Oscillatory and Periodic Solutions of Delay Differential Equations with Piecewise Constant Argument

Abstract

ABSTRACT. Oscillatory and periodic solutions of retarded functional differential equations are investigated.The study concerns equations with piecewise constant arguments which found applications in certain biomédical problems.1. The study of oscillatory solutions of differential equations with deviating arguments has been the subject of many recent investigations.Of particular importance, however, has been the study of oscillations which are caused by the deviating arguments and which do not appear in the corresponding ordinary differential equation, see [1,[4][5][6][7][8][9][10][11]13].In this paper we study oscillatory properties of solutions of the linear delay differential equations with piecewise constant deviating argument of the typewhere a(t) and b(t) are continuous functions on [0, oo), and [•] designates the greatest integer function. Such equations are similar in structure to those found in certain "sequential-continuous" models of disease dynamics as treated by Busenberg and Cooke [2]. We give sufficient condition under which equation (1) has oscillatory solutions. We emphasize the fact that our condition is the "best possible" in the sense that when a and b are constants the condition reduces to b > ae~a/4(ea -1) which is a necessary and sufficient condition. In case of constant coefficients we find conditions under which oscillatory solutions are periodic. As it is customary, a solution is said to be oscillatory if it has arbitrarily large zeros.2. In this section we give a sufficient condition under which equation (1) (iii) equation (1) is satisfied on each interval [n,n+ 1) C [0, oo) with integral endpoints.

On Quasi-Periodic Solutions of Differential Equations with Piecewise Constant Argument

“…In [20], Yuan and Hong proved the existence of periodic solutions for the 173 0022-247X/02 $35.00 following differential equations with piecewise constant argument (EPCA, for short),ẋ t = ax t + N i=−N a i x t + i + f t N ≥ 1 (2) f ∈ T , under certain assumptions. In particular, it was needed that T be a rational number.…”

“…It is well known that KAM theory can be applied to show the existence of quasi-periodic motions for periodic ordinary differential equations (see [9] and references therewith). Equation (2) has the structure of a continuous dynamical system in intervals of unit length. Continuity of a solution at a point joining any two consecutive intervals implies a recursion relation for the values of the solution at such points.…”

Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument

“…Then, oscillation and stability of DEPCA have been studied by many authors (see [2][3][4][5][6] and the references therein). In 1994, Dai and Singh [7] studied the oscillatory motion of spring-mass systems subject to piecewise constant forces of the form…”

Advanced Impulsive Differential Equations With Piecewise Constant Arguments

“…So, they describe hybrid dynamical systems and combine the properties of both differential and difference equations. The oscillation, periodicity and some asymptotic properties for various differential equations with piecewise constant arguments without impulses were methodically demonstrated in [1,2,3,11,12,13,17,19,20,22,24,27]. Also, Wiener's book [25] is a distinguished source in this area.…”