Non-autonomous retarded differential equations: the variation of constants formulas and the asymptotic behaviour

Abstract

This paper is devoted to show a variation of constants formula for the operator solution to the non-autonomous retarded differential equation $$ x'(t)=A(t)x(t)+L(t)x_t+f(t),quad x_s=varphi, quad tgeq sgeq 0, $$ in terms of the inhomogeneous term $f$, which will allow us to study the asymptotic behaviour of this solution. We treat also the existence of fundamental solutions and the stability of semi-linear retarded differential equations.

Keywords

• Non-autonomous retarded differential equations
• variation of constants formula
• Dyson-Phillips series
• asymptotic almost periodicity.