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

Open Access Research

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

Tongxing Li1*, Blanka Baculíková2 and Jozef Džurina2

Author Affiliations

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

2 Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, Košice, 042 00, Slovakia

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Boundary Value Problems 2014, 2014:68  doi:10.1186/1687-2770-2014-68


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/2014/1/68


Received:16 January 2014
Accepted:6 March 2014
Published:24 March 2014

© 2014 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 credited.

Abstract

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

MSC: 34C10, 34K11.

Keywords:
oscillation; neutral differential equation; second-order equation; distributed deviating argument

1 Introduction

This paper is concerned with oscillation of the second-order nonlinear functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M3">View MathML</a> is a constant, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M4">View MathML</a>. Throughout, we assume that the following hypotheses hold:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M7">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M8">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M10">View MathML</a> is not eventually zero on any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M12">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M14">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M15">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>;

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M19">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M20">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M21">View MathML</a> is nondecreasing and the integral of (1.1) is taken in the sense of Riemann-Stieltijes.

By a solution of (1.1), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M22">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M23">View MathML</a>, which has the properties that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M25">View MathML</a>, and satisfies (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M26">View MathML</a>. We restrict our attention to those solutions x of (1.1) which exist on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M26">View MathML</a> and satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M28">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M29">View MathML</a>. A solution x of (1.1) is termed oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.

As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed networks containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see [1]. Hence, there has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [2-30] and the references cited therein.

In the following, we present some background details that motivate our research. Recently, Baculíková and Lacková [6], Džurina and Hudáková [12], Li et al.[15,18], and Sun et al.[22] established some oscillation criteria for the second-order half-linear neutral differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M32">View MathML</a>

Baculíková and Džurina [4,5] and Li et al.[17] investigated oscillatory behavior of a second-order neutral differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M33">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M34">View MathML</a>

(1.2)

Ye and Xu [26] and Yu and Fu [27] considered oscillation of the second-order differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M35">View MathML</a>

Assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M36">View MathML</a>, Thandapani and Piramanantham [23], Wang [24], Xu and Weng [25], and Zhao and Meng [30] studied oscillation of an equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M37">View MathML</a>

As yet, there are few results regarding the study of oscillatory properties of (1.1) under the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M38">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M39">View MathML</a>. Thereinto, Li and Thandapani [19] obtained several oscillation results for (1.1) in the case where (1.2) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M41">View MathML</a>

(1.3)

In the subsequent sections, we shall utilize the Riccati substitution technique and some inequalities to establish several new oscillation criteria for (1.1) assuming that (1.3) holds or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M42">View MathML</a>

(1.4)

All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.

2 Main results

In what follows, we use the following notation for the convenience of the reader:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M43">View MathML</a>

where h, ρ, and η will be specified later.

Theorem 2.1Assume (H1)-(H5), (1.3), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>. Suppose further that there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M49">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M54">View MathML</a>

(2.1)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M61">View MathML</a>. Applying (1.1), one has, for all sufficiently large t,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M62">View MathML</a>

Using the inequality (see [[5], Lemma 1])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M63">View MathML</a>

the definition of z, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M66">View MathML</a>

(2.2)

By virtue of (1.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M67">View MathML</a>

(2.3)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68">View MathML</a> is nonincreasing. Now we have two possible cases for the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69">View MathML</a>: (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70">View MathML</a> eventually, or (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71">View MathML</a> eventually.

(i) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. Then we have by (2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M74">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M75">View MathML</a>

Then we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M76">View MathML</a> due to (1.3), which is a contradiction.

