New Criteria for Sharp Oscillation of Second-Order Neutral Delay Differential Equations

Irena Jadlovská

doi:10.3390/math9172089

Abstract

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones.

Highlights

The aim of this work is to study the asymptotic and oscillatory properties of solutions of the second-order half-linear neutral delay differential equation r (t) z0 (t) α 0 t ≥ t0 > 0,+ q(t) x α (σ(t)) = 0, (1)Citation: Jadlovská, I
The following assumptions will be made without further mention: Hypothesis 1 (H1). α > 0 is a quotient of odd positive integers
In [43,44], we have presented an oscillation criterion for (1) with p(t) = 0 which is sharp for the Euler half-linear delay differential Equation (31)

Summary

Introduction

The aim of this work is to study the asymptotic and oscillatory properties of solutions of the second-order half-linear neutral delay differential equation r (t) z0 (t) α 0 t ≥ t0 > 0,. The following assumptions will be made without further mention: Hypothesis 1 (H1). Α > 0 is a quotient of odd positive integers The following assumptions will be made without further mention: Hypothesis 1 (H1). α > 0 is a quotient of odd positive integers

Objectives

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Results