Open Access Research

Application of the shooting method to second-order multi-point integral boundary-value problems

Huilan Wang*, Zigen Ouyang and Liguang Wang

Author Affiliations

School of Mathematics and Physics, University of South China, Hengyang, 421001, P.R. China

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Boundary Value Problems 2013, 2013:205  doi:10.1186/1687-2770-2013-205


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/205


Received:3 March 2013
Accepted:8 August 2013
Published:9 September 2013

© 2013 Wang et al.; licensee Springer

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

Abstract

In this paper, we focus on the following second-order multi-point integral boundary-value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5">View MathML</a> are given constants. The proof is based on the shooting method. By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6">View MathML</a> and the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7">View MathML</a>. The method is concise and some new criteria are established.

MSC: 34B10, 34B15, 34B18.

Keywords:
shooting method; integral boundary-value problem; positive solution

1 Introduction

For the study of nonlinear second-order multi-point boundary-value problem, many results have been obtained by using all kinds of fixed point theorems related to a completely continuous map defined in a Banach space. We refer the reader to [1-9] and the references therein. Some of the results are so classical that little work can exceed; however, most of these papers are concerned with problems with boundary conditions of restrictions either on the slope of solutions and the solutions themselves, or on the number of boundary points [2,5-8,10].

In [8], Ma investigated the existence of positive solutions of the nonlinear second-order m-point boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M8">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M9">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M15">View MathML</a>, and there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M16">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M17">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M18">View MathML</a>

The author obtained the existence of a positive solution to (1.1)-(1.2) under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M20">View MathML</a> (super-linear case) or the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22">View MathML</a> (sub-linear case) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M23">View MathML</a>.

Recently, Tariboon [9] considered three-point boundary-value problem (1.1) with the integral boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M24">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M26">View MathML</a>.

Such a boundary condition might be more realistic in the mathematical models of thermal conductivity, groundwater flow, thermoelectric flexibility and plasma physics, because it describes the fluid properties in a certain continuous medium. Under the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M27">View MathML</a>, Tariboon and the author proved that problem (1.1)-(1.3) has at least one positive solution in the super-linear case or in the sub-linear one.

However, the method used in the previous two papers is Krasnoselskii’s fixed point theorem in a cone, which relates to constructing a completely continuous cone map in a Banach space, and the proof is somewhat procedural.

Constructively, Agarwal [11] explored the solution of multi-point boundary value problems by converting BVPs to equivalent IVPs, which is called shooting method. After that Man Kam Kwong [4,12] used the shooting method to consider second-order multi-point boundary value problems. In [12], Kwong studied the existence of a positive solution to the following three-point boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M28">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M29">View MathML</a>

(1.5)

The principle of the shooting method used in [12] is converting BVP (1.4)-(1.5) into finding suitable initial slopes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M30">View MathML</a> such that the solution of equation (1.4) with the initial value condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M31">View MathML</a>

(1.6)

vanishes for the first time after <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M32">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33">View MathML</a> the solution of (1.4)-(1.6) provided it exists. Then solving the boundary value problem is equivalent to finding m such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M34">View MathML</a>

If we can find two solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36">View MathML</a> of (1.4) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M37">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M41">View MathML</a>, then there must exist a number m between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33">View MathML</a> is the solution of (1.4)-(1.5). By constructing two sine functions as the shooting objects and combining with the comparison principle, the author obtained some better results than those via fixed point techniques for the existence of positive solutions to (1.4)-(1.5).

In this paper, we try to employ the shooting method to establish the existence results of positive solutions for (1.1) with the more generalized multi-point integral boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M45">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M3">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M5">View MathML</a> are given constants. Following the principle of the shooting method, there are two obstacles we encounter. The first one is that the boundary condition involves integral from 0 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M50">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M51">View MathML</a>), so we transform the integral problem into a single-point problem by using the integral mean value theorem. The other difficulty is that we cannot obtain the existence results by constructing two sine functions as in [12] because of the particularity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M52">View MathML</a> in [12]. Therefore, we construct a quadratic function and a sine function as the objective ones.

