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

Open Access Research

Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

Simona Fišnarová and Robert Mařík*

Author Affiliations

Department of Mathematics, Mendel University in Brno, Zemědělská 1, Brno, 613 00, Czech Republic

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Boundary Value Problems 2014, 2014:83  doi:10.1186/1687-2770-2014-83

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/83


Received:10 January 2014
Accepted:1 April 2014
Published:10 April 2014

© 2014 Fi¿narová and Ma¿ík; 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 the second-order neutral delay half-linear differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M4">View MathML</a>. We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M5">View MathML</a>.

MSC: 34K11, 34K40.

Keywords:
half-linear differential equation; oscillation criteria; Riccati technique; delay equation; neutral equation; Euler type equation

1 Introduction

In the paper we study the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M6">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M2">View MathML</a> is the power type nonlinearity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a>, which ensures that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M9">View MathML</a> is a convex function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M10">View MathML</a>. The coefficients r and p are subject of usual conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M12">View MathML</a> and the coefficient q is positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M13">View MathML</a>. Further we suppose that the deviating arguments are unbounded and sufficiently smooth functions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M19">View MathML</a> and the deviating arguments from the differential term and potential satisfy either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M20">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M21">View MathML</a> (in the latter case we use stronger condition on the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22">View MathML</a> and the conclusion is weaker than in the commutative case).

By the solution of (1) we understand any differentiable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23">View MathML</a> which does not identically equal zero eventually, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> is differentiable and (1) holds for large t.

The solution of (1) is said to be oscillatory if it has infinitely many zeros tending to infinity. Equation (1) is said to be oscillatory if all its solutions are oscillatory. In the opposite case, i.e., if there exists an eventually positive solution of (1), (1) is said to be nonoscillatory.

The neutral equations naturally arise in the mathematical models where the rate of the growth depends not only on the current state and the state in the past, but also on the rate of change in the past. The paper [1] suggests to imagine a child, which begins to grow more rapidly at the age of about 12 years, growing more and more rapidly until a certain height is approached, at which time there is a rapid slowing of the growth, stopping at the adult height dictated by genes. This process can easily be modeled by neutral equation. Similarly, the paper [2] suggests one to use a logistic neutral differential equation to model a population of Daphnia magna. If in the system modeled by first-order neutral differential equation the mature individuals produce some toxin which inhibits the rate the growth and if the production of this toxin is constant per capita and unit time, then the rate of the growth is inhibited by the term like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M25">View MathML</a> and we naturally obtain the second-order neutral differential equation.

This research is motivated by the paper [3], where the main results of [3] are illustrated by an example of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M26">View MathML</a>

(2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M30">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M31">View MathML</a>, then (2) is proved to be oscillatory if (see [[3], Example 5.2])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M32">View MathML</a>

(3)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M33">View MathML</a>, then (2) is proved to be oscillatory if (see [[3], Example 5.4])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M34">View MathML</a>

(4)

Note that if we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M36">View MathML</a>, both oscillation constants (3) and (4) are worse than the constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M37">View MathML</a>

which is well known to be an optimal oscillation constant for the Euler type equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M38">View MathML</a>

(5)

see [[4], Chapter 1.4.2]. The aim of this paper is to develop sharper results than those of [3]. The main idea is to use the classical approach based on Riccati type inequality. Due to the neutral nature of (1) we have to work with (1) and with the same equation shifted from t to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39">View MathML</a>. In contrast to the results of some other authors [3,5,6], we do not simply sum up the arising Riccati equations, but we develop an advanced technique based on suitable linear combination and careful comparison of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41">View MathML</a>. We show that an application of this technique allows to remove the above mentioned disadvantages of the paper [3] and allows to derive sharper results comparing the results published in the literature. Finally, we also discuss the case when the usual assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M42">View MathML</a> is broken. This step opens applications to the neutral delay differential equations with proportional delay σ and constant delay τ.