(ii) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. It follows from (2.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M79">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M80">View MathML</a>

(2.4)

We define a Riccati substitution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M81">View MathML</a>

(2.5)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M82">View MathML</a>. From (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M84">View MathML</a>

(2.6)

Differentiating (2.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M85">View MathML</a>

(2.7)

Therefore, by (2.5), (2.6), and (2.7), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M86">View MathML</a>

(2.8)

Similarly, we introduce another Riccati transformation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M87">View MathML</a>

(2.9)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M88">View MathML</a>. From (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M90">View MathML</a>

(2.10)

Differentiating (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M91">View MathML</a>

(2.11)

Therefore, by (2.9), (2.10), and (2.11), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M92">View MathML</a>

(2.12)

Combining (2.8) and (2.12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M93">View MathML</a>

It follows from (2.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M94">View MathML</a>

Integrating the latter inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95">View MathML</a> to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M96">View MathML</a>

(2.13)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M97">View MathML</a>

Using the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M98">View MathML</a>

(2.14)

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M99">View MathML</a>

On the other hand, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M100">View MathML</a>

Then we have by (2.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M101">View MathML</a>

Thus, from (2.13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M102">View MathML</a>

which contradicts (2.1). This completes the proof. □

Assuming (1.2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M104">View MathML</a> are constants, we obtain the following result.

Theorem 2.2Suppose (H1)-(H5), (1.2), (1.3), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M46">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M47">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M111">View MathML</a>

(2.15)

then (1.1) is oscillatory.

Proof As above, let x be an eventually positive solution of (1.1). Proceeding as in the proof of Theorem 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77">View MathML</a>, (2.3), and (2.4) for all sufficiently large t. Using (1.2), (2.3), and (2.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M113">View MathML</a>

(2.16)

The remainder of the proof is similar to that of Theorem 2.1, and hence it is omitted. □

Theorem 2.3Suppose we have (H1)-(H5), (1.3), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>. Assume also that there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M49">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M50">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M122">View MathML</a>

(2.17)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. As in the proof of Theorem 2.1, we obtain (2.3) and (2.4). In view of (2.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68">View MathML</a> is nonincreasing. Now we have two possible cases for the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69">View MathML</a>: (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70">View MathML</a> eventually, or (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71">View MathML</a> eventually.

(i) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. Then, with a proof similar to the proof of case (i) in Theorem 2.1, we obtain a contradiction.

(ii) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. We define a Riccati substitution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M137">View MathML</a>

(2.18)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M82">View MathML</a>. From (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M140">View MathML</a>

(2.19)

Differentiating (2.18), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M141">View MathML</a>

(2.20)

Therefore, by (2.18), (2.19), and (2.20), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M142">View MathML</a>

(2.21)

Similarly, we introduce another Riccati substitution:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M143">View MathML</a>

(2.22)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M88">View MathML</a>. Differentiating (2.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M145">View MathML</a>

(2.23)

Therefore, by (2.22) and (2.23), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M146">View MathML</a>

(2.24)

Combining (2.21) and (2.24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M147">View MathML</a>

It follows from (2.4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M149">View MathML</a>

Integrating the latter inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95">View MathML</a> to t, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M151">View MathML</a>

(2.25)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M152">View MathML</a>

Using inequality (2.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M153">View MathML</a>

On the other hand, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M154">View MathML</a>

Then, by (2.14), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M155">View MathML</a>

Thus, from (2.25), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M156">View MathML</a>

which contradicts (2.17). This completes the proof. □

Assuming we have (1.2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M104">View MathML</a> are constants, we get the following result.

Theorem 2.4Suppose we have (H1)-(H5), (1.2), (1.3), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M163">View MathML</a>

(2.26)

then (1.1) is oscillatory.

Proof Assume again that x is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77">View MathML</a>, (2.3), and (2.4) for all sufficiently large t. By virtue of (1.2), (2.3), and (2.4), we have (2.16) for all sufficiently large t. The rest of the proof is similar to that of Theorem 2.3, and so it is omitted. □

In the following, we present some oscillation criteria for (1.1) in the case where (1.4) holds.

