SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Asymptotic problems for fourth-order nonlinear differential equations

Miroslav Bartušek and Zuzana Došlá*

Author Affiliations

Faculty of Science, Masaryk University Brno, Kotlářská 2, Brno, 611 37, The Czech Republic

For all author emails, please log on.

Boundary Value Problems 2013, 2013:89  doi:10.1186/1687-2770-2013-89

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/89


Received:13 November 2012
Accepted:16 March 2013
Published:12 April 2013

© 2013 Bartušek and Došlá; licensee Springer

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

Abstract

We study vanishing at infinity solutions of a fourth-order nonlinear differential equation. We state sufficient and/or necessary conditions for the existence of the positive solution on the half-line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a> which is vanishing at infinity and sufficient conditions ensuring that all eventually positive solutions are vanishing at infinity. We also discuss an oscillation problem.

Dedication

Dedicated to Jean Mawhin on occasion of his seventieth birthday.

1 Introduction

In this paper we study the fourth-order nonlinear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5">View MathML</a> for large t, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M6">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M7">View MathML</a> for large t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M8">View MathML</a>.

Jointly with (1), we consider a more general equation

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M10">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12">View MathML</a>, and the associated linear second-order equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M13">View MathML</a>

(2)

By a solution of (1) we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M15">View MathML</a>, which satisfies (1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M16">View MathML</a>. A solution is said to be nonoscillatory if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M17">View MathML</a> for large t; otherwise, it is said to be oscillatory. Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, according to [[1], Theorem 11.5], all nontrivial solutions of (1) satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M19">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M20">View MathML</a>, on the contrary to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M21">View MathML</a>, when nontrivial solutions satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M22">View MathML</a> for large t may exist.

Fourth-order differential equations have been investigated in detail during the last years. The periodic boundary value problem for the superlinear equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M23">View MathML</a> has been studied in [2]. In [3], the fourth-order linear eigenvalue problem, together with the nonlinear boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M24">View MathML</a>, has been investigated. Oscillatory properties of solutions for self-adjoint linear differential equations can be found in [4]. Equation (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M25">View MathML</a> can be viewed as a prototype of even-order two-term differential equations, which are the main object of monographs [1,5,6].

Equation (1′) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M26">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a> is a special case of higher-order differential equations investigated in [7]. Equation (1′) with q near to a nonzero constant as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28">View MathML</a> has been considered in [8] as a perturbation of the linear equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M29">View MathML</a>, and the existence of oscillatory solutions of (1′) has been proved. In [9], necessary and sufficient conditions for the existence of asymptotically linear solutions of (1′) have been given.

In the recent paper [10], the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M34">View MathML</a> has been investigated and applications to the biharmonic PDE’s can be found there. In particular, the so called homoclinics solutions, which are defined as nontrivial solutions x such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M35">View MathML</a>, are studied.

The goal of this paper is to investigate asymptotic problems associated with (1) and the asymptotic boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M36">View MathML</a>

(3)

A solution x of (1) satisfying (3) is said to be vanishing at infinity.

We start with the Kneser problem for (1). The Kneser problem is a problem concerning the existence of solutions of (1) subject to the boundary conditions on the half-line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M38">View MathML</a>

(4)

We establish necessary and/or sufficient conditions for the solvability of the boundary value problem (1), (3), (4). In the light of these results, as the second problem, we study when all eventually positive solutions x of (1) are vanishing at infinity assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a> and (2) is oscillatory. As a consequence, we give a bound for the set of all nonoscillatory solutions. Finally, we discuss when problem (1), (3) is not solvable and solutions to (1) are oscillatory.

A systematic analysis of solutions of (1) satisfying (3) is made according to whether (2) is nonoscillatory or oscillatory. If (2) is nonoscillatory, then the following approach will be used. Equation (1) can be rewritten as the two-term equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M40">View MathML</a>

(5)

where h is a positive solution of (2). According to [11], a solution h of (2) is said to be a principal solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M41">View MathML</a>, and such a solution is determined uniquely up to a multiple constant. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5">View MathML</a>, every eventually positive solution of (2) is nondecreasing for large t. Hence there exists a principal solution h of (2) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M44">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M45">View MathML</a>

(6)

Therefore, we can use the known results [12,13] stated for systems of differential equations or in [14] for fourth order differential equations.

