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ON NUMERICAL APPROXIMATION USING DIFFERENTIAL EQUATIONS WITH PIECEWISE-CONSTANT ARGUMENTS ´n Gyo ˝ ri1 and Ferenc Hartung2 Istva

Dedicated to the memory of Professor Mikl´ os Farkas

1

Department of Mathematics and Computing, University of Pannonia 8201 Veszpr´em, P.O. Box 158, Hungary E-mail: [email protected]

2

Department of Mathematics and Computing, University of Pannonia 8201 Veszpr´em, P.O. Box 158, Hungary E-mail: [email protected] (Received September 17, 2007; Accepted September 24, 2007)

Abstract

In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments (EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and oscillation of a class of linear delay equations with continuous and piecewise constant arguments.

1. Introduction Delay differential equations (DDEs) provide a mathematical model for physical, biological systems in which the rate of change of the system depends upon their past history. The general theory of DDEs with continuous arguments has been thoroughly investigated by now, the number of the papers devoted to this area of research continues to grow very rapidly. Mathematics subject classification number : 34K28, 92D25. Key words and phrases: delay equations; equations with piecewise constant arguments; numerical approximation, nonlinear age-dependent population model, hybrid system. This research was partially supported by Hungarian NFSR Grant No. T046929. 0031-5303/2008/$20.00

c Akad´emiai Kiad´o, Budapest 

Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht

56

˝ I. GYORI and F. HARTUNG

This paper is devoted to a generalized class of DDEs, namely delay differential equations with piecewise constant arguments (EPCAs). EPCAs include, as particular cases, impulsive DDEs and some equations of control theory, and are similar to those found in some biomedical models, hybrid control systems and numerical approximation of differential equations with discrete difference equations. The general theory and basic results for DDEs can be found for instance in the book of Hale [20] (see also Bellman and Cooke [3], Hale and Lunel [21] and Myshkis [30]), and subsequent articles by many authors. The study of EPCAs has been initiated by Wiener [39], [40], Cooke and Wiener [6], [7], Shah and Wiener [31]. A survey of the basic results has been given in [8], [41]. A typical EPCA is of the form x(t) ˙ = f (t, x(h(t)), x(g(t))),

(1.1)

where the argument h is a continuous function and the argument g has intervals of constancy. For example g(t) = [t] or g(t) = [t − n], where n is a positive integer, and [·] denotes the greatest-integer function. Note that if f (t, u, v) ≡ f1 (t, u), then (1.1) is a classical DDE with continuous argument. If f (t, u, v) ≡ f2 (t, v), and, for instance, g(t) = [t], then the solution x of (1.1) is piecewise linear and at t