L2-Perturbation of a Linear Delay Differential Equation

In this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation ẋ(t) = (a0 + a(t))x(t) + (b0 + b(t))x(t − τ), assuming that it is an L2-perturbation of the autonomous equation ẋ(t) = a0x(t) + b0x(t − τ). Our theorem is an essential generalization of the corresponding result in the classical book by R. Bellman and K. L. Cooke. The proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation.