Oscillation Criteria for Odd Order Neutral Equations

In this paper four main results are obtained for oscillation of all solutions of the odd order neutral differential equation [formula] where pi(t) ≥ 0, σi, ∈ (0, ∞), ƒi: R → R (i = 1, 2, ..., m), and p, τ ∈ [0, ∞). Theorem 1 shows that if, in addition to the above, 0 ≤ p < 1, ƒ(i)(x) = x, n > 1, and, for some μ ∈ (0, 1), all solutions of the first order delay equation [formula] are oscillatory, then all solutions of (*) are oscillatory. In particular, when m = 1 and p1(t) = p1 ∈ (0, ∞) then p1 σn1 > ((1 − p)(n)!/e)(n/(n − 1))n−1 implies that all solutions of (*) are oscillatory. In such case Theorem 4.1, due to Gopalsamy, et al. [Czech. Math. J.42 (1992), 313-323] reduces to p1 σn1 > (1 − p)(n)! which, in view of the known inequality (1/e)(n/(n − 1))n−1 < 1, is a stronger condition. Some results of Zhang [J. Math. Anal. Appl.139 (1989), 311-318], Graef et al. [J. Math Anal. Appl.155 (1991), 562-571], and Ladas et al. [Proc. Amer. Math. Soc.113 (1991), 123-133] have been generalized in Theorems 2, 3, and 4, respectively.