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Unformatted text preview: , , and such that kG( )k 1 Ke
e ( ; x) (x; ) if
x:
if > x The BVP has dichotomy if (7.1.6) holds with = = 0. 5 (7.1.6) Remark 1. The notions of dichotomy and exponential dichotomy correspond to stability and asymptotic stability, respectively, for an IVP
It is relatively easy to show that the stability constant for Example 7.1.1 has a modest
size and that this problem has exponential dichotomy (Problem 2). The problem, once
again, is that the IVP is unstable and this renders Q ill conditioned.
There is a alternative shooting procedure for linear systems that is slightly more
e cient than (7.1.2  7.1.3) when m is large. We'll present it, although it will not solve
our di culties. 1. Obtain a particular solution v that satis es v = Av + b
0 Lv(a) = l: (7.1.7) We'll have to describe a means of computing v(a) so that it satis es the initial
condition (7.1.1b), but let's postpone this.
2. Obtain linearlyindependent solutions u(1) u(2) : : : u(r), to the IVPs
(u(i) ) = Au(i)
0 Lu(i) (a) = 0 i...
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 Spring '14
 JosephE.Flaherty

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