Nonoscillation of a class of Neutral Differential Equations
by
Yijun Sun
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39759
Coauthors: Qingkai Kong and Binggen Zhang

Numerous results have been obtained for the oscillation of the neutral differential equation (x(t)-c(t)x(t-\tau))⁽ⁿ⁾ + p(t)  x(t-\sigma) = 0, and its variations, where n is odd, p(t) is continuous, \tau > 0, and \sigma in R. It is known that c(t)=1 is a critical case. That is, the properties of solutions of the above equation in the case c(t) > 1 are essentially different from those in the case 0 <= c(t) < 1. However, very little is known so far for the properties of nonoscillatory solutions in the case c(t)=1. In this paper, we will give a complete classification of nonoscillatory solutions for the case of c(t)=1 by dividing all positive solutions into three types. We show that every positive solution must be one of these three types. We give necessary and sufficient conditions for the existence of the first two types of solutions. We also give sufficient conditions, and necessary conditions for the existence of the third type of solutions, respectively.

Date received: March 29, 2003