SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

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

Open Access Research Article

Properties of solutions of fourth-order differential equations with boundary conditions

Samir H Saker1, Ravi P Agarwal23* and Donal O’Regan4

Author Affiliations

1 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, Texas, 78363, USA

3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:278  doi:10.1186/1029-242X-2013-278

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/278


Received:14 March 2013
Accepted:11 May 2013
Published:3 June 2013

© 2013 Saker et al.; licensee Springer

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

Abstract

In this paper, we establish some sufficient conditions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1">View MathML</a>-disconjugacy and study the distribution of zeros of nontrivial solutions of fourth-order differential equations. The results are extended to cover some boundary value problems in bending of beams. The main results are proved by making use of a generalization of Hardy’s inequality and some Opial-type inequalities. Some examples are considered to illustrate the main results.

MSC: 34K11, 34C10.

Keywords:
fourth-order differential equations; bending of beams; Opial and Wirtinger inequalities

1 Introduction

Linear differential equations subject to some boundary conditions arise in the mathematical description of some physical systems. For example, mathematical models of deflection of beams. These beams, which appear in many structures, deflect under their own weight or under the influence of some external forces. For example, if a load is applied to the beam in a vertical plane containing the axis of symmetry, the beam undergoes a distortion, and the curve connecting the centroids of all cross sections is called the deflection curve or elastic curve. In elasticity it is shown that the deflection of the curve, say <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M2">View MathML</a> measured from the x-axis, approximates the shape of the beam and satisfies the linear fourth-order differential equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M3">View MathML</a>

(1.1)

on an interval, say <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4">View MathML</a>, with some boundary conditions. Boundary conditions associated with these types of differential equations depend on how the ends of the beams are supported (see [1,2]). For example, the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M5">View MathML</a> correspond to a beam clamped at each end, the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M6">View MathML</a> correspond to a beam clamped at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7">View MathML</a> and free at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8">View MathML</a>, the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M9">View MathML</a> correspond to a beam clamped at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8">View MathML</a> and free at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7">View MathML</a>, and the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M12">View MathML</a> correspond to a beam hinged or supported at both ends. In this paper, we consider the fourth-order differential equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M13">View MathML</a>

(1.2)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M14">View MathML</a> are continuous measurable functions such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M15">View MathML</a>, and I is a nontrivial interval of reals. By a solution of (1.2) on the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M16">View MathML</a>, we mean a nontrivial real-valued function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M17">View MathML</a>, which has the property that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M18">View MathML</a> and satisfies equation (1.2) on J. We assume that (1.2) possesses such a nontrivial solution on I. Equation (1.2) is said to be <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M19">View MathML</a>-disconjugate if i and j are positive integers such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M20">View MathML</a>, and no solution of (1.2) has an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M19">View MathML</a>-distribution of zeros, i.e., no nontrivial solution has a pair of zeros of multiplicities i and j respectively. If no nontrivial solution of (1.2) has more than three zeros, the equation is termed disconjugate. Equation (1.2) and its generalization were studied in [3-17]. Most of the results in these papers dealt with obtaining sufficient conditions for oscillation, nonoscillation and the types of zeros of solutions, i.e., the solutions that have at least four zeros, two or more of which are distinct zeros and one can determine the disconjugacy type of the equations. To the best of the authors knowledge, there are a few papers [18-20] which studied the distributions (gaps between zeros) of zeros of a special case of (1.2), when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M22">View MathML</a>. Our paper is a continuation of these papers. In particular, we establish some sufficient conditions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1">View MathML</a>-disconjugacy and establish lower bounds on the distance between zeros of nontrivial solutions and also lower bounds on the distance between zeros of a solution and/or its derivatives. The results in [18,19] extended a result in the literature for disconjugacy of differential equations due to de la Vallée Poussin [21]. This result was established for the linear nth-order differential equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M24">View MathML</a>

(1.3)

with real continuous coefficients <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M25">View MathML</a>. In [21] de la Vallée Poussin asserted that equation (1.3) is disconjugate on any interval sufficiently short with respect to the magnitude of the coefficients of the equation. More precisely, he proved that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M26">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4">View MathML</a> and the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M28">View MathML</a>

(1.4)

holds, then (1.3) is disconjugate.