The paper is organized as follows. In the following section we formulate inequalities which are used to prove the main results. Section 3 contains main results of the paper and examples which prove that we provide sharp oscillation constant for Euler type differential equation. These criteria are expressed in terms of positive mutually conjugate numbers l and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, the multiplicative factor φ, and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M44">View MathML</a>.

2 Preliminary results

First we derive some technical lemmas - inequalities which are necessary to reduce second-order differential equation into a combination of two first-order Riccati type equations. Note the changes against [3] and other related papers.

(i) We use convex linear combination (6) instead of arithmetic mean in Lemma 2. To achieve this, we consider two positive mutually conjugate numbers l and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47">View MathML</a>.

(ii) We relax the condition on the commutativity of the composition of σ and τ if x is an increasing function in Lemma 3.

(iii) We use new multiplicative factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a> in the definition of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M44">View MathML</a> in Lemma 4. The factor φ allows us to make terms involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41">View MathML</a> closer (or even equal, as in Examples 1 and 4 below) when looking for a smaller one.

As far as we know, these ideas have never been used in the context of Riccati technique even in the linear case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M52">View MathML</a> and in the section with main results we show that these points are crucial points of the paper which allow to derive sharper results than the results published in the literature.

Throughout the paper <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M53">View MathML</a> denotes the positive part of A, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M54">View MathML</a>.

Lemma 1The following inequality holds for everyAand every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M56">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M57">View MathML</a>

Proof The inequality is trivial if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M58">View MathML</a> and a special case of the Young inequality if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M59">View MathML</a>. □

Lemma 2The following inequality holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a>, positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, and nonnegativeaandb:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M62">View MathML</a>

(6)

Proof The proof follows immediately from the convexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M63">View MathML</a>. □

Lemma 3Suppose that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M64">View MathML</a>

(7)

or suppose thatxis an increasing function and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M65">View MathML</a>

(8)

The inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M66">View MathML</a>

(9)

holds for positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, and everytwhich satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M69">View MathML</a>.

Proof From the previous lemma using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M71">View MathML</a> and also from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M72">View MathML</a> and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M73">View MathML</a>. □

Lemma 4Letxbe solution of (1). Suppose that either (7) holds or suppose thatxis an increasing function and (8) holds. The inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M74">View MathML</a>

(10)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M75">View MathML</a>

(11)

is valid for positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, and a positive function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M79">View MathML</a>are nonnegative. Moreover, if there exist numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M81">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M83">View MathML</a>, we have also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M84">View MathML</a>

(12)

whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M79">View MathML</a>are nonnegative and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M87">View MathML</a>is negative.

Proof To obtain the second term from the definition of z we shift (1) from t to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39">View MathML</a> and multiply by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M89">View MathML</a>. We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M90">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M91">View MathML</a>.

Now we multiply (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M92">View MathML</a>, (13) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M93">View MathML</a>, and we add. We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M94">View MathML</a>

Now (10) follows from the definition of Q and from (9). Inequality (12) follows from (10) and from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M87">View MathML</a> is negative. □

3 Main results with applications to Euler type equation

Now we are ready to prove the main results of the paper. The function Q which appears in these criteria is a function defined by (11).

We will distinguish two cases: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M97">View MathML</a>. Let us start with the first case.

Theorem 1Suppose that (7) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M98">View MathML</a>

(14)

hold. Further suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M100">View MathML</a>and there exist positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M104">View MathML</a>

(15)

Then (1) is oscillatory.

Proof Suppose, by contradiction, that all of the assumptions of the theorem hold and there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23">View MathML</a> of (1) and a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M106">View MathML</a> which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M107">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M108">View MathML</a>.

Condition (14) ensures that the corresponding function z is eventually increasing. In fact, from (1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M109">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M110">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> is decreasing and either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M112">View MathML</a>

for large t.

