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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Oscillation results for second-order nonlinear neutral differential equations

Tongxing Li1, Yuriy V Rogovchenko2* and Chenghui Zhang1

Author Affiliations

1 School of Control Science and Engineering, Shandong University, Jinan, Shandong, 250061, P.R. China

2 Department of Mathematical Sciences, University of Agder, Post Box 422, Kristiansand, N-4604, Norway

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Advances in Difference Equations 2013, 2013:336  doi:10.1186/1687-1847-2013-336

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/336


Received:17 February 2013
Accepted:24 October 2013
Published:21 November 2013

© 2013 Li et al.; 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 cited.

Abstract

We obtain several oscillation criteria for a class of second-order nonlinear neutral differential equations. New theorems extend a number of related results reported in the literature and can be used in cases where known theorems fail to apply. Two illustrative examples are provided.

MSC: 34K11.

Keywords:
oscillation; second-order; neutral differential equation; integral averaging

1 Introduction

In this paper, we are concerned with the oscillation of a class of nonlinear second-order neutral differential equations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M2">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4">View MathML</a> is a quotient of two odd positive integers. In what follows, it is always assumed that

(H1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M5">View MathML</a>;

(H2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M6">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M7">View MathML</a> is not identically zero for large t;

(H3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M9">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M10">View MathML</a> and for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M11">View MathML</a>;

(H4) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M12">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M13">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M14">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M15">View MathML</a>.

By a solution of equation (1) we mean a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M16">View MathML</a> defined on an interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M18">View MathML</a> is continuously differentiable and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M16">View MathML</a> satisfies (1) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M20">View MathML</a>. We consider only solutions satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M21">View MathML</a> and tacitly assume that equation (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M17">View MathML</a>; otherwise, it is called nonoscillatory. We say that equation (1) is oscillatory if all its continuable solutions are oscillatory.

During the past decades, a great deal of interest in oscillatory and nonoscillatory behavior of various classes of differential and functional differential equations has been shown. Many papers deal with the oscillation of neutral differential equations which are often encountered in applied problems in science and technology; see, for instance, Hale [1]. It is known that analysis of neutral differential equations is more difficult in comparison with that of ordinary differential equations, although certain similarities in the behavior of solutions of these two classes of equations are observed; see, for instance, the monographs [2-4], the papers [5-22] and the references cited there.

Oscillation results for (1) have been reported in [2,4,6,8,11,14,18-20]. A commonly used assumption is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M23">View MathML</a>

(2)

although several authors were concerned with the oscillation of equation (1) in the case where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M24">View MathML</a>

(3)

In particular, Xu and Meng [[19], Theorem 2.3] established sufficient conditions for the oscillation of (1) assuming that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M25">View MathML</a>

(4)

Further results in this direction were obtained by Ye and Xu [20] under the assumptions that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M26">View MathML</a>

(5)

see also the paper by Han et al.[8] where inaccuracies in [20] were corrected and new oscillation criteria for (1) were obtained [[8], Theorems 2.1 and 2.2]. We conclude this brief review of the literature by mentioning that Li et al.[13] and Sun et al.[18] extended the results obtained in [8] to Emden-Fowler neutral differential equations and neutral differential equations with mixed nonlinearities.

Our principal goal in this paper is to derive new oscillation criteria for equation (1) without requiring restrictive conditions (4) and (5). Developing further ideas from the paper by Hasanbulli and Rogovchenko [9] concerned with a particular case of equation (2) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27">View MathML</a>, we study the oscillation of (1) in the case where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4">View MathML</a>.

2 Oscillation criteria

In what follows, all functional inequalities are tacitly assumed to hold for all t large enough, unless mentioned otherwise. As usual, we use the notation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M29">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M30">View MathML</a>. Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M31">View MathML</a>

We say that a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M32">View MathML</a> belongs to a class <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M33">View MathML</a> if

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M34">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M35">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M36">View MathML</a>;

(ii) H has a nonpositive continuous partial derivative with respect to the second variable satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M37">View MathML</a>

for a locally integrable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M38">View MathML</a>.

In what follows, we assume that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M39">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M40">View MathML</a>

(6)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M41">View MathML</a>

In order to establish our main theorems, we need the following auxiliary result. The first inequality is extracted from the paper by Jiang and Li [[11], Lemma 5], whereas the second one is a variation of the well-known Young inequality [23].

Lemma 1 (i) Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M4">View MathML</a>be a ratio of two odd integers. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M43">View MathML</a>

(7)

for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M44">View MathML</a>.

(ii) For any two numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M45">View MathML</a>and for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M46">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M47">View MathML</a>

the equality holds if and only if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M48">View MathML</a>.

