Open Access Research

Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions

Huihui Pang1* and Yulong Tong2

Author Affiliations

1 College of Science, China Agricultural University, Beijing, 100083, P.R. China

2 The School of Statistics, Renmin University of China, Beijing, 100872, P.R. China

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Boundary Value Problems 2013, 2013:150  doi:10.1186/1687-2770-2013-150


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/150


Received:15 February 2013
Accepted:3 June 2013
Published:25 June 2013

© 2013 Pang and Tong; 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

This paper investigates the existence of concave symmetric positive solutions and establishes corresponding iterative schemes for a second-order boundary value problem with integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.

Keywords:
integral boundary conditions; iterative; monotone positive solution; symmetric; completely continuous

1 Introduction

The theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlinear problems with integral boundary conditions; we refer readers to [1-3] for examples and references.

At the same time, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary value problems as special cases.

Hence, increasing attention is paid to boundary value problems with integral boundary conditions [4-8]. Generally, the Guo-Krasnosel’ skii fixed point theorem in a cone, the Leggett-Williams fixed point theorem, the method of upper and lower solutions and the monotone iterative technique play extremely important roles in proving the existence of solutions to boundary value problems. In particular, we would like to mention some excellent results.

In [4], Ma studied the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M2">View MathML</a>, h and f are continuous. The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones.

In 2010, Wang et al.[7] considered the second-order boundary value problem with the integral boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M3">View MathML</a>

where ϕ, f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M5">View MathML</a> are continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M7">View MathML</a> are nonnegative constants. The existence result was obtained by applying the method of upper and lower solutions and Leray-Schauder degree theory. Theorem 1 (see [7]) supposed that the upper and lower solutions exist, and then, the theory of differential inequalities was used to prove that there is a solution to the boundary value problem between the upper and lower solutions.

Different from [7], [9] is not based on the assumption that the upper and lower solutions to the boundary value problem should exist, but constructs the specific form of the symmetric upper and lower solutions. The author in [9] investigated a second-order Sturm-Liouville boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M8">View MathML</a>

And by applying monotone iterative techniques, author proved the existence of n symmetric positive solutions.

To the best of our knowledge, no contribution exists concerning the existence of solutions for a boundary value problem with integral boundary conditions by applying monotone iterative techniques. Inspired by the work mentioned above, we concentrate on the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M9">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M10">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M11">View MathML</a>. We construct a specific form of the symmetric upper and lower solutions, and by applying monotone iterative techniques, we construct successive iterative schemes for approximating solutions.

The difficulty of this paper is that the nonlinear term f depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M12">View MathML</a>, which leads to complexities to prove the properties of the operator T, especially the monotonicity of the operator T. In Lemma 2.2, we skillfully use the cone’s character to overcome the mentioned obstacle. In addition, it is worth stating that the first term of our iterative scheme is a simple function or a constant function. Therefore, the iterative scheme is feasible. Under the appropriate assumptions on nonlinear term, this paper aims to establish a new and general result on the existence of a symmetric positive solution to BVP (1.1) and (1.2).

2 Preliminaries

Definition 2.1 Let E be a Banach space, a nonempty convex closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M13">View MathML</a> is said to be a cone provided the following hypotheses are satisfied:

(i) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M15">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M16">View MathML</a>;

(ii) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M18">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M19">View MathML</a>.

Every cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M13">View MathML</a> induces a partial ordering ‘⩽’ on E defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M21">View MathML</a>

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M22">View MathML</a> be an ordered Banach space. An operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M23">View MathML</a> is said to be nondecreasing (nonincreasing) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M24">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M25">View MathML</a>) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M26">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M27">View MathML</a>. If the inequality is strict, then φ is said to be strictly nondecreasing (nonincreasing).

Definition 2.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M29">View MathML</a> is said to be concave on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M30">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M31">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M33">View MathML</a>.

We consider the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M28">View MathML</a> equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M36">View MathML</a>. In this paper, a symmetric positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M37">View MathML</a> of (1.1) means a function which is symmetric and positive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M38">View MathML</a> and satisfies equation (1.1) as well as the boundary conditions (1.2).

In this paper, we always suppose that the following assumptions hold:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M41">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M42">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M43">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M44">View MathML</a> is nondecreasing for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M46">View MathML</a> is nondecreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M47">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M11">View MathML</a> is nonnegative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M49">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M50">View MathML</a>.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M51">View MathML</a>

It is easy to see that P is a cone in E.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M52">View MathML</a>, suppose that u is a solution of the following BVP:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M53">View MathML</a>

Then we can easily get the solution:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M54">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M55">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M56">View MathML</a>

During the process of getting the above solution, we can also know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M57">View MathML</a>

(2.2)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58">View MathML</a>.