Suppose that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M113">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M114">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M115">View MathML</a>. There exists a positive constant M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M116">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M117">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M115">View MathML</a>. Integrating this inequality over the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M119">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M120">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M121">View MathML</a> we have a negative upper bound for the function z and large t. However, the positivity of both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M123">View MathML</a> implies positivity of z. This contradiction proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M125">View MathML</a> eventually.

Consequently, we will work on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M126">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M128">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M129">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M130">View MathML</a>

(16)

Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M131">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M132">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99">View MathML</a> and from the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M135">View MathML</a>

and combining these computations we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M136">View MathML</a>

(17)

Further we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M137">View MathML</a>

(18)

we use the obvious fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M138">View MathML</a> and differentiate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M139">View MathML</a>

Using the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M142">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M143">View MathML</a>

(19)

Multiplying (17) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M92">View MathML</a>, (19) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M145">View MathML</a>, adding the resulting inequalities and using (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M146">View MathML</a>

Using the product rule for derivatives we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M147">View MathML</a>

and Lemma 1 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M148">View MathML</a>

Integrating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127">View MathML</a> to t

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M150">View MathML</a>

Multiplying by −1 and taking into account the fact that both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M152">View MathML</a> are nonnegative we get a finite upper bound for the integral from (15), which contradicts (15). □

Remark 1 Under the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M154">View MathML</a> we can obtain [[3], Theorem 3.1] as a corollary of Theorem 1, since the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M155">View MathML</a>

holds.

The following corollary is in fact a variant of Theorem 1 if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22">View MathML</a> is bounded above by a nonnegative number and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M157">View MathML</a> is bounded below by a positive number. Since (15) is not simply monotone with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39">View MathML</a>, we have to include the corresponding estimates in the opening part of the proof.

Corollary 1Suppose that (7), (14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a>are satisfied and there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M163">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M165">View MathML</a>. If there exist positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M169">View MathML</a>

(20)

then (1) is oscillatory.

Proof The proof is the same as the proof of Theorem 1, we just use (12) instead of (10) and in the remaining part of the proof we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M157">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M81">View MathML</a>. □

Example 1 For the Euler type equation (2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M174">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M181">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M183">View MathML</a>. With this setting we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M184">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M185">View MathML</a>. Further <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M188">View MathML</a> and (20) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M189">View MathML</a>

and (2) is oscillatory if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M190">View MathML</a>

(21)

Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M153">View MathML</a>, then this condition becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M192">View MathML</a>

and since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M193">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M194">View MathML</a>, this oscillation constant is smaller than the oscillation constant from (3).

Further, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M196">View MathML</a>, then (1) becomes (5). Condition (21) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M197">View MathML</a>

and, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M46">View MathML</a> is arbitrary, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M199">View MathML</a>

which is well known to be an optimal and non-improvable oscillation constant for (5). In this sense we consider our result as reasonably sharp.

Finally, taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47">View MathML</a>, condition (21) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M201">View MathML</a>

A simple computation shows that the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M202">View MathML</a>

(22)

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M203">View MathML</a>

and has a global minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M204">View MathML</a>. Thus the choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M204">View MathML</a> in (21) produces the smallest oscillation constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M206">View MathML</a>

Example 2 Baculíková et al. [[5], Example 2.1] considered the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M207">View MathML</a>

(23)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M211">View MathML</a>. They proved that under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M212">View MathML</a> (23) is oscillatory if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M213">View MathML</a>

(24)

(note that this condition is misprinted in [5]). This condition naturally produces poor oscillation constant if β is close to ω. In our notation we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M220">View MathML</a>. We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M222">View MathML</a>. Thus (20) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M223">View MathML</a>

Taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M47">View MathML</a> and that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M225">View MathML</a> has a local minimum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M226">View MathML</a> at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M227">View MathML</a>, we find that (23) is oscillatory if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M228">View MathML</a>

(25)

This condition completes condition (24). It is possible to find constants ω and β for which (25) is better than (24), as well as constants where the opposite is true. The fact that both estimates depend heavily on the parameters is illustrated by Figure 1.

thumbnailFigure 1. Bounds for oscillation constantbfrom (23) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M229">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M230">View MathML</a>.

