On Neutral Functional–Differential Equations with Proportional Delays

AbstractIn this paper we develop a comprehensive theory on the well-posedness of the initial-value problem for the neutral functional-differential equationy′(t)=ay(t)+∑i=1∞biy(qit)+∑i=1∞ciy′(pit),t>0,y(0)=y0,and the asymptotic behaviour of its solutions. We prove that the existence and uniqueness of solutions depend mainly on the coefficientsci,i=1,2,…, and on the smoothness of functions in the solution space. As far as the asymptotic behaviour of analytic solutions is concerned, thecihave little effect. We prove that if Rea>0 then the solutiony(t) either grows exponentially or is polynomial. The most interesting result is that if Rea≤0 anda≠0 then the asymptotic behaviour of the solution depends mainly on the characteristic equationa+∑i=1∞biqλi=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