Nonlinear Equations - UFRJ
Nonlinear Equations - UFRJ
Nonlinear Equations - UFRJ
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
146 [CH. 10: HOMOTOPY<br />
Lemma 10.13. Assume the conditions of Lemma 10.8, and suppose<br />
furthermore that<br />
2ɛ 2<br />
min Φ<br />
t ti,σ(X i ) ≤<br />
i≤σ≤t i+1 D 3/2 µ F (F ti , X i )<br />
with equality for σ = t i+1 . Then,<br />
Proof.<br />
L(f t , z t ; t i , t i+1 ) =<br />
≥<br />
≥<br />
≥<br />
L(f t , z t ; t i , t i+1 ) ≥<br />
∫ ti+1<br />
t i<br />
∫ ti+1<br />
t i<br />
1<br />
CD 3/2√ n<br />
µ(f s , z s )‖( f ˙ s , z˙<br />
s )‖ fs,z s<br />
ds<br />
µ(f s , z s )‖ z˙<br />
s ‖ zs ds<br />
∫ ti+1<br />
µ<br />
1 + ɛ 1 + π(1 − ɛ 1 )αr 0 (α)/ √ ‖ z˙<br />
s ‖ zs ds<br />
D t i<br />
µ<br />
1 + ɛ 1 + π(1 − ɛ 1 )αr 0 (α)/ √ D d proj(z ti+1 , z ti ).<br />
At this point we use triangular inequality:<br />
d proj (z ti+1 , z ti ) ≥d proj (N(F ti+1 , X i ), X i ) − d proj (X i , z ti )<br />
− d proj (N(F ti+1 , X i ), z ti+1 )<br />
The first norm is precisely β(F ti+1 , X i ). From (10.10),<br />
d proj (N(F ti+1 , X i ), X i ) ≥ 2 (ɛ 2 − a 0 ) √ 1 − ɛ 2 1<br />
.<br />
D 3/2 (1 + ɛ 1 )µ<br />
The second and third norms are distances to a zero. From Theorem<br />
9.9 applied to F ti , X i ,<br />
d proj (X i , z ti ) ≤ r 0 (a 0 )β ≤ 2<br />
D 3/2 µ a 0r 0 (a 0 ).<br />
Applying the same theorem to F ti+1 , X i with α(F ti+1 , X i ) < α<br />
by (10.11), and estimating ‖N(F ti+1 , X i )‖ ≥ 1 − β(F ti+1 , X i ),<br />
β(F ti+1 , X i )<br />
d proj (N(F ti+1 , X i ), z ti+1 ) ≤ r 1 (α)<br />
1 − β(F ti+1 , X i )
Change language