The following corollary suggests another modification of the proof of Theorem 1: we replace condition (7) by weaker condition (8) and add conditions which ensure that x possesses the same type of monotonicity as z.

Corollary 2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M231">View MathML</a>, (8), (14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a>hold. If (20) holds for some mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>, then every solution of (1) is either oscillatory, or the first derivative of this solution is oscillatory.

Proof Suppose, by contradiction, that the assumptions are satisfied and x is an eventually positive solution of (1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M237">View MathML</a> is not oscillatory.

We proceed as in Theorem 1 with modifications mentioned in the proof of Corollary 1. To ensure that Lemma 3 can be applied even though (7) need not to hold note that from the fact that z is eventually increasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M22">View MathML</a> constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23">View MathML</a> not oscillatory we conclude easily that x is also eventually increasing. □

In the following example we show an application of Corollary 2 to the equation where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M240">View MathML</a>.

Example 3 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M241">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M246">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a> for large t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M254">View MathML</a>. We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M183">View MathML</a>. With this setting the condition (20) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M257">View MathML</a>

Using this computation and using the fact that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M258">View MathML</a> takes global minimum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M259">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M261">View MathML</a> we see that the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M262">View MathML</a>

guarantees that either every solution or derivative of every solution of the equation is oscillatory.

In the following theorem we drop the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M96">View MathML</a> and use the opposite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a>. In this case we modify the denominator in the Riccati type substitutions (16) and (18).

Theorem 2Suppose that (7), (14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a>hold. Further suppose that there exist positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M270">View MathML</a>

Then (1) is oscillatory.

Proof Suppose, by contradiction, that all the conditions are satisfied and an eventually positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M23">View MathML</a> of (1) exists. As in the proof of Theorem 1, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> is decreasing eventually and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M273">View MathML</a> increasing eventually. Let us work on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M126">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M127">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M276">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M129">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M278">View MathML</a>

As in the proof of Theorem 1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M131">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M280">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265">View MathML</a> and from the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M24">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M283">View MathML</a>

and combining these computations we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M284">View MathML</a>

Further we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M285">View MathML</a>

differentiate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M286">View MathML</a>

and conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M287">View MathML</a>

Similarly as in the proof of Theorem 1 and using the fact that monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M72">View MathML</a> and inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M290">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M291">View MathML</a>

The remaining part of the proof is the same as in Theorem 1. □

Remark 2 Similarly as in Remark 1, [[3], Theorem 3.3] is a corollary of Theorem 2.

Corollary 3Suppose that (7), (14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M292">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a>hold. Furthermore, suppose that there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M163">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M165">View MathML</a>. If there exist positive mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>, and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M301">View MathML</a>

(26)

then (1) is oscillatory.

Proof The proof is he same as the proof of Corollary 1. We just use Theorem 2 instead of Theorem 1. □

Example 4 Consider (2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M302">View MathML</a>. We choose the functions ρ and φ as in Example 1 and find that (1) is oscillatory if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M303">View MathML</a>

(27)

Let us compare this result with (4). The inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M162">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M306">View MathML</a> imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M307">View MathML</a>

where f is defined by (22) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M308">View MathML</a> is a global minimum of f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M309">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M310">View MathML</a>

and (27) is sharper than (4).

The following corollary is a variant of Corollary 2 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a>.

Corollary 4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M231">View MathML</a>, (8), (14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M265">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M264">View MathML</a>hold. If (26) holds for some mutually conjugate numbersl, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a>and positive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a>, then every solution of (1) is either oscillatory, or the first derivative of this solution is oscillatory.

