Higher Order Linear Autonomous Differential Differential Equations,

Consider the following equation (?)whose characteristic equation is (?) where p, q,σandτare constants, andτ,σ>0, n is a positive integer.In the present paper, we study higher order linear time-invariant differential difference equations, which are theoretically important and practically often used. The problem of oscillations of the equations has received a large lot of attention. In the discussion of oscillation, it is of great value to be able to predict oscilation or nonoscillation directly from the coefficients of given equations. The above mentioned methods are of geat value in this kind of applied problems in automatic control theory.In this paper, we introduce one function and describe its properlities provided in [13]. By using them, we establish oscillation criteria for all the solutions of the equation of (E) for n∈N, which can predict the oscillation or nonoscillation directly from the coefficients of the given equation. The above mentioned conditions are entire and easy to verify and apply. At the same time, our results substantially genelizate existing ones as special cases.