Positive solutions of a higher order neutral differential equation

✍ Scribed by John R. Graef; Chuanxi Qian; Bo Yang

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 120 KB
• Volume: 256
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310067

✦ Synopsis

Abstract

In this paper, we consider the higher order neutral delay differential equation

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.