The purpose of this article lies in two aspects. One is to explore the application of the shooting method in a more complicated multi-point integral boundary value problem, which demonstrates another way in studying BVPs. The other one is to establish new criteria for the existence of positive solutions to (1.1)-(1.7) under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M6">View MathML</a> and the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M7">View MathML</a>.

For the sake of convenience, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M55">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33">View MathML</a> be the solution of (1.1)-(1.6) and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M57">View MathML</a>

(1.8)

In this paper, we always assume:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M60">View MathML</a>.

Under the assumption, it is not difficult to prove that the initial problem (1.1)-(1.6) has at least one solution defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M61">View MathML</a>. In fact, after translating second-order differential equation (1.1) into one-order equations, one can draw the conclusion [13].

Further, we introduce the comparison results derived from [4,12], which evolved from the Sturm comparison theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M64">View MathML</a>be the solution of the initial value problems, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M65">View MathML</a>

and suppose thatF, G, gare nonnegative continuous functions on a certain intervalIfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66">View MathML</a>and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M67">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M68">View MathML</a>does not vanish in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M69">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M25">View MathML</a>, it yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M71">View MathML</a>

The paper is arranged as follows. In the next section, we put forward the basic principle of the shooting method used in this paper, and show that BVP (1.1)-(1.7) has no positive solution when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M72">View MathML</a>. In Section 3, the general criteria are established for the existence of positive solutions to (1.1)-(1.7) under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M73">View MathML</a>. Moreover, we present the special results in the form of corollaries corresponding to the super-linear case or the sub-linear case. Finally, we come to the conclusion and an example is presented to illustrate our results.

2 Preliminaries

Lemma 2.1If there exist two initial slopes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M74">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M75">View MathML</a>such that

(i) the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35">View MathML</a>of (1.1)-(1.6) remains positive in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M78">View MathML</a>;

(ii) the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36">View MathML</a>of (1.1)-(1.6) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M40">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M41">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M82">View MathML</a>; then multi-point boundary value problem (1.1)-(1.7) has a positive solution with the slope<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M83">View MathML</a>between<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a>.

Proof Since the solutions of (1.1)-(1.6) depend on the initial value continuously, then from (1.8), it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M86">View MathML</a> is continuous on m. In view of the intermediate value theorem of continuous functions, there exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M87">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M90">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M91">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M92">View MathML</a> is the solution of (1.1)-(1.7). □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M72">View MathML</a>, then (1.1)-(1.7) has no positive solution.

Proof Assume that (1.1)-(1.7) has a positive solution u.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M94">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M95">View MathML</a>, the convexity of u implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M97">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M98">View MathML</a>

which contradicts with the convexity of u.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M99">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M100">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M101">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M102">View MathML</a>. If there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M104">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M104">View MathML</a>, which contradicts with the convexity of u. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M101">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66">View MathML</a>.

In the rest of this paper, we always assume:

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M109">View MathML</a>.

 □

3 Main results

Theorem 3.1Assume that (H1)-(H2) holds. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M110">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M111">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M112">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M113">View MathML</a>.

Then problem (1.1)-(1.7) has a positive solution.

Proof (i) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M114">View MathML</a>, we can choose a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M116">View MathML</a>

We claim that there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M118">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66">View MathML</a>. The claim is based on the convexity of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M35">View MathML</a> and the Sturm comparison theorem (see [12]). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M121">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M122">View MathML</a>

(3.1)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M123">View MathML</a>

From (1.8), (3.1) and combining the integral mean value theorem with Theorem 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M124">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M127">View MathML</a>.

The second inequality in (i) means that there exists a number M large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M128">View MathML</a>

For this M, there exist two numbers δ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M129">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M130">View MathML</a>

(3.3)

and there exists another number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M132">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M133">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M134">View MathML</a>

(3.4)

In view of (H2) and (3.3), it is not difficult to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M135">View MathML</a>

which implies from (3.4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M136">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M137">View MathML</a>. Thus, by the convexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36">View MathML</a> and Theorem 1.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M139">View MathML</a>

(3.5)

By Lemma 2.1 and (3.2)-(3.5), there exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M87">View MathML</a> between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M92">View MathML</a> is the positive solution of (1.1)-(1.7). The proof for (i) is complete.