Theorem 2Assume that conditions (H1)-(H4), (3), and (6) are satisfied. Suppose also that there exist two functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M49">View MathML</a>such that, for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50">View MathML</a>and for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M52">View MathML</a>

(8)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M53">View MathML</a>

(9)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M54">View MathML</a>

(10)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M55">View MathML</a>

(11)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M56">View MathML</a>

(12)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M57">View MathML</a>

(13)

Then equation (1) is oscillatory.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58">View MathML</a> be a nonoscillatory solution of (1). Since γ is a quotient of two odd positive integers, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M59">View MathML</a> is also a solution of (1). Hence, without loss of generality, we may assume that there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M60">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M63">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M64">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M65">View MathML</a>, and by virtue of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M66">View MathML</a>

the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M67">View MathML</a> is nonincreasing for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M64">View MathML</a>. Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M69">View MathML</a> does not change sign eventually, that is, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M70">View MathML</a> such that either <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>. We consider each of two cases separately.

Case 1. Assume first that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>. Equation (1) and condition (H2) yield

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M76">View MathML</a>

(14)

In view of (H4), there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M77">View MathML</a> such that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M79">View MathML</a>

(15)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M80">View MathML</a>

(16)

It follows from (14) and (15) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M81">View MathML</a>

(17)

Define a generalized Riccati substitution by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M82">View MathML</a>

(18)

Differentiating (18) and using (16) and (17), one arrives at

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M83">View MathML</a>

(19)

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M84">View MathML</a>

By virtue of Lemma 1, part (i), we have the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M85">View MathML</a>

(20)

It follows now from (19) and (20) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M86">View MathML</a>

(21)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87">View MathML</a> is defined by (10). Replacing in (21) t with s, multiplying both sides by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M88">View MathML</a> and integrating with respect to s from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M89">View MathML</a> to t, we have, for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50">View MathML</a> and for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M92">View MathML</a>

(22)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M93">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M94">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M95">View MathML</a>

Application of Lemma 1, part (ii), yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M96">View MathML</a>

Hence, by the latter inequality and (22), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M97">View MathML</a>

(23)

Using monotonicity of H, we conclude that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M99">View MathML</a>

Thus,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M100">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M101">View MathML</a>

which contradicts (8).

Case 2. Assume now that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>. It follows from the inequality <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M104">View MathML</a> that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M105">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M106">View MathML</a>

Integrating this inequality from t to l, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M107">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M108">View MathML</a>

Passing to the limit as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M109">View MathML</a>, we conclude that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M110">View MathML</a>

which yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M111">View MathML</a>

Hence, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M112">View MathML</a>

It follows from (1) and the latter inequality that there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M113">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M114">View MathML</a>

(24)

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M115">View MathML</a>, define a generalized Riccati substitution by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M116">View MathML</a>

(25)

Differentiating (25), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M117">View MathML</a>

(26)

Letting in Lemma 1, part (i),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M118">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M119">View MathML</a>

It follows from (24) and (26) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M120">View MathML</a>

(27)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121">View MathML</a> is defined by (12). Replacing in (27) t with s, multiplying both sides by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M88">View MathML</a> and integrating with respect to s from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M123">View MathML</a> to t, we conclude that, for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M50">View MathML</a> and for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M126">View MathML</a>

(28)

Letting in Lemma 1, part (ii),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M128">View MathML</a>

we conclude that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M129">View MathML</a>

Using the latter inequality and (28), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M130">View MathML</a>

(29)

Proceeding as in the proof of Case 1, we obtain contradiction with our assumption (9). Therefore, equation (1) is oscillatory. □

Theorem 3Assume that conditions (H1)-(H4), (3), and (6) are satisfied. Suppose also that there exist functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M49">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M133">View MathML</a>such that, for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M134">View MathML</a>and for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M136">View MathML</a>

(30)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M137">View MathML</a>

(31)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M138">View MathML</a>

(32)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M141">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M142">View MathML</a>are as in Theorem 2. If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M143">View MathML</a>

(33)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M144">View MathML</a>

(34)

equation (1) is oscillatory.

Proof Without loss of generality, assume again that (1) possesses a nonoscillatory solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M63">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M149">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M60">View MathML</a>. From the proof of Theorem 2, we know that there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M70">View MathML</a> such that either <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>.

