Nonlinear Equations - UFRJ

[SEC. 10.2: PROOF OF THEOREM 10.5 143
Proof. Because of (10.1), ‖F ti ‖, ‖F s ‖ ≥ 1 and
F ti
∥‖F ti ‖ − F s
‖F s ‖ ∥ ≤ ‖F t i
− F s ‖ ≤ ɛ 1
µ
Then Lemma 8.22 with u = 0, v = ɛ 1 implies (10.8).
For (10.9) and (10.10), we write
β(F s , X i ) = DF s (X i ) † DF ti (X i ) ( DF ti (X i ) † F ti (X i )+
+DF ti (X i ) † (F s (X i ) − F ti (X i )) ) .
Let v = ‖Fs−Ft i ‖
‖F ti ‖
µ. By (10.2) ‖F ti ‖ > 1 so that v ≤ ɛ 1 . From
Lemma 9.5, we deduce that
√
1 − ɛ
2
1
1 + ɛ 1
( 2ɛ2
D 3/2 µ − β )
≤ β(F s , X i ) ≤ β +
(10.11) is ob-
Now equation (10.3) implies (10.9) and (10.10).
tained by multiplying (10.8) and (10.9).
Lemma 10.9. Under the conditions of Lemma 10.8,
and
D 3/2
µ(f s , [N(F s , X i )])
β(F s , N(F s , X i ))
≤
≤
2ɛ2
D 3/2 µ
1 − ɛ 1
µ
1 − ɛ 1 − πa 0 / √ D
(10.12)
2 1 − ɛ 1 1 − α
D 3/2 µ ψ(α) α2 (10.13)
2 β(F s, N(F s , X i ))µ(f s , [N(F s , X i )]) ≤ (1 − (1 − ɛ 1 )α/2) a 0
(10.14)
Proof. The proof of (10.12) is similar to the one of (10.8). We need
to keep in mind that X ti is scaled but N(F s , X ti ) is not assumed
scaled. Anyway, we know that
‖X ti − N(F s , X ti )‖ = β.