A Hilbert space approach to difference equations

Abstract

We consider general difference equations $u_{n+1}=F(u{)}_n$ for $n\in \mathbb{Z}$ on exponentially weighted ${\ell }_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold theorem for causal nonlinear difference equations.

23 pages. Reasons for replacement: Added reference, corrected typos