Asymptotic and Oscillatory Behavior of Solutions to Nonlinear Delay Equations

It is known that the general solution of a linear delay differential equation is given by the renewal equation involving the fundamental solution. The latter is a special solution satisfying discontinuous initial data. In this paper we represent the fundamental solution of a linear delay differential equation as a sum of solutions with smooth initial data. Using this result, we derive an asymptotic approximation for the solution of a nonlinear delay differential equation in the case when its characteristic equation has only roots with negative real parts. As a corollary we state a sufficient condition for the solutions of such equations to have oscillatory behavior.