Lemma 2.1If (H3) is satisfied, the following results are true:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M59">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M60">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M61">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M62">View MathML</a>.

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M64">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M60">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M67">View MathML</a> is defined

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M68">View MathML</a>

(2.3)

Lemma 2.2If (H3) is satisfied, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M69">View MathML</a>is completely continuous, i.e., T is continuous and compact. Moreover, Tis nondecreasing provided that (H2) holds.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M70">View MathML</a>, from the definition of Ty, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M71">View MathML</a>

Obviously, Ty is concave. From the expression of Ty, combining with Lemma 2.1, we know that Ty is nonnegative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M30">View MathML</a>. We now prove that Ty is symmetric about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M75">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M76">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M77">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M78">View MathML</a>. The continuity of T is obvious. We now prove that T is compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M79">View MathML</a> be a bounded set. Then there exists R such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M80">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M81">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M82">View MathML</a>

Therefore, from (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M83">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M84">View MathML</a> is uniformly bounded. Now we prove Ty is equi-continuous. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M85">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M86">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M87">View MathML</a>

And the similar results can be obtained for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M89">View MathML</a>.

The Arzelà-Ascoli theorem guarantees that TΩ is relatively compact, which means T is compact.

Finally, we show that Ty is nondecreasing about y.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M90">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M91">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M92">View MathML</a>. By the properties of a cone, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M93">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M95">View MathML</a> is concave and symmetric about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M97">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74">View MathML</a>, by (H2) and the definition of Ty, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M99">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M100">View MathML</a>

In order to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M101">View MathML</a> is concave, we need to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M102">View MathML</a> is nonincreasing. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M103">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M104">View MathML</a>

A similar result can be obtained for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M41">View MathML</a>. And it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M101">View MathML</a> is symmetric about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M73">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M108">View MathML</a> and thus T is nondecreasing. □

3 Existence and iterative of solutions for BVP (1.1) and (1.2)

Theorem 3.1Assume that (H1)-(H3) hold. If there exist two positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M109">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M110">View MathML</a>

(3.1)

whereaand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M111">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M112">View MathML</a>

(3.2)

then problem (1.1) and (1.2) has a concave symmetric positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M113">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M114">View MathML</a>

Proof We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M115">View MathML</a>. In what follows, we first prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M117">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M119">View MathML</a>.

By assumption (H2) and (3.1), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M74">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M121">View MathML</a>

(3.3)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M122">View MathML</a>, by Lemma 2.2, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M123">View MathML</a> and, as a result,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M124">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M125">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M126">View MathML</a>. Thus, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M128">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M131">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M132">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M133">View MathML</a>. We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M134">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M116">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M136">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137">View MathML</a>). From Lemma 2.2, T is compact, we assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M138">View MathML</a> has a convergent subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M139">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M140">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M141">View MathML</a>. From the definition of T, (3.1) and (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M142">View MathML</a>

On the other hand, we notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M143">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M144">View MathML</a>. By Lemma 2.2, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M145">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147">View MathML</a>. By induction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137">View MathML</a>). Hence, we assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M151">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M152">View MathML</a> in (3.4) to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M153">View MathML</a> since T is continuous. It is well known that the fixed point of the operator T is the solution of BVP (1.1) and (1.2). Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M154">View MathML</a> is a concave symmetric positive solution of BVP (1.1) and (1.2).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M157">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M158">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M159">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M160">View MathML</a>

(3.5)

Similarly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M161">View MathML</a>, we assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M161">View MathML</a> has a convergent subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M163">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M164">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M165">View MathML</a>. Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M166">View MathML</a>, by Lemma 2.2, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M167">View MathML</a>, which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147">View MathML</a>. By induction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M147">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M137">View MathML</a>). Hence, we assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M174">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M176">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M177">View MathML</a> is a concave symmetric positive solution of BVP (1.1) and (1.2). □

Remark The existence of a solution under the assumptions of Theorem 3.1 is just a consequence of Schauder’s fixed point theorem. The monotone iterative technique adds the information about the approximation sequences.

Example Consider the following second-order boundary value problem with integral boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M178">View MathML</a>

(3.6)

And we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M179">View MathML</a>

It is easy to check that the assumptions (H1)-(H3) hold and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M180">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M182">View MathML</a>. Then we can verify that condition (3.1) is satisfied. Then applying Theorem 3.1, BVP (3.6) has a concave symmetric positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M113">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/150/mathml/M184">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Chinese Universities Scientific Fund (Project No. 2013QJ004).

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