If (2) is oscillatory, then our approach is based on the choice of a suitable transformation. The main idea is based on a transformation of (1) to the fourth-order quasilinear equation and the use of the estimates for positive solutions of such an equation on a compact interval stated in [15]. This, together with an energy function associated with (1), enables us to state an oscillation theorem. In the final section, some extensions of our results to (1′) are given.

2 The Kneser problem

In this section we present necessary and/or sufficient conditions for solvability of boundary value problem (1), (3), (4).

2.1 Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a>

Proposition 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, (2) be disconjugate on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Then boundary value problem (1), (4) is solvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a>.

To prove this theorem, we use Chanturia’s result [[12], Theorem 1] for the system of differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M52">View MathML</a>

(7)

where we restrict to the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M53">View MathML</a> are continuous functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M58">View MathML</a>. Then this result reads as follows.

Theorem A ([12])

Let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M60">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64">View MathML</a>). Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M65">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64">View MathML</a>), where functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M71">View MathML</a>) are continuous and nondecreasing in the second argument such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M72">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M73">View MathML</a>is a continuous function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M74">View MathML</a>is a continuous nondecreasing function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M75">View MathML</a>

Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M76">View MathML</a>, system (7) has a solution satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M77">View MathML</a>

(8)

Proof of Proposition 1 Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Since (2) is disconjugate, it has a positive solution h on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>, and (1) can be written as (5) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M81">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>. Let x be a solution of (5) and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M83">View MathML</a>

(9)

Then (5) is equivalent to the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M84">View MathML</a>

(10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a> be from (4). We apply Theorem A choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M86">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M89">View MathML</a>. By this result, system (10) has a solution such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M90">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, system (10) has no solutions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M92">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93">View MathML</a> and large t; see [16] or [[17], Lemma 2, Theorem 2]. Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a>, equation (1) has a solution x such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M98">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a>. □

Now we state conditions for the existence of a solution for problem (1), (3), (4).

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M102">View MathML</a>

(11)

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M104">View MathML</a>

(12)

then problem (1), (3), (4) is solvable for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a>.

In addition, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M106">View MathML</a>

(13)

then the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M107">View MathML</a>

(14)

is necessary and sufficient for the solvability of problem (1), (3), (4).

For the proof, the following lemma will be needed.

Lemma 1Consider system (10) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M108">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M109">View MathML</a>), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M43">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>andhis a principal solution of (2). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112">View MathML</a>be a solution of (10) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M113">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M114">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M117">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64">View MathML</a>, and if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M119">View MathML</a>

(15)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M120">View MathML</a>, too. Vice versa, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M120">View MathML</a>, then (15) holds.

Proof In view of the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M93">View MathML</a>. Since h is the principal solution, (6) holds, and integrating the first three equations in (10) from a to t, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M126">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M64">View MathML</a>. Now integrating (10) from t to ∞, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M128">View MathML</a>

Let (15) hold and assume, by contradiction, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M129">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M130">View MathML</a>

(16)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28">View MathML</a> and using the change of the order of integration, we get a contradiction with the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M132">View MathML</a>. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M133">View MathML</a>.

Let the integral in (15) be convergent and assume, by contradiction, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M133">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M135">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M136">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, then using the change of the order of integration, we get a contradiction for large t. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M129">View MathML</a>. □

Proof of Theorem 1 In view of (11), (2) is disconjugate on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>. By Proposition 1, equation (1) has a solution x satisfying (4). Therefore, system (10) has a solution such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M141">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Choose h in (10) as a principal solution of (2). The Euler equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M143">View MathML</a>

(17)

is the majorant of (2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a> and has the principal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M145">View MathML</a>. By the comparison theorem, for the minimal solution of the Riccati equation related to (2) and (17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M146">View MathML</a>

for large t; see, e.g., [11]. Thus there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M147">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M148">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a>. Assume (12). Then (15) holds, and by Lemma 1 a solution x satisfies (3).