Proof The proof is the same as the proof of Corollary 2; we only replace Theorem 1 by Theorem 2 and Corollary 1 by Corollary 3. □

Remark 3 There are two main approaches how to handle Riccati type transformation in the oscillation theory of neutral differential equations. The first applies if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M318">View MathML</a> and the shift in the differential term is handled by utilizing the estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M319">View MathML</a>; see e.g.[7-9]. Thus the results of this type depend on term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M320">View MathML</a>. Another frequent approach which has been used in [3,10] and also in this paper is summing up the equation at t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M39">View MathML</a> and working with the resulting sum. Since it is necessary to take out common factor, the oscillation criteria usually contain term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M322">View MathML</a>. Since both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41">View MathML</a> may differ significantly, we developed in this paper a method which replaces this term with the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M325">View MathML</a>, where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a> is in some sense arbitrary and may have influence on the final oscillation criterion. We also showed on examples in previous section that this idea produces nonempty extension of known results. We conjecture that a similar idea can be used to obtain new results also in the case of a series of papers by Baculíková and Džurina [5,11,12], where a sum of two equations (in the original variable and in the shifted variable) is used to derive a certain first-order delay differential equation and the oscillation criteria are formulated in terms of this first-order equation. However, this idea exceeds the scope of this paper and will be examined in other research.

4 Conclusion

New oscillation theorems for second-order half-linear differential equations have been obtained. The novelty is in the point that we employed general linear combination based on conjugate numbers l and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M43">View MathML</a> rather than its special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M328">View MathML</a> considered in the other papers devoted to this problem and also included a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M48">View MathML</a> which plays a role when taking minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/83/mathml/M41">View MathML</a>. These extensions are capable to produce sharper results than the results published in the literature as has been shown on examples. As a byproduct we also relaxed in Corollaries 2 and 4 the usual requirement that the composition of delays is commutative. This makes our results applicable to equations with combined constant and proportional delays.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper has been prepared by both authors. Both authors have made the same contribution and approved the final manuscript.

Acknowledgements

This research was supported by the Grant P201/10/1032 of the Czech Science Foundation.

References

  1. Burton, TA, Purnaras, IK: A unification theory of Krasnoselskii for differential equations. Nonlinear Anal.. 89, 121–133 (2013)

  2. Smith, FE: Population dynamics in Daphnia magna and a new model for population growth. Ecology. 44(4), 651–663 (1963). Publisher Full Text OpenURL

  3. Sun, S, Li, T, Han, Z, Li, H: Oscillation theorems for second order quasilinear neutral functional differential equations. Abstr. Appl. Anal.. 2012, (2012) Article ID 819342

  4. Došlý, O, Řehák, P: Half-Linear Differential Equations, Elsevier, Amsterdam (2005)

  5. Baculíková, B, Li, T, Džurina, J: Oscillation theorems for second-order superlinear neutral differential equations. Math. Slovaca. 63, 123–134 (2013). Publisher Full Text OpenURL

  6. Li, T, Han, Z, Zhang, C, Li, H: Oscillation criteria for second-order superlinear neutral differential equations. Abstr. Appl. Anal.. 2011, (2011) Article ID 367541

  7. Dong, JG: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments. Comput. Math. Appl.. 59, 3710–3717 (2010). Publisher Full Text OpenURL

  8. Fišnarová, S, Mařík, R: Oscillation of half-linear differential equations with delay. Abstr. Appl. Anal.. 2013, (2013) Article ID 583147

  9. Hasanbulli, M, Rogovchenko, Y: Oscillation criteria for second order nonlinear neutral differential equations. Appl. Math. Comput.. 215, 4392–4399 (2010). Publisher Full Text OpenURL

  10. Han, Z, Li, T, Sun, S, Chen, W: Oscillation criteria for second-order nonlinear neutral delay differential equations. Adv. Differ. Equ.. 2010, (2010) Article ID 763278

  11. Baculíková, B, Džurina, J: Oscillation theorems for second-order neutral differential equations. Comput. Math. Appl.. 61, 94–99 (2011). Publisher Full Text OpenURL

  12. Baculíková, B, Džurina, J: Oscillation theorems for second-order nonlinear neutral differential equations. Comput. Math. Appl.. 62, 4472–4478 (2011). Publisher Full Text OpenURL