A Hilbert space approach to difference equations

TL;DR

This paper introduces a Hilbert space framework for analyzing difference equations, providing new insights into stability and stable manifolds for both linear and nonlinear cases.

Contribution

It develops a Hilbert space approach to difference equations, characterizes exponential stability, and proves a stable manifold theorem for nonlinear equations.

Findings

Characterization of exponential stability for linear difference equations

Proof of a stable manifold theorem for nonlinear difference equations

Application of Hilbert space methods to two-sided sequences

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.