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Functional differential equations of advanced type with piecewise constant argument deviations are studied. They are closely related to impulse, loaded and, especially, to difference equations, and have the structure of continuous dynamical systems within intervals of unit length

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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.

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.