Assume (13). Then the principal solution h of (2) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M150">View MathML</a> for large t (see, e.g., [11]). Hence, condition (15) reads as (14), and by Lemma 1 this condition is equivalent to the property (3). □

As a consequence of Lemma 1, we get the following result.

Corollary 1Let (2) be disconjugate on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Then any solutionxof (1) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M154">View MathML</a>

(18)

is a solution of the Kneser problem, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M155">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M157">View MathML</a>.

Proof Let h be a positive solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a> satisfying (6), and let x be a solution of (1) satisfying (18). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M132">View MathML</a> are defined by (9), is a solution of system (10). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M161">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a> and (6) holds, we have by the Kiguradze lemma (see, e.g., [1]) that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M163">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M141">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M71">View MathML</a> and large t, say for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M44">View MathML</a>. Since x is positive and tends to zero, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M167">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>, so also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M169">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. By Lemma 1, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M172">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170">View MathML</a>) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M175">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M177">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M179">View MathML</a> is positive and decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>. Hence, proceeding by the same argument, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M170">View MathML</a>) is positive and decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M80">View MathML</a>. Now the conclusion follows from (9). □

2.2 Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>

First we show that the sign condition posed on r is necessary for the solvability of problem (1), (4).

A function g, defined in a neighborhood of infinity, is said to change sign if there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M185">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M186">View MathML</a>.

Theorem 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>for larget. Then problem (1), (4) has no solution and the following hold:

(a) If (2) is nonoscillatory, then every nonoscillatory solutionxof (1) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M188">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M189">View MathML</a>is of one sign for larget.

(b) If (2) is oscillatory, then every nonoscillatory solutionxof (1) satisfies either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M190">View MathML</a>, or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M191">View MathML</a>changes sign. In addition, if a solutionxsatisfies (3), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192">View MathML</a>changes sign.

Proof Claim (a). Let x be a positive solution of (1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193">View MathML</a>, or, equivalently, of (5) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193">View MathML</a>, where h satisfies (6). Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M195">View MathML</a>

(19)

Then (5) is equivalent to the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M196">View MathML</a>

(20)

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M161">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. Assume by contradiction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M202">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M179">View MathML</a> is nonincreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M204">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M205">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M28">View MathML</a>, we get a contradiction with the positiveness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M207">View MathML</a>. The remaining case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M208">View MathML</a> can be eliminated in a similar way using (6). Observe that system (20) is a special case of the Emden-Fowler system investigated in [13], and the proof follows also from [[13], Lemma 2.1].

Claim (b). Without loss of generality, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M210">View MathML</a> and there exists a solution x of (1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M212">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214">View MathML</a>. Borůvka [18] proved that if (2) is oscillatory, then there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M215">View MathML</a>, called a phase function, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M216">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M217">View MathML</a>

(21)

Using this result, we can consider the change of variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M218">View MathML</a>

(22)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M221">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M222">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M223">View MathML</a>

Substituting into (1), we obtain the second-order equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M224">View MathML</a>

From here and (21), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M225">View MathML</a>

(23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M226">View MathML</a>, (22) yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M227">View MathML</a> and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M228">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M229">View MathML</a> is decreasing. If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M230">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M231">View MathML</a>, X becomes eventually negative, which is a contradiction. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M233">View MathML</a> is nondecreasing. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M234">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M235">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M236">View MathML</a>. Thus, using (23) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M237">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M238">View MathML</a>, which contradicts the nonnegativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M239">View MathML</a>. Finally, the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M240">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M241">View MathML</a> cannot occur, because if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M242">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M213">View MathML</a>, then from (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M245">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M246">View MathML</a>, which is a contradiction.

Finally, let x be a positive solution of (1) satisfying (3). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192">View MathML</a> is either oscillatory or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M248">View MathML</a> for large t. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M248">View MathML</a> on some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M250">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M251">View MathML</a> is decreasing and either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M252">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M253">View MathML</a> for large t. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M252">View MathML</a> for large t, then we get a contradiction with (3). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M253">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M256">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M257">View MathML</a> and x becomes negative for large t. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192">View MathML</a> must be oscillatory. □

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>, the analogous result to Theorem 1 is the following oscillation result.

