Oscillation and Asymptotic Behaviour of a Higher‐Order Nonlinear Neutral‐Type Functional Differential Equation with Oscillating Coefficients

Mustafa Kemal Yildiz,Hülya Durur,Emrah Karaman

doi:10.1155/2011/158060

Abstract

We will study oscillation of bounded solutions of higher‐order nonlinear neutral delay differential equations of the following type: [y(t) + p(t)f(y(τ(t)))] (n) + q(t)h(y(σ(t))) = 0, t ≥ t0, t ∈ ℝ, where p ∈ C([t0, ∞), ℝ), limt→∞p(t) = 0, q ∈ C([t0, ∞), ℝ+), τ(t), σ(t) ∈ C([t0, ∞), ℝ), τ(t), σ(t) < t, lim t→∞τ(t), σ(t) = ∞, and f, h ∈ C(ℝ, ℝ). We obtain sufficient conditions for the oscillation of all solutions of this equation.

Highlights

We are concerned with the oscillation of the solutions of a certain more general higher-order nonlinear neutral-type functional differential equation with an oscillating coefficient of the form yt ptf yτt n qthyσt
A solution y t is said to be oscillatory if y t is not eventually positive or not eventually negative
The purpose of this paper is to study oscillatory behaviour of solutions of 1.1

Summary

Introduction

We are concerned with the oscillation of the solutions of a certain more general higher-order nonlinear neutral-type functional differential equation with an oscillating coefficient of the form yt ptf yτt n qthyσt0, t ≥ t0, t ∈ R, 1.1 where p ∈ C t0, ∞ , R is oscillatory and limt → ∞p t 0, q ∈ C t0, ∞ , R , τ t , σ t ∈ C t0, ∞ , R , τ t , σ t < t, limt → ∞τ t ∞, limt → ∞σ t ∞, and f, h ∈ C R, R. We will study oscillation of bounded solutions of higher-order nonlinear neutral delay differential equations of the following type: ytptfyτtnqthyσt We obtain sufficient conditions for the oscillation of all solutions of this equation.

Objectives

Results

Conclusion