Oscillatory behavior of third-order nonlinear neutral delay differential equations

Ying Jiang,Cuimei Jiang,Tongxing Li

doi:10.1186/s13662-016-0902-7

Abstract

We provide an oscillation criterion for a class of third-order nonlinear neutral delay differential equations by using the double generalized Riccati substitutions. Our theorem complements and improves previous results. Two illustrative examples are included.

Highlights

1 Introduction This article is concerned with the oscillation and asymptotic behavior of a nonlinear thirdorder neutral delay differential equation r(t) z (t) α + q(t)f x σ (t) =, ( . )
We suppose that the following assumptions hold: (A ) r ∈ C ([t, ∞), (, ∞)), p, q ∈ C([t, ∞), [, ∞)), τ ∈ C ([t, ∞), R), σ ∈ C([t, ∞), R), and q is not identically zero for large t; (A )
We provide some background details that motivated our study

Summary

Introduction

We suppose that the following assumptions hold: (A ) r ∈ C ([t , ∞), ( , ∞)), p, q ∈ C([t , ∞), [ , ∞)), τ ∈ C ([t , ∞), R), σ ∈ C([t , ∞), R), and q is not identically zero for large t; (A ) ) which satisfy condition sup{|x(t)| : t ≥ T} > for all T ≥ Tx and assume that Baculíková et al [ ] and Li and Rogovchenko [ ] established several oscillation theorems for a second-order neutral differential equation r(t) z (t) α– z (t) + q(t)f x σ (t) = , z := x + p · (x ◦ τ )

Results

Conclusion