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# The exact solution of the linear 4 bvp 711 can be

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Unformatted text preview: ) y(x) = Y(x)Q ; 1 l + Z b G(x )b( )d r a (7.1.4a) where Q was given by (7.1.3b) and G(x ) is the Green's function 8 > > < L Y(a)Y 1( ) if x 0 G(x ) = > : > ;Y(x)Q 1 0 Y(b)Y 1 ( ) if > x : R Y(x)Q 1 ; ; ; (7.1.4b) ; The Green's function behaves like the inverse of the di erential operator. We may use it to de ne the stability of BVP. De nition 7.1.1. The BVP (7.1.1) is stable if there exists a constant such that ky( )k jlj + jrj + 1 1 Zb jb( )j d ] (7.1.5a) jbj = 1max jbij: im 1 (7.1.5b) a 1 where ky( )k = amaxb jy(x)j x 1 1 1 From (7.1.4a), we see that = max(kY( )Q 1k kG( )k ) ; 1 1 (7.1.5c) assuming that all quantities are bounded. Thus, the solution of a stable BVP is bounded by its data. Following the techniques used for IVPs, we may show that this also applies to a perturbation of the data associated with the BVP (7.1.1) ( 2], Chapter 6) hence, the reason for the name stability. De nition 7.1.2. The BVP (7.1.1) has exponential dichotomy if there are positive constants K...
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