De nition 712 the bvp 711 has exponential dichotomy

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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 linearly-independent solutions u(1) u(2) : : : u(r), to the IVPs (u(i) ) = Au(i) 0 Lu(i) (a) = 0 i...
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