SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Upper and lower solution method for nth-order BVPs on an infinite interval

Hairong Lian1*, Junfang Zhao1 and Ravi P Agarwal2

Author Affiliations

1 School of Science, China University of Geosciences, Beijing, 100083, PR China

2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, Texas, 78363, USA

For all author emails, please log on.

Boundary Value Problems 2014, 2014:100  doi:10.1186/1687-2770-2014-100


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/100


Received:9 January 2014
Accepted:7 April 2014
Published:7 May 2014

© 2014 Lian 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 credited.

Abstract

This work is devoted to the study of nth-order ordinary differential equations on a half-line with Sturm-Liouville boundary conditions. The existence results of a solution, and triple solutions, are established by employing a generalized version of the upper and lower solution method, the Schäuder fixed point theorem, and topological degree theory. In our problem the nonlinearity depends on derivatives, and we allow solutions to be unbounded, which is an extra interesting feature.

MSC: 34B15, 34B40.

Keywords:
higher order; boundary value problem; half-line; upper solution; lower solution

1 Introduction

In this paper, we study nth-order ordinary differential equations on a half-line,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M1">View MathML</a>

(1)

together with the Sturm-Liouville boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M2">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M4">View MathML</a> are continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M8">View MathML</a>.

Higher-order boundary value problems (BVPs) have been studied in many papers, such as [1-3] for two-point BVP, [4,5] for multipoint BVP, and [6-9] for infinite interval problem. However, most of these works have been done either on finite intervals, or for bounded solutions on an infinite interval. The authors in [1,2,5,10-15] assumed one pair of well-ordered upper and lower solutions, and then applied some fixed point theorems or a monotone iterative technique to obtain a solution. In [5,11,16], the authors assumed two pairs of upper and lower solutions and showed the existence of three solutions.

Infinite interval problems occur in the study of radially symmetric solutions of nonlinear elliptic equations; see [6,7,17]. A principal source of such problems is fluid dynamics. In boundary layer theory, Blasius-type equations lead to infinite interval problems. Semiconductor circuits and soil mechanics are other applied fields. In addition, some singular boundary value problems on finite intervals can be converted into equivalent nonlinear problems on semi-infinite intervals [7]. During the last few years, fixed point theorems, shooting methods, upper and lower technique, etc. have been used to prove the existence of a single solution or multiple solutions to infinite interval problems; see [6-13,17-24] and the references therein.

When applying the upper and lower solution method to infinite interval problems, the solutions are always assumed to be bounded. For example, in [10], Agarwal and O’Regan discussed the following second-order Sturm-Liouville boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M11">View MathML</a>. They established existence criteria by using a diagonalization argument and existence results of appropriate boundary value problems on finite intervals.

Eloe et al.[11] studied the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M12">View MathML</a>

They employed the technique of lower and upper solutions and the theory of fixed point index to obtain the existence of at least three solutions.

The problems related to global solutions, especially when the boundary data are prescribed asymptotically and the solutions may be unbounded, have been briefly discussed in [7,9]. Recently, Yan et al.[14] developed the upper and lower solution theory for the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M15">View MathML</a>. By using the upper and lower solutions method and a fixed point theorem, they presented sufficient conditions for the existence of unbounded positive solutions; however, their results are suitable only to positive solutions. In [12,13], Lian et al. generalized their existence results to unbounded solutions, and somewhat weakened the conditions in [14]. In 2012, Zhao et al.[15] similarly investigated the solutions to multipoint boundary value problems in Banach spaces on an infinite interval.

Inspired by the works listed above, in this paper, we aim to discuss the nth-order differential equation on a half-line with Sturm-Liouville boundary conditions. To the best of our knowledge, this is the first attempt to find the unbounded solutions to higher-order infinite interval problems by using the upper and lower solution technique. Since, the half-line is noncompact, the discussion is rather involved. We begin with the assumption that there exist a pair of upper and lower solutions for problem (1)-(2), and the nonlinear function f satisfies a Nagumo-type condition. Then, by using the truncation technique and the upper and lower solutions, we estimate a-priori bounds of modified problems. Next, the Schäuder fixed point theorem is used which guarantees the existence of solutions to (1)-(2). We also assume two pairs of upper and lower solutions and show that this infinite interval problem has at least three solutions. In the last section, an example is included which illustrates the main result.

2 Preliminaries

In this section, we present some definitions and lemmas to be used in the main theorem of this paper.