In [18] the authors studied a special case of (1.2) and considered the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M29">View MathML</a>

(1.5)

and proved that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30">View MathML</a> is a nontrivial solution of (1.5) which satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M31">View MathML</a>

(1.6)

then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M32">View MathML</a>

(1.7)

In [19] the authors established some sufficient conditions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M33">View MathML</a>-disconjugacy of (1.5) and also proved some results on bending of beams. In particular in [19] the authors established: if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30">View MathML</a> is a nontrivial solution of (1.5) which satisfies (1.6), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M35">View MathML</a>

(1.8)

Our aim in this paper is to employ some new inequalities of Hardy’s type and some Opial-type inequalities with weighted functions to establish some new results for the general equation (1.2).

In particular, in this paper, we are concerned with the following problems for the general equation (1.2):

(i) obtain lower bounds for the spacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36">View MathML</a>, where y is a solution of (1.2) satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M39">View MathML</a>,

(ii) obtain lower bounds for the spacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36">View MathML</a>, where y is a solution of (1.2) satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M42">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M43">View MathML</a>,

(iii) obtain lower bounds for the spacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M36">View MathML</a>, where y is a solution of (1.2) satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M45">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M46">View MathML</a>.

We also establish some new results related to some boundary value problems in bending of beams. In particular, we consider the boundary conditions (1.6) which correspond to a beam clamped at each end. The case of the boundary conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M47">View MathML</a>

(1.9)

which correspond to a beam clamped at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7">View MathML</a> and free at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M49">View MathML</a> and the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M50">View MathML</a> which correspond to a beam clamped at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M8">View MathML</a> and free at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M7">View MathML</a>, and the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M12">View MathML</a> which correspond to a beam hinged or supported at both ends will be left to the interested reader.

The rest of the paper is divided into two sections. In Section 2, we prove several results related to the problems (i)-(iii) and also prove some results related to the boundary value problems of the bending of beams with the boundary conditions (1.6) and (1.9). In Section 3, we give some illustrative examples. The results in this paper yield sufficient conditions for disconjugacy.

2 Main results

In this section, we prove the main results by using some generalizations of Hardy’s inequality and Opial’s inequality.

Theorem 2.1 [[22], Theorem 3.9.1]

Assume that the functionsϑandϕare non-negative and measurable on the interval<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54">View MathML</a>, m, nare real numbers such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M55">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M56">View MathML</a>but fixed. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M57">View MathML</a>be such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M58">View MathML</a>absolutely continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M62">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M63">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M64">View MathML</a>

(2.2)

If we replace<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>, then (2.1) holds where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M67">View MathML</a>is replaced by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M68">View MathML</a>which is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M69">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M70">View MathML</a>

From Theorem 2.1, we note that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M71">View MathML</a>, then we have an inequality of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M72">View MathML</a>

(2.4)

where in this case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M73">View MathML</a> is determined from the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M74">View MathML</a>

(2.5)

In the following, we present a Hardy-type inequality [23,24] that will be used to prove some new results related to the boundary value problems of the bending of beams. If y is absolutely continuous on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M75">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77">View MathML</a>, then the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M78">View MathML</a>

(2.6)

where q, r the weighted functions are measurable positive functions in the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M79">View MathML</a> and m, n are real parameters satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M80">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M81">View MathML</a>. The constant C satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M82">View MathML</a>

(2.7)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M83">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M84">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M85">View MathML</a>

An inequality of type (2.6) also holds when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M86">View MathML</a>. In this case, we see that (2.6) is satisfied if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M88">View MathML</a>

(2.8)

For simplicity, we let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M89">View MathML</a>

(2.9)

Now, we are ready to state and prove the main results. For simplicity, we introduce the following notations:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M90">View MathML</a>

(2.10)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M91">View MathML</a>

(2.11)

Theorem 2.2Suppose thatyis a nontrivial solution of (1.2). If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M95">View MathML</a>

(2.12)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M97">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M101">View MathML</a>

(2.13)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M103">View MathML</a>.

Proof We prove (2.12). Multiplying (1.2) by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M104">View MathML</a> and integrating by parts yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M105">View MathML</a>

(2.14)

This with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M106">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M108">View MathML</a>

(2.15)

Integrating by parts the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M109">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M110">View MathML</a>

Using the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M111">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M112">View MathML</a>

(2.16)

Integrating by parts the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M113">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M114">View MathML</a>

Using the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M115">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M116">View MathML</a>

(2.17)

Substituting (2.17) and (2.16) into (2.15) yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M117">View MathML</a>

(2.18)

Applying the inequality (2.1) on the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M118">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M119">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M121">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M122">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M123">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M125">View MathML</a>, we get (note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38">View MathML</a>) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M128">View MathML</a>

(2.19)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M129">View MathML</a> is defined as in (2.10). Applying the inequality (2.1) again on the integral

