Variation of constants formula and exponential dichotomy for non autonomous non densely defined Cauchy problems

TL;DR

This paper develops a variation of constants formula for non autonomous, non densely defined Cauchy problems and characterizes exponential dichotomy, with applications to non local boundary conditions.

Contribution

It introduces a new variation of constants formula applicable to non densely defined operators and provides criteria for exponential dichotomy in this context.

Findings

Established a variation of constants formula for non densely defined operators

Derived necessary and sufficient conditions for exponential dichotomy

Demonstrated persistence of exponential dichotomy under small perturbations

Abstract

In this paper we prove a variation of constants formula for a non autonomous and non homogeneous Cauchy problems whenever the linear part is not densely defined and is not a Hille-Yosida operator. By using this variation of constants formula we derive a necessary and sufficient conditions for the existence of exponential dichotomy for the evolution family generated by the associated non autonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally we illustrate our result by consider a parabolic equation with non local and non autonomous boundary conditions.