Characteristic functions of differential equations with deviating arguments

The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If a, b, c and {(sic)(l)} are real and gamma((sic)) is real-valued and continuous, an example with these parameters is u'(t) = {au(t) + bu(t + (sic)(1)) + cu(t + (sic)(2))} + integral((sic)4)((sic)3) gamma((sic))(s)u(t+s)ds. (star) A wide class of equations (star), or of similar type, can be written in the "canonical" form u'(t) = integral(tau max)(tau min) u(t+s)d sigma(s) (t is an element of R), for a suitable choice of tau(min), tau(max) (star star) where sigma is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (star star), there is a corresponding characteristic function chi(zeta)) := zeta-integral(tau max)(tau min) exp(zeta s)d sigma(s) (zeta is an element of C), (star star star) whose zeros (if one considers appropriate subsets of equations (star star) - the literature provides additional information on the subsets to which we refer) play a role in the study of oscillatory or non-oscillatory solutions, or of bounded or unbounded solutions. We show that the related discussion of the zeros of chi is facilitated by observing and exploiting some simple and fundamental properties of characteristic functions. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.