Oscillation of Second Order Neutral Differential Equations With Mixed Neutral Term

Abstract

This paper deals with the following second order neutral differential equation of the formwhere z(t) = x(t) + a(t)x(t − τ ) + b(t)x(t + δ), andWe obtain some new oscillation criteria which extend some known results. Examples are presented to illustrate the main results.

Second-Order Non-Canonical Neutral Differential Equations with Mixed Type: Oscillatory Behavior

“…, and q(l) ≥ 0. Ruan [4] obtained some oscillation criteria for the Equation (2) by employing Riccati technique and averaging function method, when p 2 (l) = 0 and σ(l) = l − σ. Arul and Shobha [5] studied the oscillatory behavior of solution of (2), when 0…”

Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term

“…In [24], the present authors established some sufficient conditions for the oscillation of all solutions of Equation (1) when…”

“…Remark 3.1. The results presented in [24] are not applicable to Equations (31) and (32) since in these equations ( )…”

Oscillation criteria for second-order quasi-linear neutral functional differential equation

“…The significant practical importance and theoretical interest for qualitative study of the neutral differential equations is due to the appearance of differential equations with neutral delay appear in mechanics in problem of oscillating masses connected to an elastic bar, in solution of the problems of variations with delay in the time and the problems of electric networks which containing transmission lines without loss, as an example, lines of transmission without loss are used to interconnect switching circuits in computers with high speed, see [7]. Lately, there were a many articles dedicated to the oscillation of solutions of the differential equations with delay or neutral delay, see [1,2,3,5,6,8,10,11,12,14,17,19].…”

“…Actually, under the assumption η (t 0 ) = ∞, there are many studies in the literature that have been concerned with the oscillation and nonoscillation criteria of solutions of Eq. 1, see for example [2,10,17,19].…”