Case 1. Assume first that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M71">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>. Proceeding as in the proof of Theorem 2, we arrive at inequality (23), which yields, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M157">View MathML</a> and for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M159">View MathML</a>

The latter inequality implies that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M157">View MathML</a> and for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M162">View MathML</a>

Consequently,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M163">View MathML</a>

(35)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M164">View MathML</a>

(36)

Assume now that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M165">View MathML</a>

(37)

Condition (30) implies existence of a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M166">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M167">View MathML</a>

(38)

It follows from (37) that, for any positive constant η, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M168">View MathML</a> such that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M169">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M170">View MathML</a>

(39)

Using integration by parts and (39), we have, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M169">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M172">View MathML</a>

By virtue of (38), there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M173">View MathML</a> such that, for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M174">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M175">View MathML</a>

which implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M176">View MathML</a>

Since η is an arbitrary positive constant,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M177">View MathML</a>

but the latter contradicts (36). Consequently,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M178">View MathML</a>

and, by virtue of (35),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M179">View MathML</a>

which contradicts (33).

Case 2. Assume now that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M73">View MathML</a>. It has been established in Theorem 2 that (29) holds. Using (29) and proceeding as in Case 1 above, we arrive at the desired conclusion. □

As an immediate consequence of Theorem 3, we have the following result.

Theorem 4Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M121">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M141">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M142">View MathML</a>be as in Theorem 3, and assume that conditions (H1)-(H4), (3), and (6) are satisfied. Suppose also that there exist functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M51">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M187">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M188">View MathML</a>such that (30), (33), and (34) hold. If, for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M134">View MathML</a>and for some<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M191">View MathML</a>

(40)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M192">View MathML</a>

(41)

equation (1) is oscillatory.

3 Examples

Efficient oscillation tests can be easily derived from Theorems 2-4 with different choices of the functions H, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M193">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M194">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M195">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M196">View MathML</a>. In this section, we illustrate possible applications with two examples.

Example 5 For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M197">View MathML</a>, consider the second-order nonlinear neutral delay differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M198">View MathML</a>

(42)

Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M199">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M200">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M201">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M202">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M203">View MathML</a>, whereas <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M204">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M207">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M208">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M209">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M210">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M211">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M212">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M213">View MathML</a>, and a straightforward computation shows that all assumptions of Theorem 2 are satisfied. Hence, equation (42) is oscillatory.

Example 6 For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M197">View MathML</a>, consider the second-order neutral delay differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M215">View MathML</a>

(43)

Here, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M216">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M217">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M218">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M219">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M220">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M221">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M27">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M207">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M225">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M210">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M227">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M228">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M229">View MathML</a>. It is not difficult to verify that all assumptions of Theorem 2 hold. Hence, equation (43) is oscillatory. In fact, one such solution is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M230">View MathML</a>.

4 Conclusions

Most oscillation results reported in the literature for neutral differential equation (1) and its particular cases have been obtained under the assumption (2) which significantly simplifies the analysis of the behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M231">View MathML</a> for a nonoscillatory solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M232">View MathML</a> of (1). In this paper, using a refinement of the integral averaging technique, we have established new oscillation criteria for second-order neutral delay differential equation (1) assuming that (3) holds.

We stress that the study of oscillatory properties of equation (1) in the case (3) brings additional difficulties. In particular, in order to deal with the case when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M72">View MathML</a> (which is simply eliminated if condition (2) holds), we have to impose an additional assumption <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M234">View MathML</a>. In fact, it is well known (see, e.g., [6,14]) that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M58">View MathML</a> is an eventually positive solution of (1), then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M236">View MathML</a>

(44)

One of the principal difficulties one encounters lies in the fact that (44) does not hold when (3) is satisfied, cf.[8]. Since the sign of the derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M69">View MathML</a> is not known, our criteria for the oscillation of (1) include a pair of assumptions as, for instance, (8) and (9). On the other hand, we point out that, contrary to [8,13,18,19], we do not need in our oscillation theorems quite restrictive conditions (4) and (5), which, in a certain sense, is a significant improvement compared to the results in the cited papers. However, this improvement has been achieved at the cost of imposing condition (6).

Therefore, two interesting problems for future research can be formulated as follows.

(P1) Is it possible to establish oscillation criteria for (1) without requiring conditions (4), (5), and (6)?

(P2) Suggest a different method to investigate (1) in the case where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/336/mathml/M238">View MathML</a> (and thus inequality (7) does not hold).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The research of TL and CZ was supported in part by the National Basic Research Program of PR China (2013CB035604) and the NNSF of PR China (Grants 61034007, 51277116, and 51107069). YR acknowledges research grants from the Faculty of Science and Technology of Umeå University, Sweden and from the Faculty of Engineering and Science of the University of Agder, Norway. TL would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for support and useful advices. Last but not least, the authors are grateful to two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies.

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