The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hospital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.

Basic Calculus on Time Scales and some of its Applications

On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays

On etudie les solutions oscillatoires et periodiques des equations differentielles fonctionnelles retardees

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

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Comparison theorems for functional differential equations with deviating arguments

On even-order functional differential equations with advanced and retarded arguments

Oscillations of functional differential equations with retarded argument

Consider the n -th order delay differential equation In the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q ( t ). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.

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A comparison theorem and the forced oscillation

Oscillatory property of ordinary differential equations of arbitrary order

On Oscillatory Properties of Differential Equations with Deviating Argument

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Hiroshi Onose

Papers

Oscillations of functional differential equations with retarded argument

A comparison theorem and the forced oscillation

Oscillatory property of ordinary differential equations of arbitrary order

On Oscillatory Properties of Differential Equations with Deviating Argument

Oscillation of solutions of nonlinear differential delay equations of arbitrary order