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M130">View MathML</a>

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M131">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M134">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M135">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M125">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M137">View MathML</a>

(2.20)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M138">View MathML</a> is defined as in (2.10). Applying the inequality (2.6) on the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M139">View MathML</a> (note <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M140">View MathML</a>), we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M141">View MathML</a>

(2.21)

Substituting (2.21) into (2.20), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M142">View MathML</a>

(2.22)

Applying the inequality (2.1) on the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M143">View MathML</a>, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M144">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M121">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M123">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M150">View MathML</a>, we get (note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M152">View MathML</a>) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M153">View MathML</a>

(2.23)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M154">View MathML</a> is defined as in (2.10). Applying the inequality (2.1) again on the integral

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M155">View MathML</a>

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M156">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M120">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M158">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M160">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M150">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M162">View MathML</a>

(2.24)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M163">View MathML</a> is defined as in (2.10). Substituting (2.19), (2.22), (2.23) and (2.24) into (2.18) and canceling the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M164">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M165">View MathML</a>

which is the desired inequality (2.12). The proof of (2.13) is similar to the proof of (2.12) by using integration by parts, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M166">View MathML</a> is replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M167">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M168">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M169">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M170">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M171">View MathML</a> which are defined in (2.11) instead of the constants <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M173">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M154">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M163">View MathML</a>. The proof is complete. □

In the following, we apply a special case of an inequality with two functions proved by Agarwal and Pang [25]. In this case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M176">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M169">View MathML</a> will be replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M179">View MathML</a> defined below. This inequality is given in the following theorem.

Theorem 2.3[25]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M180">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181">View MathML</a>be non-negative measurable functions on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M182">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M56">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M62">View MathML</a>) but fixed. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M185">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M186">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M188">View MathML</a>is absolutely continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M190">View MathML</a>

(2.25)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M191">View MathML</a>

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M185">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M61">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M188">View MathML</a>is absolutely continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M54">View MathML</a>, then (2.25) holds with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M197">View MathML</a>is replaced by<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M198">View MathML</a>where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M199">View MathML</a>

Now we apply the inequality (2.25). Suppose that the solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30">View MathML</a> of (1.2) satisfies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M201">View MathML</a>. Applying the inequality (2.25) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M202">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M133">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M124">View MathML</a> on the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M205">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M206">View MathML</a>

(2.26)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M207">View MathML</a>

If instead <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M208">View MathML</a>, then (2.26) holds where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178">View MathML</a> is replaced by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M210">View MathML</a>

Using <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M178">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M179">View MathML</a> instead of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M138">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M214">View MathML</a> in the proof of Theorem 2.2, we obtain the following result.

Theorem 2.4Suppose thatyis a nontrivial solution of (1.2). If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M218">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M220">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M38">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M223">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102">View MathML</a>.

If the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181">View MathML</a> is non-increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M4">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M227">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M228">View MathML</a>

Substituting these last two inequalities into Theorem 2.4, we have the following result.

Theorem 2.5Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181">View MathML</a>is a non-increasing function. Ifyis a nontrivial solution of (1.2) which satisfies<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M233">View MathML</a>

If instead<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M237">View MathML</a>

In the following, we apply an inequality due to Boyd [26] to obtain new results. The Boyd inequality states that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M238">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76">View MathML</a> (or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M241">View MathML</a>

(2.27)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M242">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M243">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M244">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M245">View MathML</a>

(2.28)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M246">View MathML</a>

Note that an inequality of type (2.27) also holds when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M86">View MathML</a>. Choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M248">View MathML</a> and apply (2.27) to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M249">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M250">View MathML</a> and then add to obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M251">View MathML</a>

(2.29)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M252">View MathML</a> is defined as in (2.28). An inequality of type (2.27) holds when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M253">View MathML</a> when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M76">View MathML</a> (or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M77">View MathML</a>). In this case, equation (2.27) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M256">View MathML</a>

(2.30)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M257">View MathML</a>

(2.31)

and Γ is the gamma function. To apply (2.30) on the term

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M258">View MathML</a>

we first apply the Schwarz inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M259">View MathML</a>

(2.32)

to get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M260">View MathML</a>

(2.33)

Now, we apply the inequality (2.30) on the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M261">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M262">View MathML</a> (note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M263">View MathML</a>). This gives us that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M264">View MathML</a>

(2.34)

where we assumed that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181">View MathML</a> is a non-increasing function. Substituting (2.34) into (2.33), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M266">View MathML</a>

Applying the inequality (2.6) on the integral <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M267">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M268">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M269">View MathML</a>

This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M270">View MathML</a>

(2.35)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96">View MathML</a>. If we replace <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M272">View MathML</a> by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M273">View MathML</a>, then (2.35) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M274">View MathML</a>

(2.36)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102">View MathML</a>.

