Notes on solution maps of abstract FDEs

TL;DR

This paper demonstrates that solution maps of abstract functional differential equations are -contractions under certain conditions, enabling their analysis in phase spaces with equivalent norms, with applications to delayed reaction-diffusion systems.

Contribution

It establishes -contraction properties of solution maps for abstract FDEs, extending the analysis to systems with time delay.

Findings

Solution maps are -contractions in phase space.

Applicable to time-delayed reaction-diffusion equations.

Provides a framework for analyzing evolution systems with delays.

Abstract

It is shown that the solution maps of an abstract functional differential equations (FDEs) are $\alpha$-contractions in the phase space equipped with an equivalent norm under appropriate assumptions. This result can be applied to time-delayed reaction-diffusion equations and other evolution systems with time delay.