Positive Solutions of a Higher Order Neutral Differential Equation
In this paper, we consider the higher order neutral delay differential equation (x(t) - x(t - r))(n) + p (t)x(t - σ) = 0, t ≥ 0, (*) where p: [0, ∞) → (0, ∞) is a continuous function, r > 0 and σ > 0 are constants, and n > 0 is an odd integer. A positive solution x(t) of Eq. (*) is called a Class-I solution if y(t) > 0 and y′(t) < 0 eventually, where y(t) = x(t) - x(t - r). We divide Class-I solutions of Eq. (*) into four types. We first show that every positive solution of Eq. (*) must be of one of these four types. For three of these types, a necessary and sufficient condition is obtained for the existence of such solutions. A necessary condition for the existence of a solution of the fourth type is also obtained. The results are illustrated with examples.
Graef, J. R., Qian, C., & Yang, B. (2003). Positive solutions of a higher order neutral differential equation. Mathematische Nachrichten, 256(1), 17-28. doi:10.1002/mana.200310067