Proposition 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>for larget. Assume either (11) for larget, (12), or (13), (14). Then all the solutions of (1) are oscillatory.

Proof Let x be a solution of (1) and h be the principal solution of (2). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M112">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M123">View MathML</a> are given by (19), is a solution of system (20). Proceeding by the similar way as in the proof of Theorem 1, we have that (15) holds. Using the change of the order of integration in (15), we can check that conditions of Theorem 4.3 in [13] applied to system (20) are verified. Hence by this result all the solutions of (20) are oscillatory, which gives the conclusion. □

The following result follows from [[7], Theorem 1.5] and completes Proposition 2 in the case when (2) is oscillatory.

Proposition 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Then the condition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M268">View MathML</a>is necessary and sufficient for every solution of (1) to be oscillatory.

In the light of these results, in the sequel, we study asymptotic and oscillation problems to (1) when (2) is oscillatory.

3 Vanishing at infinity solutions

In this section we study when all nonoscillatory solutions of (1) are vanishing at infinity.

Theorem 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>and (2) be oscillatory. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270">View MathML</a>for largetand some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M271">View MathML</a>, the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M272">View MathML</a>

(24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M273">View MathML</a>

(25)

Then any eventually positive solution of (1) is vanishing at infinity.

The proof of Theorem 3 is based on the following auxiliary results.

Consider the fourth-order quasi-linear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M274">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M275">View MathML</a> and R are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M276">View MathML</a>. In [[15], Theorem 2.4], the following uniform estimate for positive solutions of (26) with a common domain was proved.

Proposition 4 ([[15], Theorem 3.4, Corollary 3.6])

Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>. Letybe a positive solution of (26) defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M279">View MathML</a>

(27)

on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278">View MathML</a>for some constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M281">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M282">View MathML</a>. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M283">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M284">View MathML</a>

(28)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M285">View MathML</a>

Remark 1 In [[15], Theorem 3.4] the constant M is explicitly calculated.

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>. Assume that (27) holds on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M287">View MathML</a>. Then any positive solution of (26) defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M289">View MathML</a>

(29)

whereαandMare constants from Proposition 4.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M290">View MathML</a>. By Proposition 4, applied on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M278">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M292">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M293">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M294">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M295">View MathML</a>, we get (29). □

The next lemma describes the transformation between solutions of (1) and a certain quasi-linear equation.

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M5">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M298">View MathML</a>

and consider the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M299">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M300">View MathML</a>is a solution of equation (1) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M301">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M302">View MathML</a>is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M303">View MathML</a>

(30)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M304">View MathML</a>is the inverse function to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M305">View MathML</a>.

Proof We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M306">View MathML</a>

Substituting into (1), we get the conclusion. □

Proof of Theorem 3 Let x be a positive solution of (1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M307">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M308">View MathML</a>). Suppose, for simplicity, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M111">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M311">View MathML</a>

(31)

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M1">View MathML</a> for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M315">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M316">View MathML</a>

(32)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M317">View MathML</a>

(33)

Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M318">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M319">View MathML</a>

(34)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M320">View MathML</a>. Then, according to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M322">View MathML</a>

(35)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M323">View MathML</a> be nondecreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M324">View MathML</a> and put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M325">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M326">View MathML</a> arbitrarily fixed. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M270">View MathML</a>, we can consider the transformation from Lemma 3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M328">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M329">View MathML</a>

(36)

Then equation (1) is transformed into equation (30) which is a quasilinear equation of the form (26), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M330">View MathML</a>

and Q is defined by (31) and (32). Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M331">View MathML</a> arbitrarily. We apply Lemma 2 to equation (30) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M332">View MathML</a>

Hence estimate (29) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M333">View MathML</a> reads as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M334">View MathML</a>

(37)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M335">View MathML</a>. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M336">View MathML</a>, we have by (35) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M337">View MathML</a> and the conclusion follows from (25) and (37). □

From the proof of Theorem 3, we get the estimate for the set of all nonoscillatory solutions of (1) which will be used in the next section.

