Averaging Technique and Oscillation for Even Order Damped Delay Differential Equations

By using the averaging technique, some integral oscillation criteria are obtained for even order damped delay differential equations. 1 – Introduction This paper deals with the oscillatory behavior of the even order damped delay differential equation (1.1) ( Φ(xn−1)(t)) )′ + p(t)Φ(x(n−1)(t)) + f ( t, x[ τ01(t) ], ..., x[ τ0m(t) ], ..., x(n−1)[ τn−11(t) ], ..., x (n−1)[ τn−1m(t) ] ) = 0 for t ≥ t0 > 0 , where Φ(s) = |s|α−1s with α > 0 a fixed constant, and n is an even number. Throughout this paper, we assume that (A1) p ∈ C(I,R0) and lim t→∞ ∫ t t̄ [