Cauchy Problem for a Singularly Perturbed Delay Equation

Abstract

In the study, the Cauchy problem for a singularly perturbed first-order ordinary differential equation is considered, with, generally speaking, a nonlinear right-hand side that depends not only on the desired function but also on this same function taken with a time delay. The problem under consideration is singularly perturbed due to the presence of a small parameter in front of the time derivative. For such problems, solutions that possess a large gradient in the vicinity of the initial time moment and in the vicinity of the moment equal to the delay time are typical. The aim of the work is to construct an asymptotic approximation and to prove the existence of a smooth solution to the problem.