Corollary 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M339">View MathML</a>, (24) and (25) hold. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M341">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214">View MathML</a>such that every nonoscillatory solutionxof (1) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M343">View MathML</a>

(38)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340">View MathML</a> be fixed and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M214">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M346">View MathML</a>

where α is given by (33). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M210">View MathML</a> be fixed. Using estimate (37) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M348">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M349">View MathML</a>

(39)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M350">View MathML</a> is given by (34), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M351">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M352">View MathML</a> and estimate (38) follows from (25) and (39). □

Example 1 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M353">View MathML</a>

(40)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M354">View MathML</a> and by Theorem 3 all eventually positive solutions are vanishing at infinity. One can check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M355">View MathML</a> is such a solution of (40).

Open problem It is an open problem to find conditions for the solvability of boundary value problem (1), (3), (4) in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M356">View MathML</a> and (2) is oscillatory.

In view of Theorem 2, Corollary 2 and Proposition 1, it is a question whether (1) can have vanishing at infinity solutions in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a> and (2) is oscillatory.

In the next section, we show that under certain additional assumptions the answer is negative.

4 Oscillation

Here we consider (1) in case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a> for large t. When (2) is nonoscillatory, we have established the oscillation criterion in Proposition 2. When (2) is oscillatory, the following oscillation theorem holds.

Theorem 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M184">View MathML</a>and assumptions (24), (25) hold. Assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M361">View MathML</a>

(41)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M362">View MathML</a>

(42)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M340">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M364">View MathML</a>. Then problem (1), (3) is not solvable and all the solutions of (1) are oscillatory.

Proof Suppose that (25) holds on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193">View MathML</a>. First, observe that the assumption (42) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M366">View MathML</a>

(43)

Indeed, putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M367">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M368">View MathML</a>

and thus, in view of (42), we get (43). Consider a solution x of (1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M369">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M370">View MathML</a>. According to Corollary 2, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M371">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M372">View MathML</a>

(44)

and in view of (25) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M373">View MathML</a>. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M374">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M375">View MathML</a>

and in view of (41) the function F is increasing for large t. Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M376">View MathML</a> such that either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M377">View MathML</a>

(45)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M378">View MathML</a>

(46)

According to Theorem 2(b), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192">View MathML</a> oscillates. Define by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M380">View MathML</a> an increasing sequence of zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M192">View MathML</a> tending to ∞ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M382">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M383">View MathML</a>

(47)

In view of (44) and (43) the function Z is well defined and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M384">View MathML</a>

(48)

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M385">View MathML</a>. Moreover, we have from (42), (44) and (47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M386">View MathML</a>

(49)

If (45) holds, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M387">View MathML</a> and because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M388">View MathML</a>

we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M389">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M390">View MathML</a>. This is a contradiction with (49), so (45) is impossible.

If (46) holds, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M392">View MathML</a>. This is again a contradiction with (49), so also this case is impossible. □

Example 2 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M393">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M394">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M396">View MathML</a>, then by Theorem 1 this equation has a solution satisfying (3) and (4). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M394">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M398">View MathML</a>, then by Theorem 3 any nonoscillatory solution (if any) satisfies (3).

5 Extensions

As it was mentioned in [10], a certain nonlinear PDE leads to the fourth-order equation with the exponential nonlinearity. In the sequel, we show that the results of this paper can be extended to the nonlinear equation

where q, r are as for (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M402">View MathML</a>

(50)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M404">View MathML</a>. The prototype of such an extension is the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M405">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M406">View MathML</a>.

Theorems 1-4 read for (1′) as follows.

Theorem 1′Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M46">View MathML</a>and (11) hold for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M27">View MathML</a>. Assume that either (i) (12), or (ii) (13) and (14) hold. Then problem (1′), (3), (4) has a solution for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M51">View MathML</a>.

Proof of Theorem 1′ It is analogous to the proofs of Proposition 1 and Theorem 1 replacing the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M411">View MathML</a> in system (10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M412">View MathML</a>. Lemma 1 remains to hold as a sufficient condition for (3). □

Theorem 2′Theorem 2 remains to hold for (1′) without assuming (50).

