Abstract
We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities:rtzn-1tα-1zn-1t'+q0(t)(xτ0tα-1x(τ0(t))+q1t(x(τ1(t))β-1x(τ1(t))+q2t(x(τ2(t))γ-1x(τ2(t))=0, t≥t0,wherez(t)=x(t)+p(t)x(σ(t)),nis even integer, andγ>α>β>0.Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities.
Highlights
We are concerned with oscillation behavior of the certain even order neutral delay differential equations with mixed nonlinearities: (r (t)
We assume that there exists a function τ ∈ C1([t0, ∞), R), such that τ(t) ≤ τi(t), i = 0, 1, 2, τ(t) ≤ t, τ(t) > 0, and limt → ∞τ(t) = ∞
Some new oscillation criteria are established for the secondorder nonlinear neutral delay differential equations:
Summary
We are concerned with oscillation behavior of the certain even order neutral delay differential equations with mixed nonlinearities:. Xu and Xia [7], by means of Riccati transformation technique, established some oscillation criteria for certain even order delay differential equations:. Some new oscillation criteria are established for the secondorder nonlinear neutral delay differential equations:. Meng and Xu [19], by using the Riccati transformation technique and inequalities, considered the oscillation for even order quasilinear neutral differential equations:. Some new criteria are established that improve a number of related results reported in the literature and can be used in cases where known theorems fail to apply. The purpose of this paper is to extend and improve the abovementioned oscillation theorems for certain even order neutral delay differential equations with mixed nonlinearities (1).
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