Asymptotic behaviour of the solution of a functional-differential equation

Abstract

The asymptotic behaviour as t→∞ of the solution of the functional- differential equation y'(t) = -y(t/k), with y(0) = 1 and k > 1 , is derived from an integral representation by the method of steepest descents. It is shown that the solution oscillates (that is, has arbitrarily large zeros), that the amplitude of the oscillations growsfasterthan any polynomialbut slower thananyexponential, and that the ratios of successive zeros of the solution decrease to the limiting value k.