Now, we prove for (ii).

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M144">View MathML</a>, we can choose a number N large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M145">View MathML</a>

For this N, there exist a number ϵ small enough and a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M42">View MathML</a> large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M148">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M149">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M150">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M151">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M152">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M148">View MathML</a> approximately for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M152">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M66">View MathML</a>. Similar to (3.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M158">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M161">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M162">View MathML</a>, then there exist two positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a> and σ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M164">View MathML</a>

By the convexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M36">View MathML</a>, for these σ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M43">View MathML</a>, there exists a positive number τ small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M167">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M168">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M169">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M170">View MathML</a>

(3.7)

From (3.6) and (3.7), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M136">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M137">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M174">View MathML</a>

By Lemma 2.1, the proof for (ii) is complete. □

Theorem 3.2Assume that (H1)-(H2) holds. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M175">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M176">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M177">View MathML</a>

Then problem (1.1)-(1.7) has a positive solution under the case

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M112">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M179">View MathML</a>.

Proof Note the computation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M180">View MathML</a> in Theorem 3.1. In (3.2), if we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M181">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M182">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M78">View MathML</a>, and all the steps in the following are the same as in Theorem 3.1. □

Now, let us consider the special super-linear case or the sub-linear case. It is not difficult to verify the following corollaries.

Corollary 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M184">View MathML</a>and

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M186">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22">View MathML</a>.

Then problem (1.1)-(1.7) has a positive solution.

Corollary 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M175">View MathML</a>and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M190">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M191">View MathML</a>

Then, problem (1.1)-(1.7) has a positive solution under the case

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M186">View MathML</a> ; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M22">View MathML</a>.

4 Conclusion and examples

The tool which we used for the analysis in this article is the shooting method derived from [4,12]; however, we considered a more general problem which involves integral boundary-value and multiplicity of boundary-point. The meaningful work that we have done lies in the following three aspects. The first one is that we transform the integral problem into a single-point value one by using the integral mean value theorem. The other one is that we construct a quadratic function and a sine function as the comparison functions because it does not take effect to construct two sine functions as in [12]. Finally, we established the new criteria for the existence of positive solutions to (1.1)-(1.7) under the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M196">View MathML</a> and the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M197">View MathML</a>. Obviously, (1.7) vanishes to (1.3) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M198">View MathML</a> and the sup-linear case or the sub-linear case is sufficient for the conditions in Theorem 3.1 and Theorem 3.2, so some of our results are more general or better than those via fixed point techniques. However, in Theorem 3.2, whether the transcendental equation has a solution is somewhat difficult to verify. It can be seen that each method has its pros and cons.

Example 4.1 Consider the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M199">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M200">View MathML</a>

(4.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M201">View MathML</a>

It is not difficult to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M202">View MathML</a>

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M203">View MathML</a>, Matlab software gives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M205">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M206">View MathML</a>

Therefore, the condition (ii) of Theorem 3.2 is satisfied. A numerical simulation (Figure 1) for Example 4.1 demonstrates that BVP (4.1)-(4.2) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M207">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/205/mathml/M208">View MathML</a>.

thumbnailFigure 1. Numerical simulation for Example 4.1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work was carried out in collaboration between all authors. HL practised the methods and organized this paper. ZG found the topic of this paper and suggested the methods. LG finished the Matlab program of numerical simulation. All authors have contributed to, seen and approved the manuscript.

Acknowledgements

The authors would like to thank the editors and the anonymous referees for their valuable suggestions on the improvement of this paper. First author was partially supported by the Scientific Research Fund of Hunan Provincial Educational Department (1200361), Project of Science and Technology Bureau of Hengyang, Hunan Province (2012KJ2). Second author was partially supported by the Doctor Foundation of University of South China ( No. 5-XQD-2006-9), the Foundation of Science and Technology Department of Hunan Province (No. 2009RS3019), the Natural Science Foundation of Hunan Province (No. 13JJ3074) and the Subject Lead Foundation of University of South China (No. 2007XQD13).

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