On the Asymptotic Solution of Initial Value Problems for Differential Equations with Small Delay

This paper obtains asymptotic expansions for initial value problems of the form \[ \begin{gathered} (1)\qquad \dot x(t) = f(t,x(t), x(t - \mu ),\dot x(t - \mu )),\quad t \geqq 0, \hfill \\ (2)\qquad x(t) = \phi (t), \quad - \mu \leqq t \leqq 0 , \hfill \\ \end{gathered} \] as the positive delay parameter $\mu $ tends to zero. The critical hypotheses are: (i) that the reduced problem ((1)–(2) with $\mu = 0$) has a unique solution $X_0 (t)$, and (ii) that $| {f_u (t,x,y,u)} | < a < 1$ everywhere. Then $x(t)$ will converge to $X_0 (t)$ as $\mu \to 0$ and boundary layer behavior will occur in higher order approximations near $t = 0$.