.5, a remark is in
order.
Both Newton iteration and γ are invariant with respect to translation
and to linear changes of coordinates: let g(x) = Af(x − ζ),
where A is a continuous and invertible linear operator from F to E.
Then
N(g, x + ζ) = N(f, x) + ζ and γ(g, x + ζ) = γ(f, x).
Also, distances in E are invariant under translation.
Proof of Theorem 7.5. Assume without loss of generality that ζ = 0
and Df(ζ) = I. Set γ = γ(f, x), u 0 = ‖x 0 ‖γ, and let h γ and the
sequence (u i ) be as in Lemma 7.10.
We will bound
‖N(f, x)‖ = ∥ ∥x − Df(x) −1 f(x) ∥ ∥ ≤ ‖Df(x) −1 ‖‖f(x) − Df(x)x‖.
(7.6)