Exponential dichotomy of difference equations and applications to evolution equations on the half-line
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 \(U(t,s)\), \(0\leq s\leq t<\infty\).
scientific article; zbMATH DE number 1755252