Consider the space X defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M16">View MathML</a>

(3)

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M17">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M18">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M19">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M20">View MathML</a>

(5)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M21">View MathML</a> is a Banach space.

To obtain a solution of the BVP (1)-(2), we need a mapping whose kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M22">View MathML</a> is the Green function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M23">View MathML</a> with the homogeneous boundary conditions (2), which is given in the following lemma.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M24">View MathML</a>. Then the linear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M25">View MathML</a>

(6)

has a unique solution given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M26">View MathML</a>

(7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M27">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M28">View MathML</a>

(8)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M29">View MathML</a>. Then from (6), we obtain the Sturm-Liouville boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M30">View MathML</a>

(9)

and the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M31">View MathML</a>

(10)

Clearly, (9) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M32">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M33">View MathML</a>

Now integrating (10) and applying the initial conditions, we obtain (7). □

Lemma 2.2The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M22">View MathML</a>defined in (8) is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M35">View MathML</a>times continuously differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M36">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M37">View MathML</a>, itsith derivative is uniformly continuous inton any compact interval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>and is uniformly bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>.

Proof We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M40">View MathML</a>, which is also used in the later part of the paper. By direct calculations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M41">View MathML</a>

(11)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M42">View MathML</a> is uniformly continuous in t on any compact interval of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M37">View MathML</a>. Now since, for all integers k and l,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M45">View MathML</a>

from (11), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M46">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M47">View MathML</a>

and, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M48">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M49">View MathML</a>

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M50">View MathML</a>

(12)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M51">View MathML</a>. □

When applying the Schäuder fixed point theorem to prove the existence result, it is necessary to show that the operator is completely continuous. While the usual Arezà-Ascoli lemma fails here due to the non-compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>, the following generalization (see [6,13]) will be used.

Lemma 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M53">View MathML</a>is relatively compact if the following conditions hold:

1. all the functions fromMare uniformly bounded;

2. all the functions fromMare equi-continuous on any compact interval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>;

3. all the functions fromMare euqi-convergent at infinity, that is, for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M55">View MathML</a>, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M56">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M57">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M58">View MathML</a>

Finally, we define lower and upper solutions of (1)-(2), and introduce the Nagumo-type condition.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M59">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M60">View MathML</a>

(13)

is called a lower solution of (1)-(2). If the inequalities are strict, it is called a strict lower solution.

Definition 2.2 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M61">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M62">View MathML</a>

(14)

is called an upper solution of (1)-(2). If the inequalities are strict, it is called a strict upper solution.

Definition 2.3 Let α, β be the lower and upper solutions of BVP (1)-(2) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M63">View MathML</a>

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>. We say f satisfies a Nagumo condition with respect to α and β if there exist positive functions ψ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M65">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M66">View MathML</a>

(15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M67">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M68">View MathML</a>

3 The existence results

Our existence theory is based on using the unbounded lower and upper solution technique. Here we list some assumptions for convenience.

H1: BVP (1)-(2) has a pair of upper and lower solutions β, α in X with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M69">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M70">View MathML</a> satisfies the Nagumo condition with respect to α and β.

H2: For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M72">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M74">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M75">View MathML</a>

H3: There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M77">View MathML</a>

where ψ is the function in Nagumo’s condition of f.

Lemma 3.1Suppose conditions (H1) and (H3) hold. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78">View MathML</a>such that every solutionuof (1)-(2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M79">View MathML</a>

satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M80">View MathML</a>.

Proof Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M81">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M82">View MathML</a> is any arbitrary constant. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M83">View MathML</a> where C is the nonhomogeneous boundary data, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M84">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M85">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M86">View MathML</a> holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>, then the result follows immediately. If not, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88">View MathML</a> does not hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>. Otherwise, without loss of generality, we suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M90">View MathML</a>

But then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M91">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M92">View MathML</a>

which is a contraction. So there must exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M93">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M94">View MathML</a>. Furthermore, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M95">View MathML</a>

just take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M96">View MathML</a> and then the proof is completed. Finally, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M97">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M100">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M103">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M100">View MathML</a>. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M107">View MathML</a>

from which one concludes that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M108">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M110">View MathML</a> are arbitrary, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M111">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M112">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>. In a similarly way, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M114">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M115">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>.

