Exponential stability for abstract linear autonomous functional differential equations with infinite delay

Abstract

Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equation x˙(t)=L(xt) (∗) where L is a continuous linear operator on some abstract phase space B into a Banach space E. We prove that the solution semigroup of (∗) is exponentially stable if and only if the fundamental operator (∗) is exponentially stable and the phase space B is an exponentially fading memory space.

Keywords

• Exponentially stable
• solution semigroup
• abstract linear autonomous functional differential equation
• fundamental operator
• fading memory space.