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Exponential dichotomy of difference equations and applications to evolution equations on the half-line. (English) Zbl 1016.39007

This article deals with homogeneous and inhomogeneous difference equations \[ x_{n+1}= A_nx_n(n=0,1,2, \dots)\text{ and }x_{n+1}= A_n x_n+f_n (n=0,1,2, \dots), \] where \(A_n\), \(n=1,2, \dots\), is a sequence of uniformly bounded linear operators on a given Banach space \(X\). The main result is the equivalence of the exponential dichotomy property for homogeneous equation and the following properties of inhomogeneous equation: (a) the operator \[ T(u_1,\dots, u_n,\dots)= (u_1-A_0u_0, \dots, u_{n+1}-A_nu_n, \dots): \ell_\infty \to \ell_\infty \] is surjective; (b) the subspace \(X_0(0)= \{x \in X:\sup_{n\geq 0}\|A_{n-1}A_{n-2}\dots A_0 x\|<\infty\}\) is complemented in \(X\). As application of this theorem the authors present a similar result for strongly continuous and exponential bounded evolution families \(U(t,s)\), \(0\leq s\leq t<\infty\).

MSC:

39A11 Stability of difference equations (MSC2000)
47D06 One-parameter semigroups and linear evolution equations
46B20 Geometry and structure of normed linear spaces
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
39A12 Discrete version of topics in analysis
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