Theorem 2.5Suppose we have (H1)-(H5), (1.2), (1.4), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M167">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. Assume further that there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that (2.15) is satisfied. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M172">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M175">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M176">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M177">View MathML</a>

(2.27)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M55">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M57">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M58">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M61">View MathML</a>. As in the proof of Theorem 2.1, we get (2.2). By virtue of (1.1), we have (2.3). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M68">View MathML</a> is nonincreasing. Now we have two possible cases for the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M69">View MathML</a>: (i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M70">View MathML</a> eventually, or (ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M71">View MathML</a> eventually.

(i) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. Then, by the proof of Theorem 2.2, we obtain a contradiction to (2.15).

(ii) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M72">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M73">View MathML</a>. It follows from (2.2), (2.3), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M194">View MathML</a>

(2.28)

We define the function u by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M195">View MathML</a>

(2.29)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M196">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M197">View MathML</a> is nondecreasing, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M198">View MathML</a>

Integrating this inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M199">View MathML</a> to l, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M200">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M201">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M202">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M203">View MathML</a>

Thus, we get by (2.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M204">View MathML</a>

(2.30)

Similarly, we define another function v by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M205">View MathML</a>

(2.31)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M206">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M197">View MathML</a> is nondecreasing and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M209">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M210">View MathML</a>. Hence, by (2.30), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M211">View MathML</a>

(2.32)

Differentiating (2.29), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M212">View MathML</a>

By (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M214">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M215">View MathML</a>

(2.33)

Similarly, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M216">View MathML</a>

(2.34)

Combining (2.33) and (2.34), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M217">View MathML</a>

(2.35)

Using (2.28), (2.35), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M218">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M219">View MathML</a>

(2.36)

Multiplying (2.36) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M220">View MathML</a> and integrating the resulting inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M95">View MathML</a> to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M222">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M223">View MathML</a>

Using inequality (2.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M224">View MathML</a>

Similarly, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M225">View MathML</a>

Then we have by (2.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M226">View MathML</a>

Thus, from (2.30) and (2.32), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M227">View MathML</a>

which contradicts (2.27). This completes the proof. □

With a proof similar to the proof of Theorems 2.4 and 2.5, we obtain the following result.

Theorem 2.6Suppose we have (H1)-(H5), (1.2), (1.4), and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M169">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M16">View MathML</a>. Assume also that there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M53">View MathML</a>such that (2.26) is satisfied. If there exists a real-valued function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M172">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M175">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M48">View MathML</a>, and (2.27) holds, then (1.1) is oscillatory.

3 Applications and discussion

In this section, we provide three examples to illustrate the main results.

Example 3.1 Consider the second-order neutral functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M239">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M252">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M253">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M167">View MathML</a>. Moreover, letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M255">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M256">View MathML</a>

Hence, by Theorem 2.2, (3.1) is oscillatory. As a matter of fact, one such solution is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M257">View MathML</a>.

Example 3.2 Consider the second-order neutral functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M258">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M259">View MathML</a> is a constant. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M271">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M273">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M275">View MathML</a>. Further, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M276">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M277">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M278">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M279">View MathML</a>

Hence, (3.2) is oscillatory due to Theorem 2.3.

Example 3.3 Consider the second-order neutral functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M280">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M103">View MathML</a> and β are positive constants. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M286">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M249">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M292">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M271">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M115">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M275">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M300">View MathML</a>. Further,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M301">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M302">View MathML</a>

Hence, by Theorem 2.6, (3.3) is oscillatory when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M303">View MathML</a>.

Remark 3.1 In this paper, we establish some new oscillation theorems for (1.1) in the case where p is finite or infinite on . The criteria obtained extend the results in [22] and improve those reported in [19]. Similar results can be presented under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M305">View MathML</a>. In this case, using [[5], Lemma 2], one has to replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M306">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M307">View MathML</a> and proceed as above. It would be interesting to find another method to investigate (1.1) in the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/68/mathml/M308">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. They all read and approved the final version of the manuscript.

Acknowledgements

The authors express their sincere gratitude to the 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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