SC-90-09.pdf - ZIB

This shows that both (2.27) and (2.40) are satisfied for all j sufficiently
large.
•
The integer (2.37) cannot be exactly computed in practice because the quantity
11x° — x* || is in general unknown. In this case we may use the last inequality
from the statement of Theorem 1.5 to conclude that (2.36) is satisfied whenever
\ x k - x k +i\\ <
1 + we/2
(2.44)
By using (1.9) and proceeding the same as in the deduction of (2.34) it follows
that (2.44) is satisfied for all k > k(x°,e), where
k{x°,e) log 2 In ue
2 + ue
In h (2.45)
Once more, u> may be replaced by wq whenever q ^ 1. In particular (2.36) is
also satisfied for all k > k(x°,e). Similarly, if (2.24) holds, and if the sequence
(2.3) converges to x*-, then inequality (2.31) is satisfied for all k > fc,(x°,e),
where
^('"(ö^l/toM*?)) • ( 2 - 46 )
*;(*?>*) = 2 + Uje,
Suppose that the constants a and h in Theorem 1.1 are determined exactly, i.e.:
a = a(x°) := WF^x^Fix 0 )]], h = h{x°) = a{x°)uj/2 . (2.47)
Then from assumption Co and (2.11) it follows that
By applying (2.19), we obtain
which leads to the following corollary:
lim ajfax 0 ) = a . (2.48)
j—>oo
lim hjfax 0 ) = h , (2.49)
j-»oo
Corollary 2.4 Under the hypothesis of Theorem 2.3, there is a ji > j\, such
that (2.24) holds for x° = ITJX 0 , j > ji and
\k j (7r j x°,e)-k(x°,e)\ j 2 . (2.50)
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