This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral...

and Applied Analysis 3 Throughout this paper, we assume that R −∞, ∞ , R 0, ∞ , C t0, ∞ ,R denotes the Banach space of all continuous and bounded functions on t0, ∞ with the norm ‖x‖ supt≥t0 |x t | for each x ∈ C t0, ∞ ,R and A N,M {x ∈ C t0, ∞ ,R : N ≤ x t ≤ M, t ≥ t0} for M > N > 0. 1.8 It is easy to see that A N,M is a bounded closed and convex subset of C t0, ∞ ,R . By a solution of 1.7 , w...

where is the forward difference operator defined by is a sequence of nonnegative real numbers, is a sequence of nonnegative real numbers and is not identically zero for large values of n, is a sequence of positive real numbers and are nonnegative integers, is the ratio of two odd integers and is a continuous real valued function. Throughout the paper the following conditions are assumed to be h...