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On positive solutions to equations involving the one-dimensional p-Laplacian

Ruyun Ma*, Yanqiong Lu and Ahmed Omer Mohammed Abubaker

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China

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Boundary Value Problems 2013, 2013:125  doi:10.1186/1687-2770-2013-125

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/125


Received:15 October 2012
Accepted:29 April 2013
Published:15 May 2013

© 2013 Ma et al.; licensee Springer

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

Abstract

We consider equations involving the one-dimensional p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M1">View MathML</a>

with the Dirichlet boundary conditions. By using time map methods, we show how changes of the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M2">View MathML</a> lead to multiple positive solutions of the problem for sufficiently large λ.

MSC: 34B10, 34B18.

Keywords:
positive solutions; one-dimensional p-Laplacian; uniqueness; time map

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M3">View MathML</a> be continuous and change its sign. Let Ω be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M4">View MathML</a> with smooth boundary Ω. The semi-positone problems and their special cases

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M5">View MathML</a>

(1.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M6">View MathML</a>

(1.2)

(and their finite difference analogues) have been extensively studied since early 1980s. Several different approaches such as variational methods, bifurcation theory, lower and upper solutions method and quadrature arguments have been successfully applied to show the existence of multiple solutions. See Brown and Budin [1], Peitgen et al.[2], Peitgen and Schmitt [3], Hess [4], Ambrosetti and Hess [5], Cosner and Schmitt [6], Dancer and Schmitt [7], Espinoza [8], Anuradha and Shivaji [9], Anuradha et al.[10], de Figueiredo [11], Lin and Pai [12], Clément and Sweers [13] and the references therein.

Very recently, Loc and Schmitt [14] considered the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M7">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M8">View MathML</a> is the p-Laplace operator for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M9">View MathML</a>. They assumed that the nonlinearity f is a continuous function on ℝ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M10">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M11">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M12">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M14">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M15">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M16">View MathML</a>. They proved that, for λ sufficiently large, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M17">View MathML</a>

(1.4)

then the problem (1.3) has at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M18">View MathML</a> positive bounded solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M19">View MathML</a> which belong to the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M20">View MathML</a> and are such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M21">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M22">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M23">View MathML</a>

In the special case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M25">View MathML</a>, Brown and Budin [1] applied the quadrature arguments to get the following more detailed results.

Theorem A [[1], Theorem 3.8]

Assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M26">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M27">View MathML</a>;

(H3) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M28">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M30">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M31">View MathML</a>;

(H4) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M32">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M33">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M34">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M36">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M38">View MathML</a>.

Then:

(a) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39">View MathML</a>, there exists a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>of (1.2).

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M41">View MathML</a>, there exist at least two solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>of (1.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M43">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M44">View MathML</a>

(1.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M45">View MathML</a>

(1.6)

(c) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>is any solution of (1.2) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M48">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M49">View MathML</a>.

Of course the natural question is whether or not the similar results still hold for the corresponding problem involving the one-dimensional p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M50">View MathML</a>

(1.7)

We shall answer these questions in the affirmative if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M51">View MathML</a>. More precisely, we get the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M51">View MathML</a>and let (H1), (H3), (H4) hold. Assume that

(H2′) either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M27">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M54">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M55">View MathML</a>

(1.8)

Then:

(a) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M56">View MathML</a>, there exists a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>of (1.7), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M58">View MathML</a>is the least eigenvalue of BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M59">View MathML</a>

(1.9)

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M41">View MathML</a>, there exist at least two solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>of (1.7) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M62">View MathML</a>

(c) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>is any solution of (1.7) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M65">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M66">View MathML</a>

(1.10)

We shall apply the time map method to show how changes of the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M2">View MathML</a> lead to multiple positive solutions of (1.7) for sufficiently large λ.

In the following, we extend f so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M68">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M69">View MathML</a>, then all the solutions of (1.7) are positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70">View MathML</a>.

2 Preliminaries

To prove our main results, we use the uniqueness results due to Reichel and Walter [15] on the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M71">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M73">View MathML</a>.

Lemma 2.1Let (H1) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M74">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M75">View MathML</a>, then the initial value problem (2.1) has a unique local solution. The extension<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M76">View MathML</a>remains unique as long as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M77">View MathML</a>.

Proof It is an immediate consequence of Reichel and Walter [[15], Theorem 2]. □

Lemma 2.2Let (H1) hold. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M78">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M79">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M80">View MathML</a>

Then the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M81">View MathML</a>

(2.2)

has a unique local solution.

Proof (H1) implies that f is locally Lipschitzian. This together with the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M82">View MathML</a> and using [[15], (iii) and (v) in the case (β) of Theorem 4] yields that (2.2) has a unique solution in some neighborhood of a. □

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M83">View MathML</a>be continuous. Letube a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M84">View MathML</a>

(2.3)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M85">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M86">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M87">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M88">View MathML</a>

(2.4)

Proof Since g is independent of t, both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M90">View MathML</a> satisfy the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M91">View MathML</a>

(2.5)

By Lemmas 2.1 and 2.2, (2.5) has a unique solution defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M92">View MathML</a>. Therefore, (2.4) is true. □

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M93">View MathML</a>be a positive solution of the problem

(2.6)

(2.7)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M98">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M87">View MathML</a>. Then

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M100">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M101">View MathML</a>is the unique point on whichuattains its maximum;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M103">View MathML</a>.

