Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments

Tongxing Li,Jozef Džurina,Blanka Baculíková

doi:10.1186/1687-2770-2014-68

Abstract

We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results. MSC:34C10, 34K11.

Highlights

1 Introduction This paper is concerned with oscillation of the second-order nonlinear functional differential equation b r(t) z (t) α– z (t) + q(t, ξ ) x g(t, ξ ) α– x g(t, ξ ) dσ (ξ ) =, ( . )
By a solution of ( . ), we mean a function x ∈ C([tx, ∞), R) for some tx ≥ t, which has the properties that z ∈ C ([tx, ∞), R), r|z |α– z ∈ C ([tx, ∞), R), and satisfies ( . ) on [tx, ∞)
We restrict our attention to those solutions x of ( . ) which exist on [tx, ∞) and satisfy sup{|x(t)| : t ≥ T} > for any T ≥ tx

Summary

Introduction

1 Introduction This paper is concerned with oscillation of the second-order nonlinear functional differential equation b r(t) z (t) α– z (t) + q(t, ξ ) x g(t, ξ ) α– x g(t, ξ ) dσ (ξ ) = , There has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [ – ] and the references cited therein. Baculíková and Džurina [ , ] and Li et al [ ] investigated oscillatory behavior of a second-order neutral differential equation r(t) x(t) + p(t)x τ (t) + q(t)x σ (t) = , where

Results

Conclusion