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Symmetric positive solutions of higher-order boundary value problems

Yan Luo

Author Affiliations

Department of Mathematics, Hunan University of Science and Technology, Xiangtan, Hunan, 411201, China

Boundary Value Problems 2014, 2014:78  doi:10.1186/1687-2770-2014-78

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/78


Received:31 October 2013
Accepted:20 March 2014
Published:3 April 2014

© 2014 Luo; licensee Springer.

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

Abstract

We study the higher-order boundary value problems. The existence of symmetric positive solutions of the problem is discussed. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.

MSC: 34B15, 34B18.

Keywords:
higher-order boundary value problems; symmetric positive solutions; existence; monotone iterative technique

1 Introduction

We study the boundary value problem (BVP)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M2">View MathML</a> is an integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M3">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M5">View MathML</a> are nonnegative constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8">View MathML</a> may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M10">View MathML</a> (and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M11">View MathML</a>). If a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M12">View MathML</a> is continuous and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M13">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M14">View MathML</a>, then we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M15">View MathML</a> is symmetric on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M16">View MathML</a>. By a symmetric positive solution of BVP (1.1) we mean a symmetric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M18">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M15">View MathML</a> satisfies (1.1).

In recent years, many authors have studied BVP (1.1), they only considered that f is nondecreasing or nonincreasing in u, or the boundary condition depends only on derivatives of even orders; see [1-8] and references cited therein. To the best of the author’s knowledge, there is no such results involving (1.1). In this note, we intend to fill in such gaps in the literature.

The organization of this paper is as follows. After this introduction, in Section 2, we state the assumptions and some preliminary lemmas. By applying the monotone iterative technique, we discuss the existence of symmetric positive solutions for (1.1) and obtain the main results in Section 3.

2 Preliminaries

For convenience, in this paper we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M22">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M23">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M24">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7">View MathML</a>, and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M26">View MathML</a>

Remark 2.1 The set P is not a cone as it is not closed.

Throughout this paper, we assume the following:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M5">View MathML</a> are nonnegative constants, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M31">View MathML</a> is continuous and symmetric in t, i.e., f satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M32">View MathML</a>

(2.3)

(H2) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8">View MathML</a> is nondecreasing in u and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M35">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M37">View MathML</a>

(2.4)

(H2′) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8">View MathML</a> is nonincreasing in u and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M35">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M42">View MathML</a>

(2.5)

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M43">View MathML</a>.

Example 2.1 Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M44">View MathML</a>

It is easy to see that the function f satisfies assumptions (H1) and (H2). In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36">View MathML</a>, there exists constant λ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M47">View MathML</a>.

Remark 2.2 It is easy to see that (H2) implies that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M48">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M49">View MathML</a>

(2.6)

and (H2′) implies that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M48">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M51">View MathML</a>

(2.7)

Now, we present several lemmas that will be used in the proof of our results. By routine calculations we have the following results.

Lemma 2.1Letvbe integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M52">View MathML</a>, then the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M53">View MathML</a>

has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M54">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M55">View MathML</a>are defined by (2.1).

Lemma 2.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M56">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M57">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M58">View MathML</a>

(2.9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M7">View MathML</a>.

3 Main results

Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M61">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M62">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M55">View MathML</a> are defined by (2.1). It is clear that u is a solution of (1.1) if and only if u is a fixed point of T.

Theorem 3.1Assume (H1)-(H3) hold. Then BVP (1.1) has at least one symmetric positive solution.

Proof

Claim 3.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64">View MathML</a>is completely continuous and nondecreasing.

In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M65">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M67">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M69">View MathML</a>. (2.3), (2.9) and a change of variables imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M70">View MathML</a>

(3.2)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M65">View MathML</a>, from (2.4), (2.6), (2.8), and (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M72">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M73">View MathML</a>

(3.4)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M75">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M76">View MathML</a>

Thus, it follows from (3.3) and (3.4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M77">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64">View MathML</a>. Next by a standard method and the Ascoli-Arzela theorem one can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64">View MathML</a> is completely continuous, we omit it here. From (H2), it is easy to see that T is nondecreasing in u. Hence, Claim 3.1 holds.

Claim 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M80">View MathML</a>be fixed number satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M81">View MathML</a>

(3.5)

whereλis defined in (H2) in which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M82">View MathML</a>, and assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M83">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M84">View MathML</a>

(3.7)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M85">View MathML</a>

(3.8)

and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M87">View MathML</a>

(3.9)

In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M88">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M89">View MathML</a>. So, from (3.5) and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M80">View MathML</a>. From (3.6), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M93">View MathML</a>.

On the other hand, from (2.4) and (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M94">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M93">View MathML</a> and T is nondecreasing, by induction, (3.8) holds.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M96">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M97">View MathML</a>. It follows from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M98">View MathML</a>

that, for any natural number n,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M99">View MathML</a>

Thus, for all natural numbers n and p, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M100">View MathML</a>

which implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86">View MathML</a> such that (3.9) holds, and Claim 3.2 holds.

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M102">View MathML</a> in (3.7), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M103">View MathML</a>, which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem.  □

Theorem 3.2Assume (H1), (H2′) and (H3) hold. Then BVP (1.1) has at least one symmetric positive solution.

Proof

Claim 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M64">View MathML</a>is completely continuous and nonincreasing.

The proof of Claim 3.3 is similar to the proof of Claim 3.1, so this is omitted.

Claim 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M105">View MathML</a>be fixed number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M106">View MathML</a>be sufficiently large constant satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M107">View MathML</a>

(3.10)

whereλis defined in (H2′) in which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M108">View MathML</a>, and assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M109">View MathML</a>

(3.11)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M110">View MathML</a>

(3.12)

and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M112">View MathML</a>

(3.13)

In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M113">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M106">View MathML</a>. So from (3.11),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M116">View MathML</a>

(3.14)

From (2.5), (3.10), (3.14), and noting that T is nonincreasing in u, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M117">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M118">View MathML</a>

(3.16)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M119">View MathML</a>

(3.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M120">View MathML</a>

(3.18)

From (3.15), (3.17), (3.18), and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M121">View MathML</a> is nondecreasing, by induction, (3.12) holds.

On the other hand, from (2.5) and (2.7), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M122">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M123">View MathML</a>

(3.19)

Then from (3.16) and (3.19), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M124">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M125">View MathML</a>

Therefore, for all natural numbers n and p, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M126">View MathML</a>

(3.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M127">View MathML</a>

(3.21)

From (3.20) and (3.21), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M86">View MathML</a> such that (3.13) holds, and Claim 3.4 holds.

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M102">View MathML</a> in (3.11), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M103">View MathML</a>, which is a symmetric positive solution of BVP (1.1), and this completes the proof of the theorem.  □

Remark 3.1[3,5] only considered that f is nondecreasing or nonincreasing in u, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M132">View MathML</a> in (1.1), so our results extend the work in the literature.

Example 3.1 Consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M133">View MathML</a>

(3.22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M134">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M138">View MathML</a>.

It is easy to see that function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M8">View MathML</a> satisfies (H1) and (H3). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M36">View MathML</a>, there exists constant λ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M141">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/78/mathml/M47">View MathML</a>, so (H2) is also satisfied. Therefore, from Theorem 3.1, (3.22) has at least one symmetric positive solution.

Competing interests

The author declares that she has no competing interests.

Acknowledgements

Research supported by the Scientific Research Fund of Hunan Provincial Education Department (13C319).

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