The distribution of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations

Limei Feng,Zhenlai Han

doi:10.1186/s13662-019-2220-3

Abstract

In this paper, the distributions of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations are studied. By means of inequality techniques, specific function sequences and non-increasing solutions for corresponding first-order difference inequality, some new estimates for the distribution of the zeros of oscillatory solutions are presented, which extend and improve some well-known results.

Highlights

In recent years, the study of oscillation of differential equations has become more and more perfect, including various sufficient conditions, necessary conditions, the existence of non-oscillatory solutions, and even the zeros distribution of oscillatory solutions.In 2017, Li et al [1] studied the distribution of zeros of oscillatory solutions for secondorder nonlinear neutral delay differential equation a(t)z (t) + q(t)f x(t – σ ) = 0, t ≥ t0, and obtained a sufficient condition for oscillation of differential equation
Two theorems on the distribution of oscillation zeros for second-order nonlinear neutral delay difference equations are obtained by means of inequality techniques, specific function sequences and non-increasing solutions for corresponding first-order difference inequality
We study a second-order equation under the canonical form, and it is of great significance for the study of non-canonical forms

Summary

Introduction

In 2017, Li et al [1] studied the distribution of zeros of oscillatory solutions for secondorder nonlinear neutral delay differential equation a(t)z (t) + q(t)f x(t – σ ) = 0, t ≥ t0, and obtained a sufficient condition for oscillation of differential equation. We obtain the oscillation criteria of difference equations by studying the distribution of zeros. Most oscillatory results for second-order neutral dynamic equations are sufficient conditions for oscillation; see [12–19].

Results

Conclusion