Oscillations of higher order neutral differential equations

Abstract

Consider the nth-order neutral differential equationwhere n> 1 , 8 = ± 1 , / , 31 are initial segments of natural numbers , p ( , T ; , a k 6 R and q k > 0 for i g / and k e 3T . Then a necessary and sufficient condition for the oscillation of all solutions of (E) is that its characteristic equation has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of Equation (E).

Oscillations of higher order neutral differential equations

“…Necessary and sufficient conditions (in terms of the characteristic equations) for the oscillations of all solutions of higher order neutral differential equations have been established by Bilchev, Grammatikopoulos and Stavroulakis [1], Ladas, Partheniadis and Sficas [3] and Wang [5]. The oscillation criteria obtained in this paper improve noticeably the results in [2] and [4].…”

Oscillations in Higher-Order Neutral Differential Equations

“…In the oscillation theory of neutral differential equations one of the most important problems is to obtain necessary and sufficient conditions for the oscillation via the characteristic equation. Among numerous papers dealing with this problem we refer in particular to [1], [2], [5], [7], [12]- [15], [20]- [22], [24], [25], [28], and [30]. Sufficient conditions for the oscillation of the solutions of neutral equations have been obtained in many recent papers; we choose to refer in particular to [3], [4], [19], and [23].…”

“…Sufficient conditions for the oscillation of the solutions of neutral equations have been obtained in many recent papers; we choose to refer in particular to [3], [4], [19], and [23]. Most of the papers mentioned above concern the case of first order neutral equations and two of these papers (see [21] and [22]) are referred to second order equations; the papers [1], [7], [19], [23], [28], and [30] are concerned with the oscillation of neutral differential equations of higher order.…”

“…is defined by (4). From (5), (7) and (11) it follows that there exists a Ao < 0 such that F(X) > 0 for all A < Ao. This contradicts (10) and so (viii) has been proved.…”

“…x be a solution of (E) which is n-times continuously differentiable, positive and strictly monotone and on the interval [7,00). Then:…”