On the asymptotic behavior of solutions of some delay differential systems which have a first integral
This paper extends a previous publication by the first author and \textit{P. Seguier} [ibid. 96, 420-436 (1983; Zbl 0527.34066)] who considered asymptotic behaviour of the solutions of scalar nonautonomous delay differential equations which have a first integral. Earlier examples of that kind were introduced by K. L. Cooke and J. Yorke in the study of epidemiological problems. These are autonomous equations \(dx/dt=g(x(t))- g(x(t-T)).\) The most general (and elegant) result about asymptotic behaviour of such equations appeared in 1978 in a joint paper by J. Kaplan, M. Sorg and J. Yorke. Other related results were obtained, notably by J. Haddock, T. Krisztin and by C. Jehu who indirectly inspired the present work. While the work with P. Seguier mainly uses comparison results due to monotonicity with respect to the delay terms, the main tool here is a Lyapunov functional on the pairs of solutions, coupled with a first integral. For systems, the main field of applications is compartmental analysis and, indeed, the class of systems under consideration here is a non-autonomous version of a compartmental model originally derived by I. Gyori. Partial asymptotic results (for example, not considering periodic dependence) were previously obtained by I. Gyori.
scalar nonautonomous delay differential equations
scientific article; zbMATH DE number 3994068
scientific article