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On neutral functional-differential equations with proportional delays. (English) Zbl 0873.34066

The paper deals with the well-posedness of the initial value problem for the neutral functional-differential equation

${y}^{\text{'}}\left(t\right)=ay\left(t\right)+\sum _{i=1}^{\infty }{b}_{i}y\left({q}_{i}t\right)+\sum _{i=1}^{\infty }c{y}^{\text{'}}\left({p}_{i}t\right),\phantom{\rule{2.em}{0ex}}t>0,\phantom{\rule{1.em}{0ex}}y\left(0\right)={y}_{0}$

and the asymptotic behaviour of its solutions. The authors proved that the existence and uniqueness of solutions depend mainly on the coefficients ${c}_{i}$, $i=1,2,\cdots$ and on the smoothness of the functions in the solution sets. As far as the asymptotic behaviour of analytic solutions is concerned, the ${c}_{i}$ have little effect. If $\text{Re}\phantom{\rule{4.pt}{0ex}}a\le 0$ and $a\ne 0$ the asymptotic behaviour of the solutions depends mainly on the characteristic equation

$a+\sum _{i=1}^{\infty }{b}_{i}{q}_{i}^{k}=0·$

These results can be generalized to systems of equations. Some examples to illustrate the change of asymptotic behaviour in response to the variation of parameters are given. The main idea used in this paper is to express the solution in either Dirichlet or Dirichlet-Taylor series form.

MSC:
 34K40 Neutral functional-differential equations