On Exponential Dichotomy for Abstract Differential Equations with Delayed Argument

Abstract

We consider linear differential equations of the first order with delayed arguments in a Banach space. We establish conditions for the operator coefficients necessary for the existence of exponential dichotomy on the real axis. It is proved that the analyzed differential equation is equivalent to a difference equation in a certain space. It is shown that, under the conditions of existence and uniqueness of a solution bounded on the entire real axis, the condition of exponential dichotomy is also satisfied for any known bounded function. We also deduce the explicit formula for projectors, which form this dichotomy in the case of a single delay.