On Neutral Functional–Differential Equations with Proportional Delays

Abstract

In this paper we develop a comprehensive theory on the well-posedness of the initial-value problem for the neutral functional-differential equation$\text{y′(t)}\text{=}\text{ay(t)}\text{+}\text{∑}\text{i}\text{=}\text{1}\text{∞}\text{b}{}_{\mathrm{i}}\text{y(q}{}_{\mathrm{i}}\text{t)}\text{+}\text{∑}\text{i}\text{=}\text{1}\text{∞}\text{c}{}_{\mathrm{i}}\text{y′(p}{}_{\mathrm{i}}\text{t),}\text{t}\text{>}\text{0,}\text{y(0)}\text{=}\text{y}{}_0\text{,}$and the asymptotic behaviour of its solutions. We prove that the existence and uniqueness of solutions depend mainly on the coefficientsc_i,i = 1, 2,…, and on the smoothness of functions in the solution space. As far as the asymptotic behaviour of analytic solutions is concerned, thec_ihave little effect. We prove that if Re a > 0 then the solutiony(t) either grows exponentially or is polynomial. The most interesting result is that if Re a ≤ 0 anda ≠ 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation$\text{a}\text{+}\text{∑}\text{i}\text{=}\text{1}\text{∞}\text{b}{}_{\mathrm{i}}\text{q}{}^{\mathrm{\lambda }}{}_{\mathrm{i}}\text{=}\text{0.}$These results can be generalized to systems of equations. Finally, we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters. The main idea used in this paper is to express the solution in either Dirichlet or Dirichlet–Taylor series form.