Nonlinear Equations - UFRJ

130 [CH. 9: THE PSEUDO-NEWTON OPERATOR
9.4 The alpha theorem
Definition 9.8 (Approximate zero of the second kind). Let f ∈
F 1 × · · · × F n . An approximate zero of the second kind associated to
z ∈ M/H, f(z) = 0, is a point X 0 ∈ M, scaled s.t. ‖(X 0 ) 1 ‖ = · · · =
‖(X 0 ) s ‖ = 1, and satisfying the following conditions:
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.
d proj (X i+1 , X i ) ≤ 2 −2i +1 d proj (X 1 , X 0 ).
3. lim i→∞ X i = Z.
Theorem 9.9. Let f ∈ H d . Let
Define
α ≤ α 0 = 13 − 3√ 17
.
4
r 0 = 1 + α − √ 1 − 6α + α 2
4α
and r 1 = 1 − 3α − √ 1 − 6α + α 2
.
4α
Let X 0 ∈ C n+s , ‖(X 0 ) 1 ‖ = · · · = ‖(X 0 ) s ‖ = 1, be such that α(f, X 0 ) ≤
α. Then,
1. X 0 is an approximate zero of the second kind, associated to
some zero z ∈ P n of f.
2. Moreover, d proj (X 0 , z) ≤ r 0 β(f, X 0 ).
3. Let X 1 = N(f, x 0 ). Then d proj (X 1 , z) ≤ r 1
β(f,X 0)
1−β(f,X 0)) .
Proof of Theorem 9.9. Let β = β(f, X 0 ) and γ = γ(f, X 0 ). Let h βγ
and the sequence t i be as in Proposition 7.16. By construction of the
pseudo-Newton operator, d proj (X 1 , X 0 ) = β = t 1 − t 0 . We use the
following notations:
β i = β(f, X i ) and γ i = γ(f, X i ).