Nonlinear Equations - UFRJ

[SEC. 7.3: ESTIMATES FROM DATA AT A POINT 105
Table 7.2 and Figure 7.5 show how fast ‖x i − ζ‖/β decreases in
terms of α and i.
The final issue is robustness. There is no obvious modification of
the proof of Theorem 7.15 to provide a nice statement, so we will rely
on Theorem 7.12 indeed.
Theorem 7.19. Let f : D ⊆ E → F be an analytic map between
Banach spaces. Let δ, α and u0 satisfy
0 ≤ 2δ < u 0 =
with r = 1+α−√ 1−6α+α 2
4α
. Assume that
1.
√
rα
14
(1 − rα)ψ(rα) < 2 − 2
B = B (x 0 , 2rβ(f, x 0 )) ⊆ D.
2. x 0 ∈ B, and the sequence x i satisfies
rβ(f, x 0 )
‖x i+1 − N(f, x i )‖
(1 − rα)ψ(rα) ≤ δ
3. The sequence u i is defined inductively by
u i+1 =
u2 i
ψ(u i ) + δ.
Then the sequences u i and x i are well-defined for all i, x i ∈ D,
and
‖x i − ζ‖
‖x 1 − x 0 ‖ ≤ ru i
≤ r max
(2 −2i +1 , 2 δ )
.
u 0
u 0
Numerically, α 0 = 0.074, 290 · · · satisfies the hypothesis of the
Theorem. A version of this theorem (not as sharp, and another metric)
appeared as Theorem 2 in [56].
The following Lemma will be useful:
Lemma 7.20. Assume that u = γ(f, x)‖x − y‖ ≤ 1 − √ 2/2. Then,
γ(f, y) ≤
γ(f, x)
(1 − u)ψ(u) .