Proof of Theorem 2′ In the proof of claim (a) of Theorem 2, we consider system (20) where the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M411">View MathML</a> is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M412">View MathML</a>. The proof of claim (b) of Theorem 2 is the same for the nonlinearity f. □

Theorem 3′Theorem 3 remains to hold for (1′).

Proof of Theorem 3′ Let x be a positive solution of (1′) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M416">View MathML</a> is a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M417">View MathML</a>

(51)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/89/mathml/M418">View MathML</a>

(52)

Now we apply Theorem 3 to (51). □

Theorem 4′Let the assumptions of Theorem 4 hold. Then (1′) has no eventually positive solutions.

Proof of Theorem 4′ It is similar to the one of Theorem 4. In view of (52), the estimate (38) holds and the energy function F is the same. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final draft.

Acknowledgements

Supported by the grant GAP 201/11/0768 of the Czech Grant Agency.

References

  1. Kiguradze, I, Chanturia, TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic, Dordrecht (1993)

  2. Mawhin, J, Zanolin, F: A continuation approach to fourth order superlinear periodic boundary value problems. Topol. Methods Nonlinear Anal.. 2(1), 55–74 (1993)

  3. Gupta, C, Mawhin, J: Weighted eigenvalue, eigenfunctions and boundary value problems for fourth order ordinary differential equations. Recent Trends in Differential Equations, pp. 253–267. World Scientific, River Edge (1992)

  4. Ahlbrandt, CD, Hinton, DB, Lewis, R: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory. J. Math. Anal. Appl.. 81, 234–277 (1981). Publisher Full Text OpenURL

  5. Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht (2001)

  6. Elias, U: Oscillation Theory of Two-Term Differential Equations, Kluwer Academic, Dordrecht (1997)

  7. Kiguradze, I: An oscillation criterion for a class of ordinary differential equations. Differ. Uravn. (Minsk). 28, 201–214 (1992)

  8. Bartušek, M, Cecchi, M, Došlá, Z, Marini, M: Asymptotics for higher order differential equations with a middle term. J. Math. Anal. Appl.. 388, 1130–1140 doi:10.1016/j.jmaa.2011.10.059 (2012)

    doi:10.1016/j.jmaa.2011.10.059

    PubMed Abstract | Publisher Full Text OpenURL

  9. Bartušek, M, Cecchi, M, Došlá, Z, Marini, M: Fourth-order differential equation with deviating argument. Abstr. Appl. Anal.. 2012, Article ID 185242 (2012)

  10. Berchio, E, Ferrero, A, Gazzola, F, Karageorgis, P: Qualitative behavior of global solutions to some nonlinear fourth order differential equations. J. Differ. Equ.. 251, 2696–2727 (2011). Publisher Full Text OpenURL

  11. Hartman, P: Ordinary Differential Equations, Wiley, New York (1964)

  12. Chanturia, TA: On monotone solutions of a system of nonlinear differential equations. Ann. Pol. Math.. 37, 59–70 (in Russian) (1980)

  13. Kusano, T, Naito, M, Wu, F: On the oscillation of solutions of 4-dimensional Emden-Fowler differential systems. Adv. Math. Sci. Appl.. 11(2), 685–719 (2001)

  14. Naito, M, Wu, F: A note on the existence and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations. Acta Math. Hung.. 102, 177–202 (2004)

  15. Astashova, IV: Uniform estimates for positive solutions of quasi-linear ordinary differential equations. Izv. Ross. Akad. Nauk, Ser. Mat.. 72(6), 85–104 (Russian); translation in Izv. Math. 72(6), 1141-1160 (2008) (2008)

  16. Chanturia, TA: On singular solutions of nonlinear systems of ordinary differential equations. Colloq. Math. Soc. János Bolyai. 15, 107–119 (1976)

  17. Bartušek, M, Došlá, Z: Remark on Kneser problem. Appl. Anal.. 56(3-4), 327–333 (1995). Publisher Full Text OpenURL

  18. Borůvka, O: Linear Differential Transformationen 2. Ordung, VEB, Berlin (1967)