Nonlinear Equations - UFRJ

132 [CH. 9: THE PSEUDO-NEWTON OPERATOR
Thus,
By (7.14) and induction,
β i+1 ≤ β2 i γ i(1 − β i γ i )
ψ(β i γ i )
(9.6)
β i+1 ≤ ˆβ 2 i ˆγ i(1 − ˆβ iˆγ i )
ψ( ˆβ iˆγ i )
= ˆβ i+1 .
Now the second part of the induction hypothesis:
Df(X i ) † D l f(X i+1 ) = ∑ k≥0
1 Df(X i ) † D k+l f(X i )(X i+1 − X i ) k
k!
k + l
Passing to norms and invoking the induction hypothesis,
‖Df(X i ) † D l f(X i+1 )‖ ≤ ∑ k≥0
and then using Lemma 9.4 and (7.14),
− h(k+l) βγ
(t i ) ˆβ i
k
k!h ′ βγ (t i)
‖Df(X i+1 ) † D l f(X i+1 )‖ ≤ (1 − ˆβ iˆγ i ) 2 ∑
ψ( ˆβ
− h(k+l) βγ
(t i ) ˆβ i
k
iˆγ i ) k!h ′ βγ (t i) .
k≥0
A direct computation similar to (7.14) shows that
− h(k+l) βγ
(t i+1 )
k!h ′ βγ (t i+1) = (1 − ˆβ iˆγ i ) 2
ψ( ˆβ iˆγ i )
∑
k≥0
− h(k+l) βγ
(t i ) ˆβ i
k
k!h ′ βγ (t i) .
and since the right-hand-terms of the last two equations are equal,
the second part of the induction hypothesis proceeds. Dividing by
l!, taking l − 1-th roots and maximizing over all l, we deduce that
γ i ≤ ˆγ i .
Proposition 7.17 then implies that X 0 is an approximate zero.
Let Z = lim k→∞ N k (f, Z). The second statement follows from
d proj (X 0 , Z) ≤ ‖X 0 − Z‖ ≤ β 0 + β 1 + · · · = r 0 β.