Proof (a) Suppose on the contrary that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M104">View MathML</a>, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M105">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M106">View MathML</a>

However, this is impossible since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M108">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M100">View MathML</a>.

(b) Suppose on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M111">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M112">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M113">View MathML</a>

We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M114">View MathML</a>. The other case can be treated in a similar way.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M115">View MathML</a> in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M116">View MathML</a>, then Lemma 2.3 yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M117">View MathML</a>

This contradicts the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M118">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M119">View MathML</a> in any subinterval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70">View MathML</a>.

So, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M121">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M122">View MathML</a>

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M123">View MathML</a>

Multiplying both sides of the equation in (2.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M124">View MathML</a> and integrating from t to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M101">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M126">View MathML</a>

(2.8)

and subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M127">View MathML</a>

This contradicts the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M129">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M130">View MathML</a>

Similarly, we can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M131">View MathML</a>

(c) Suppose on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M132">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M133">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M134">View MathML</a>

This together with (2.8) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M135">View MathML</a>

This contradicts the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M137">View MathML</a>. □

3 Proof of the main results

To prove Theorem 1.1, we need the following preliminary results.

Lemma 3.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128">View MathML</a>, there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M39">View MathML</a>such that

(3.1)

(3.2)

has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M143">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144">View MathML</a>is a continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M145">View MathML</a>.

Proof By Lemma 2.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a> is a positive solution of (3.1), (3.2) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a> is a positive solution of

(3.3)

(3.4)

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a> is a solution of (3.3), (3.4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M152">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M153">View MathML</a>

(3.5)

Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M154">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M155">View MathML</a>

(3.6)

Hence λ (if exists) is uniquely determined by ρ.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M128">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M157">View MathML</a> by (3.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M76">View MathML</a> by (3.5). It is straightforward to verify that u is twice differentiable, u satisfies (3.3), (3.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M108">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M161">View MathML</a>. The continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M162">View MathML</a> is implied by (3.6) and this completes the proof. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M163">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M164">View MathML</a>.

Lemma 3.2Let (H1) and (H2′) hold, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M165">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M166">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M58">View MathML</a>is the least eigenvalue of (1.9).

Proof We only deal with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M168">View MathML</a>. The other one can be treated by the same method.

To this end, we divide the proof into two cases.

Case 1. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M169">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M170">View MathML</a>.

In this case, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M171">View MathML</a>, there is a positive number R such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M172">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M173">View MathML</a>. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M174">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M175">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M176">View MathML</a>. From (3.6), we have that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M177">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M178">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M179">View MathML</a>

see Zhang [16]. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M180">View MathML</a>

Case 2. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M181">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M182">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M183">View MathML</a>

(3.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M184">View MathML</a>

(3.8)

In fact, (3.6) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M185">View MathML</a>

(3.9)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M186">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M187">View MathML</a>

We will show that the last integral in (3.9) converges to zeros as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M188">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M189">View MathML</a>, using l’Hospital’s rule, it follows that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M188">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M191">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M192">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M193">View MathML</a>

uniformly in v. Therefore, (3.9) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M194">View MathML</a>

(3.10)

Therefore, (3.7) holds. □

From the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M196">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M198">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M199">View MathML</a>. Moreover, we have the following.

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M200">View MathML</a>. Then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M201">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M202">View MathML</a>.

Proof (i) Suppose firstly that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M203">View MathML</a>. Since S is open, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M204">View MathML</a> and so there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M205">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M206">View MathML</a>

Clearly k must be a local maximum for F and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M207">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M208">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M209">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M210">View MathML</a>

Then if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M211">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M212">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M213">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M215">View MathML</a> is a nondecreasing sequence of measurable functions. Therefore, by the monotone convergence theorem and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M216">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M217">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M218">View MathML</a>.

Suppose next that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M219">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M220">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M221">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M222">View MathML</a>

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M224">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M225">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M226">View MathML</a>

(3.11)

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M227">View MathML</a>, then it follows from (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M228">View MathML</a>

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M229">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M230">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M232">View MathML</a> denotes the characteristic function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M233">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M234">View MathML</a> is a nondecreasing sequence of measurable functions, by the monotone convergence theorem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M235">View MathML</a>

 □

Proof of Theorem 1.1 (a) follows from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144">View MathML</a> and Lemma 3.2.

(b) follows from the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M144">View MathML</a> and Lemma 3.3.

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M40">View MathML</a> is any solution of (3.1), (3.2) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M239">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M240">View MathML</a>

Now, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M47">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M242">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/125/mathml/M243">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RM completed the main study, carried out the results of this article. YL drafted the manuscript. AOMA checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), SRFDP (No. 20126203110004) and Gansu Provincial National Science Foundation of China (No. 1208RJZA258).

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