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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Asymptotic behavior of an odd-order delay differential equation

Tongxing Li1 and Yuriy V Rogovchenko2*

Author Affiliations

1 Qingdao Technological University, Feixian, Shandong, 273400, P.R. China

2 Department of Mathematical Sciences, University of Agder, P.O. Box 422, Kristiansand, N-4604, Norway

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Boundary Value Problems 2014, 2014:107  doi:10.1186/1687-2770-2014-107


Dedicated to Professor Ivan Kiguradze

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/107


Received:29 January 2014
Accepted:22 April 2014
Published:9 May 2014

© 2014 Li and Rogovchenko; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

We study asymptotic behavior of solutions to a class of odd-order delay differential equations. Our theorems extend and complement a number of related results reported in the literature. An illustrative example is provided.

MSC: 34K11.

Keywords:
asymptotic behavior; odd-order; delay differential equation; oscillation

1 Introduction

Professor Ivan Kiguradze is widely recognized as one of the leading contemporary experts in the qualitative theory of ordinary differential equations. His research has been partly summarized in the monograph written jointly with Professor Chanturia [1] where many fundamental results on the asymptotic behavior of solutions to important classes of nonlinear differential equations were collected. In particular, the Kiguradze lemma and Kiguradze classes of solutions are well known to researchers working in the area and are extensively used to advance the knowledge further.

In this tribute to Professor Kiguradze, we are concerned with the asymptotic behavior of solutions to an odd-order delay differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M3">View MathML</a> is an odd natural number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M4">View MathML</a> is a ratio of odd natural numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M12">View MathML</a>.

By a solution of (1.1) we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M14">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M16">View MathML</a> satisfies (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M17">View MathML</a>. We consider only those extendable solutions of (1.1) that do not vanish eventually, that is, condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M18">View MathML</a> holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M19">View MathML</a>. We tacitly assume that (1.1) possesses such solutions. As customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros on the ray <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M17">View MathML</a>; otherwise, we call it non-oscillatory.

Analysis of the oscillatory and non-oscillatory behavior of solutions to different classes of differential and functional differential equations has always attracted interest of researchers; see, for instance, [1-19] and the references cited therein. One of the main reasons for this lies in the fact that delay differential equations arise in many applied problems in natural sciences, technology, and automatic control, cf., for instance, Hale [20]. In particular, (1.1) may be viewed as a special case of a more general class of higher-order differential equations with a one-dimensional p-Laplacian, which, as mentioned by Agarwal et al.[4], have applications in continuum mechanics.

Let us briefly comment on a number of closely related results which motivated our study. In [2,5-8,14], the authors investigated asymptotic properties of a third-order delay differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M21">View MathML</a>

Using a Riccati substitution, Liu et al.[11], Zhang et al.[16], and Zhang et al.[18] studied oscillation of (1.1) assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M22">View MathML</a> is even, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M11">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M24">View MathML</a>

(1.2)

In the special case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M25">View MathML</a>, (1.1) reduces to a two-term differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M26">View MathML</a>

(1.3)

which was studied by Zhang et al.[17] who established the following result.

Theorem 1.1 ([[17], Corollary 2.1])

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M27">View MathML</a>

and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M28">View MathML</a>. Suppose also that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M29">View MathML</a>

and, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M31">View MathML</a>

Then every solution of (1.3) is either oscillatory or converges to zero as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M32">View MathML</a>.

To the best of our knowledge, only a few results are known regarding oscillation of (1.1) for n odd. Furthermore, in this case the methods in [11,18] which employ Riccati substitutions cannot be applied to the analysis of (1.1). Therefore, the objective of this paper is to extend the techniques exploited in [17] to the study of (1.1) in the case when the integral in (1.2) is finite, that is, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M34">View MathML</a>

(1.4)

As usual, all functional inequalities considered in this paper are supposed to hold for all t large enough. Without loss of generality, we may deal only with positive solutions of (1.1), because under our assumption that γ is a ratio of odd natural numbers, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> is a solution of (1.1), so is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M36">View MathML</a>.

2 Main results

We need the following auxiliary lemmas.

Lemma 2.1Assume that (1.2) is satisfied and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a>be an eventually positive solution of (1.1). Then there exists a sufficiently large<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M40">View MathML</a>

(2.1)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> be an eventually positive solution of (1.1). Then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M42">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M45">View MathML</a>. By virtue of (1.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M46">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M47">View MathML</a>

(2.2)

which means that the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M48">View MathML</a>

is decreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M45">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M50">View MathML</a> does not change sign eventually, that is, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M51">View MathML</a> such that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M52">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M52">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>. Otherwise, there should exist a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M57">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M58">View MathML</a>

and, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M60">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M61">View MathML</a>

Inequality (2.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M62">View MathML</a>

Integrating this inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M63">View MathML</a> to t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M64">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M65">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M32">View MathML</a> and using (1.2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M67">View MathML</a>

It follows now from the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M69">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M70">View MathML</a>, which contradicts our assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M43">View MathML</a>. Finally, write (2.2) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M72">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M73">View MathML</a>. This completes the proof. □

Lemma 2.2 (Agarwal et al.[3])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M75">View MathML</a>is non-positive for all largetand not identically zero on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M76">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M77">View MathML</a>, then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78">View MathML</a>, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M79">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M80">View MathML</a>

holds on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M81">View MathML</a>.

