Oscillatory and periodic solutions of delay differential equations with piecewise constant argument

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) has oscillatory solutions.DEFINITION. A solution of equation (1) on [0, oo) is a function y(t) that satisfies the initial data y(0) = Co, y(-l) = C-x, and the conditions: (i) y(t) is continuous on [0, oo); (ii) the derivative y'(t) exists at each point t G [0, oo), with the possible exception of the points [zj] G [0, oo) where one-sided derivatives exist;(iii) equation (1) is satisfied on each interval [n,n+ 1) C [0, oo) with integral endpoints.

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

“…∈ Θ are ǫ− equivalent and write θ (1) ǫθ (2) , if there exist the representatives γ (1) and γ (2) in Γ of θ (1) and θ (2) , respectively, such that ||γ…”

“…Let γ (1) , γ (3) be the representatives of θ (1) , θ (3) , respectively, and γ (2) , γ (4) be the representatives of θ (2) such that ||γ…”

“…where [·] signifies the greatest integer function, was initiated in [13,30] and developed by many authors [1,8,9,25,28,29], [32], [36]- [39]. These systems have been under intensive investigation for the last twenty years.…”

Existence of Periodic Solutions for a Neutral Differential Equation with Piecewise Constant Argument

“…Indeed, as mentioned in [10], the term f ðt; xðtÞ; xð½t þ 1=2ÞÞ may be interpreted as the control term in the stabilization of hybrid control systems with advanced and delayed feedback, where a hybrid system is one with a continuous plant and with a discrete (sampled) controller. Furthermore, when the neutral term is missing in (1), the periodic solutions have stimulated considerable interest and have been studied in [6,9,11,17].…”

“…As mentioned in [1][2][3][4][5][6], the strong interest in such equations is motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both di¤erential and di¤erence equations.…”