On the oscillation of second-order half-linear functional differential equations with mixed neutral term

Ercan Tunç,Orhan Özdemir

doi:10.1080/16583655.2019.1595948

Abstract

In this article, the authors establish new sufficient conditions for the oscillation of solutions to a class of second-order half-linear functional differential equations with mixed neutral term. The results obtained improve and complement some known results in the relevant literature. Examples illustrating the results are included.

Highlights

This paper deals with the oscillatory behaviour of solutions to a class of second order half-linear functional differential equations with mixed neutral term of the form r(t) z (t) α + q(t)xα h(t) = 0, t ≥ t0 > 0, (1)
Oscillation and asymptotic behaviour of solutions to various classes of delay and advanced neutral differential and dynamic equations have been widely discussed in the literature; see, for example, [1–19], and the references contained therein
= lim sup c s2 + s ds = ∞, t→∞ T i.e. condition (36) holds

Summary

Introduction

This paper deals with the oscillatory behaviour of solutions to a class of second order half-linear functional differential equations with mixed neutral term of the form r(t) z (t) α + q(t)xα h(t) = 0, t ≥ t0 > 0, (1). We consider only those solutions x(t) of (1) that satisfy sup{|x(t)| : t ≥ T} > 0 for all T ≥ tx; we tacitly assume that (1) possesses such solutions Such a solution x(t) of (1) is said to be oscillatory if it has arbitrarily large zeros on [tx, ∞); otherwise, it is called nonoscillatory. In view of the observations above, we wish to develop new sufficient conditions which can be applied to the cases where limt→∞ p1(t) = ∞ and /or limt→∞ p2(t) = ∞ In this connection, the results obtained in the present paper are new, improve and complement some existing results in the relevant literature. It is hoped that the present paper will contribute significantly to the study of oscillation of solutions of second-order mixed neutral differential equations

Some preliminary lemmas

Main results