A functional analytic theory for differential equations on Banach spaces with slowly evolving parameters

Abstract This paper provides a functional analytic approach to differential equations on Banach spaces with slowly evolving parameters. We develop a Fenichel-like theory for attracting subsets of critical manifolds via a Lyapunov–Perron method. This functional analytic approach to invariant manifold theory for fast-slow systems of differential equations has not been fully developed before, especially for the case that the fast subsystem lives on an infinite-dimensional Banach space. We provide rigorous functional analytic proofs for both the persistence of attracting critical manifolds as smooth slow manifolds, as well as the validity of slow manifold reduction near slow manifolds. Several aspects of our proofs are new in the literature even for the finite-dimensional case. For instance, we require less smoothness assumptions on systems than comparable texts based on Lyapunov–Perron approaches. The theory as developed here provides a rigorous framework that allows one (for example) to derive formal statements on the dynamics of biologically meaningful spatially extended models with slowly varying parameters.