Dichotomies for linear differential equations with delays: the Carathéodory case

Abstract

This work is concerned with differential equations with delays, on [0, ∞[, of the form\(\dot u\)+Mu=r, and the corresponding homogeneous equation\(\dot u\)+Mu=0, where u and r take values in a Banach space E, and M is a « memory ». The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The « memory » M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition.