On Eventually Positive Solutions of Quasilinear Second-Order Neutral Differential Equations

Simona Fišnarová,Robert Mařík

doi:10.1155/2014/818732

Abstract

We study the second-order neutral delay differential equation[r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, whereΦα(t)=|t|α-1t,α≥1andz(t)=x(t)+p(t)x(τ(t)). Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral∫∞r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

Highlights

We give sufficient conditions which exclude the possibility that the equation possesses an eventually positive solution x(t) such that the corresponding function z(t) is eventually increasing
In the paper we study the equation [r (t) Φγ (z󸀠 (t))]󸀠 + q (t) Φβ (x (σ (t))) = 0, (1)z (t) = x (t) + p (t) x (τ (t)), where Φα(t) = |t|α−1t, α ≥ 1, is the power type nonlinearity.The coefficients r and p are subject to the usual conditions r q ∈ isC1([t0, ∞), R+), p positive, q ∈ C([t0,∞∈ )C, 1R([+t)0., ∞), R+0 )and the coefficientWe assume that limt → ∞τ(t) = ∞ = limt → ∞σ(t), σ (τ (t)) = τ (σ (t)), (2)and and there exist τ󸀠(t) ≥ τ0
The function h introduced in the following lemma plays a role in a formulation of oscillation criteria in the case β ≥ 1

Summary

Introduction

We give sufficient conditions which exclude the possibility that the equation possesses an eventually positive solution x(t) such that the corresponding function z(t) is eventually increasing. This and claim (i) prove claims (ii) and (iii) since in each case we have found an eventually positive solution y of the corresponding inequality.

Results

Conclusion