Lemma 2.3 (Agarwal et al.[4])

The equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M82">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M4">View MathML</a>is a quotient of odd natural numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M84">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M85">View MathML</a>is non-oscillatory if and only if there exist a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M86">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M87">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M89">View MathML</a>

For a compact presentation of our results, we introduce the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M90">View MathML</a>

Theorem 2.4Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M91">View MathML</a>

(2.4)

Then every solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a>of (1.1) is either oscillatory or satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M93">View MathML</a>

(2.5)

provided that either

(i) (1.2) holds or

(ii) (1.4) is satisfied and, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M95">View MathML</a>

(2.6)

Proof Assume that (1.1) has a non-oscillatory solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> which is eventually positive and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M97">View MathML</a>

(2.7)

Case (i) By Lemma 2.1, we conclude that (2.1) holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38">View MathML</a> is sufficiently large. It follows from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M100">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78">View MathML</a> and for all sufficiently large t. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M102">View MathML</a>

By virtue of (1.1), we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M103">View MathML</a> is a positive solution of a differential inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M104">View MathML</a>

However, it follows from the result due to Werbowski [[15], Corollary 1] that the latter inequality does not have positive solutions under the assumption (2.4), which is a contradiction. The proof of part (i) is complete.

Case (ii) Similar analysis to that in Lemma 2.1 leads to the conclusion that a non-oscillatory positive solution with the property (2.7) satisfies, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>, either conditions (2.1) or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M106">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38">View MathML</a> is sufficiently large. Assume first that (2.1) holds. As in the proof of the part (i), one arrives at a contradiction with the condition (2.4). Suppose now that (2.8) holds. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>, define a new function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M109">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M110">View MathML</a>

(2.9)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M111">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M39">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M113">View MathML</a>

we deduce that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M114">View MathML</a> is decreasing. Thus, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M116">View MathML</a>

(2.10)

Dividing both sides of (2.10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M117">View MathML</a> and integrating the resulting inequality from t to T, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M118">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M119">View MathML</a> and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M121">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M122">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M123">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M124">View MathML</a>

Thus, by (2.9), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M125">View MathML</a>

(2.11)

Differentiation of (2.9) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M126">View MathML</a>

It follows now from (1.1) and (2.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M127">View MathML</a>

On the other hand, it follows from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M128">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M78">View MathML</a> and for all sufficiently large t. Therefore, (2.11) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M130">View MathML</a>

(2.12)

Multiplying (2.12) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M131">View MathML</a> and integrating the resulting inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M132">View MathML</a> to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M133">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M135">View MathML</a>. Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M136">View MathML</a> and the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M137">View MathML</a>

(see Zhang and Wang [[19], Lemma 2.3] for details) and the definition of φ, we derive from (2.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M138">View MathML</a>

which contradicts (2.6). This completes the proof for the part (ii). □

Remark 2.5 For a result similar to the one established in part (i) in Theorem 2.4, see also Zhang et al. [[16], Theorem 5.3].

Remark 2.6 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M25">View MathML</a>, Theorem 2.4 includes Theorem 1.1.

In the remainder of this section, we use different approaches to arrive at the conclusion of Theorem 2.4. First, we employ the integral averaging technique to replace assumption (2.6) with a Philos-type condition.

To this end, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M140">View MathML</a>. We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M141">View MathML</a> belongs to the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M142">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M143">View MathML</a>

and H has a non-positive continuous partial derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M144">View MathML</a> with respect to the second variable satisfying the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M145">View MathML</a>

for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M146">View MathML</a>.

Theorem 2.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M147">View MathML</a>be as in Theorem 2.4 and suppose that (1.4) and (2.4) hold. Assume that there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M148">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M149">View MathML</a>

(2.13)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M38">View MathML</a>and for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30">View MathML</a>. Then the conclusion of Theorem 2.4 remains intact.

Proof Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we arrive at the inequality (2.12) which holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M153">View MathML</a>. Multiplying (2.12) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M154">View MathML</a> and integrating the resulting inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M132">View MathML</a> to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M156">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M157">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M158">View MathML</a>

Using the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M159">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M160">View MathML</a>

which contradicts assumption (2.13). This completes the proof. □

Finally, we formulate also a comparison result for (1.1) that leads to the conclusion of Theorem 2.4.

Theorem 2.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M147">View MathML</a>be as above, and assume that (1.4) and (2.4) hold. If a second-order half-linear ordinary differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M162">View MathML</a>

(2.14)

is oscillatory for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30">View MathML</a>, then the conclusion of Theorem 2.4 remains intact.

Proof Assuming again that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we obtain (2.12) which holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M165">View MathML</a>. By virtue of Lemma 2.3, we conclude that (2.14) is non-oscillatory, which is a contradiction. The proof is complete. □

3 Example

The following example illustrates possible applications of theoretical results obtained in the previous section.

Example 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M166">View MathML</a>, consider the third-order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M167">View MathML</a>

(3.1)

It is not difficult to verify that (1.4) holds and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M168">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M169">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M172">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M173">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M30">View MathML</a>. Hence, by Theorem 2.4, every solution of (3.1) is either oscillatory or satisfies (2.5). As a matter of fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M175">View MathML</a> is a solution of this equation satisfying condition (2.5).

Remark 3.2 Note that Theorems 2.4, 2.7, and 2.8 ensure that every solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M35">View MathML</a> of (1.1) is either oscillatory or satisfies (2.5) and, unfortunately, these results cannot distinguish solutions with different behaviors. Since the sign of the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/107/mathml/M177">View MathML</a> is not known, it is difficult to establish sufficient conditions which guarantee that all solutions of (1.1) are just oscillatory and do not satisfy (2.5). Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) satisfy (2.5). Therefore, these two interesting problems remain for future research.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work and are listed in alphabetical order. They both read and approved the final version of the manuscript.

Acknowledgements

The authors express their sincere gratitude to both anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.

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