Nonlinear Equations - UFRJ

[SEC. 7.2: THE γ-THEOREMS 87
7.2 The γ-Theorems
The following concept provides a good abstraction of quadratic convergence.
Definition 7.4 (Approximate zero of the first kind). Let f : D ⊆
E → F be as above, with f(ζ) = 0. An approximate zero of the first
kind associated to ζ is a point x 0 ∈ D, such that
1. The sequence (x) i defined inductively by x i+1 = N(f, x i ) is
well-defined (each x i belongs to the domain of f and Df(x i ) is
invertible and bounded).
2.
‖x i − ζ‖ ≤ 2 −2i +1 ‖x 0 − ζ‖.
The existence of approximate zeros of the first kind is not obvious,
and requires a theorem.
Theorem 7.5 (Smale). Let f : D ⊆ E → F be an analytic map
between Banach spaces. Let ζ be a non-degenerate zero of f. Assume
that
(
B = B ζ, 3 − √ )
7
⊆ D.
2γ(f, ζ)
Every x 0 ∈ B is an approximate zero of the first kind associated
to ζ. The constant (3 − √ 7)/2 is the smallest with that property.
Before going further, we remind the reader of the following fact.
Lemma 7.6. Let d ≥ 1 be integer, and let |t| < 1. Then,
1
(1 − t) d = ∑ k≥0
( ) k + d − 1
t k .
d − 1
Proof. Differentiate d−1 times the two sides of the expression 1/(1−
t) = 1 + t + t 2 + · · · , and then divide both sides by d − 1!