Using the inequalities (2.34) and (2.36) and proceeding as in the proof of Theorem 2.4, we obtain the following result.

Theorem 2.6Assume that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M181">View MathML</a>is a non-increasing function. Ifyis a nontrivial solution of (1.2) such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M278">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M280">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M96">View MathML</a>. If instead<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M283">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M285">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M102">View MathML</a>.

Remark 1 In Theorem 2.6, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M287">View MathML</a>, then the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M288">View MathML</a> changes to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M289">View MathML</a>. We note also that the inequalities in Theorem 2.6 hold for any antiderivative Q. Note

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M290">View MathML</a>

the minimum being attained at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M291">View MathML</a>. Now, one can use this in Theorem 2.6 to obtain the following.

Corollary 2.1Assume that the hypotheses of Theorem 2.6 hold. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M94">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M295">View MathML</a>

If instead<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M41">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M152">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M100">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M299">View MathML</a>

Theorem 2.7Suppose thatyis a nontrivial solution of (1.2) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M300">View MathML</a>is the antiderivative ofq. If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M301">View MathML</a>

(2.37)

then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M302">View MathML</a>

(2.38)

whereAis defined as in (2.9).

Proof Multiplying (1.2) by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30">View MathML</a> and integrating by parts the left-hand side twice and using (2.37), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M304">View MathML</a>

(2.39)

Using the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M305">View MathML</a>, we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M306">View MathML</a>

(2.40)

Integrating by parts the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M307">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M308">View MathML</a>

Using the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M309">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M310">View MathML</a>

(2.41)

Integrating by parts the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M311">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M312">View MathML</a>

Using the assumption <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M309">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M314">View MathML</a>

(2.42)

Substituting (2.41) and (2.42) into (2.40), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M315">View MathML</a>

(2.43)

Applying the inequality (2.25) on the integral

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M316">View MathML</a>

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M202">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M122">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M134">View MathML</a>, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M320">View MathML</a>

(2.44)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M321">View MathML</a>. Applying the inequality (2.6), we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M322">View MathML</a>

(2.45)

Substituting (2.44) and (2.45) into (2.43), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M323">View MathML</a>

which is the desired inequality (2.38). The proof is complete. □

Remark 2 One could consider the condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M324">View MathML</a> instead of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M321">View MathML</a> in the proof of Theorem 2.8. In this case, the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M326">View MathML</a> is replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M327">View MathML</a>, and also the term <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M328">View MathML</a> is replaced by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M329">View MathML</a>.

By using <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M330">View MathML</a> instead of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M331">View MathML</a> in the proof of Theorem 2.7, we have the following result.

Theorem 2.8Suppose thatyis a nontrivial solution of (1.2) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M300">View MathML</a>is the antiderivative of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M333">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M334">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M335">View MathML</a>

(2.46)

whereAis defined as in (2.9).

Remark 3 The contrapositive of the results in Theorems 2.7 and 2.8 yields sufficient conditions for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M1">View MathML</a>-disconjugacy of equation (1.2).

Remark 4 One can consider the boundary conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M337">View MathML</a>, which correspond to a beam hinged or supported at both ends. The proof is similar to the proof of Theorem 2.7. The details are left to the interested reader.

3 Examples

The following examples illustrate the results.

Example 1 Consider the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M338">View MathML</a>

(3.1)

where A and γ are positive constants. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M30">View MathML</a> is a solution of (3.1) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M71">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M46">View MathML</a>, then the condition (2.46) of Theorem 2.8 reads

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M342">View MathML</a>

i.e.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M343">View MathML</a>

Example 2 Consider the equation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M344">View MathML</a>

(3.2)

where λ and α are positive constants, with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M345">View MathML</a>, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M93">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M347">View MathML</a>. Then Theorem 2.2 gives us that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M348">View MathML</a>

so

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M349">View MathML</a>

i.e.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M350">View MathML</a>

From this we conclude that the interval of disconjugacy is bounded below by constant times <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M351">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M352">View MathML</a>, i.e., if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M353">View MathML</a> is the interval of disconjugacy, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M354">View MathML</a>.

Competing interests

The authors declare that they have no competing interest.

Authors’ contributions

Each author contributed equally to the paper. All authors read and approved the final version.

References

  1. Courant, R, Hilbert, D: Methods of Mathematical Physics, Wiley, New York (1989)

  2. Zill, DG, Wright, WS, Cullen, MR: Advanced Engineering Mathematics, Jones & Bartlett, Boston (2011)

  3. Ahmad, S: Asymptotic properties of linear fourth order differential equations. Proc. Am. Math. Soc.. 59, 45–51 (1976). Publisher Full Text OpenURL

  4. Amara, J, Vladimirov, AA: On oscillation of eigenfunctions of fourth order problem with spectral parameters in the boundary conditions. J. Math. Sci.. 150, 2317–2325 (2008). Publisher Full Text OpenURL

  5. Amara, J: Oscillation for fourth-order differential equations with middle term. Math. Nachr.. 285, 42–46 (2012). Publisher Full Text OpenURL

  6. Hou, C, Cheng, SS: Asymptotic dichotomy in a class of fourth-order nonlinear delay differential equations with damping. Abstr. Appl. Anal.. 2009, (2009) Article ID 484158

  7. Jaroš, J: Comparison theorems for half-linear differential equations of fourth order. Acta Math. Univ. Comen.. LXXX, 279–284 (2011)

  8. Kreith, K: Nonselfadjoint fourth order differential equations with conjugate points. Bull. Am. Math. Soc.. 80, 1190–1192 (1974). Publisher Full Text OpenURL

  9. Peterson, A: The distribution of zeros of extremal solutions of a fourth order differential equation for the n-th conjugate point. J. Differ. Equ.. 8, 502–511 (1970). Publisher Full Text OpenURL

  10. Peterson, A: Distribution of zeros of fourth order differential equations. Pac. J. Math.. 30, 751–764 (1969). Publisher Full Text OpenURL

  11. Schneider, LJ: Oscillation properties of the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/278/mathml/M355">View MathML</a> disconjugate fourth order selfadjoint differential equation. Proc. Am. Math. Soc.. 28, 545–550 (1971)

  12. Sobalová, M: On oscillatory solutions of the fourth order differential equations with the middle term. Nonlinear Anal.. 47, 3573–3578 (2001). Publisher Full Text OpenURL

  13. Taylor, WE Jr..: Asymptotic behavior of solutions of a fourth order nonlinear differential equation. Proc. Am. Math. Soc.. 65, 70–72 (1977). Publisher Full Text OpenURL

  14. Taylor, WE Jr..: Qualitative properties of solutions of certain fourth order linear differential equations. Int. J. Math. Math. Sci.. 4, 763–774 (1981). Publisher Full Text OpenURL

  15. Taylor, WE Jr..: On the oscillatory and asymptotic behavior of solutions of a certain fourth order linear differential equation. Hiroshima Math. J.. 7, 667–674 (1977)

  16. Taylor, WE Jr..: Oscillation criteria for certain nonlinear fourth order equations. Int. J. Math. Math. Sci.. 6(3), 551–557 (1983). Publisher Full Text OpenURL

  17. Zhang, M, Sun, J, Ao, J: Oscillation criteria of a class of fourth order differential equations. Math. Methods Appl. Sci. (2012) doi:10.1002/mma.1583

  18. Brown, R, Hinton, D: Lyapunov inequalities and their applications. In: Rassias T (ed.) Survey on Classical Inequalities, pp. 1–25. Kluwer Academic, Dordrecht (2000)

  19. Clark, S, Hinton, D: Some disconjugacy criteria for differential equations with oscillatory coefficients. Math. Nachr.. 278, 1476–1489 (2005). Publisher Full Text OpenURL

  20. Saker, SH: Lyapunov’s type inequalities for fourth order differential equations. Abstr. Appl. Anal.. 2012, (2012) Article ID 795825. doi:10.1155/2012/795825

  21. de la Valle Poussin, C: Sur l ’equation differentielle du second ordre. J. Math. Pures Appl.. 9, 125–144 (1929)

  22. Agarwal, RP, Pang, PYH: Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic, Dordrecht (1995)

  23. Kufner, A, Persson, L-E: Weighted Inequalities of Hardy Type, World Scientific, River Edge (2003)

  24. Kufner, A, Maligranda, L, Persson, L-E: The Hardy Inequalities: About Its History and Some Related Results, Vydavatelský Servis, Pilsen (2007)

  25. Agarwal, RP, Pang, PYH: Sharp Opial-type inequalities involving higher order derivatives of two functions. Math. Nachr.. 174, 5–20 (1995). Publisher Full Text OpenURL

  26. Boyd, D: Best constants in class of integral inequalities. Pac. J. Math.. 30, 367–383 (1969). Publisher Full Text OpenURL