Therefore there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78">View MathML</a>, just related with α, β, and ψ, h under the Nagumo condition of f, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M118">View MathML</a>. □

Remark 3.1 Similarly, we can prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M119">View MathML</a>

Remark 3.2 The Nagumo condition plays a key role in estimating the prior bound for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M35">View MathML</a>th derivative of the solution of BVP (1)-(2). Since the upper and lower solutions are in X, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M122">View MathML</a> may be asymptotic linearly at infinity.

Theorem 3.2Suppose the conditions (H1)-(H3) hold. Then BVP (1)-(2) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M123">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M124">View MathML</a>

Moreover, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M80">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M78">View MathML</a> be the same as in Lemma 3.1. Define the auxiliary functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M128">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M129">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M130">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M131">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M132">View MathML</a>

Consider the modified differential equation with the truncated function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M133">View MathML</a>

(16)

with the boundary conditions (2). To complete the proof, it suffices to show that problem (16)-(2) has at least one solution u satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M134">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M135">View MathML</a>

(18)

We divide the proof into the following two steps.

Step 1: By contradiction we shall show that every solution u of problem (16)-(2) satisfy (17) and (18).

Suppose the right hand inequality in (17) does not hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M136">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M137">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M138">View MathML</a>

Case I. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M139">View MathML</a>.

Obviously, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M140">View MathML</a>. By the boundary conditions, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M141">View MathML</a>

which is a contradiction.

Case II. There exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M142">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M143">View MathML</a>.

Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M144">View MathML</a>

(19)

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M145">View MathML</a>

Subcase i. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M146">View MathML</a>, from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M147">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M148">View MathML</a>

Subcase ii. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M149">View MathML</a>, from the conditions (H2), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M150">View MathML</a>

Similarly following the above argument, we could discuss the other two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M151">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M153">View MathML</a>, and we have the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M154">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M155">View MathML</a>

which contradicts (19).

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>. Similarly, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M71">View MathML</a>. Integrating this inequality and using the boundary conditions in (2), (13), and (14), we obtain the inequality (17). Inequality (18) then follows from Lemma 3.1. In conclusion, u is the required solution to BVP (1)-(2).

Step 2. Problem (16)-(2) has a solution u.

Consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M161">View MathML</a>

(20)

Lemma 2.1 shows that the fixed points of T are the solutions of BVP (16)-(2). Next we shall prove that T has at least one fixed point by using the Schäuder fixed point theorem. For this, it is enough to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is completely continuous.

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is well defined.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M164">View MathML</a>, by direct calculation, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M165">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M166">View MathML</a>. Further, because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M167">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M168">View MathML</a>, the Lebesgue dominated convergent theorem implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M169">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M170">View MathML</a>.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is continuous.

For any convergent sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M172">View MathML</a> in X, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M173">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M174">View MathML</a>. Thus, as in (21), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M175">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is continuous.

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is compact.

For this it suffices to show that T maps bounded subsets of X into relatively compact sets. Let B be any bounded subset of X, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M180">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M182">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M183">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M184">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M185">View MathML</a>, and thus TB is uniformly bounded. Further, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M186">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M187">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M188">View MathML</a>

that is, TB is equi-continuous. From Lemma 2.3, it follows that if TB is equi-convergent at infinity, then TB is relatively compact. In fact, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M189">View MathML</a>

and, therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M160">View MathML</a> is completely continuous. The Schäuder fixed point theorem now ensures that the operator T has a fixed point, which is a solution of the BVP (1)-(2). □

Remark 3.3 The upper and lower solutions are more strict at infinity than usually assumed. For such right boundary conditions, it is not easy to estimate the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M191">View MathML</a> even though we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M193">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M194">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M195">View MathML</a>. It remains unsettled whether this strict inequality can be weakened.

4 The multiplicity results

In this section assuming two pairs of upper and lower solutions, we shall prove the existence of at least three solutions for our infinite interval problem.

Theorem 4.1Suppose that the following condition holds.

H4: BVP (1)-(2) has two pairs of upper and lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M198">View MathML</a>inXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200">View MathML</a>strict, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M201">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M203">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M204">View MathML</a>satisfies the Nagumo condition with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206">View MathML</a>.

Suppose further that conditions (H2) and (H3) hold withαandβreplaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206">View MathML</a>, respectively. Then the problem (1)-(2) has at least three solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M210">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M211">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M212">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M203">View MathML</a>.

Proof Define the truncated function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M215">View MathML</a>, the same as F in Theorem 3.2 with α replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205">View MathML</a> and β by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206">View MathML</a>, respectively. Consider the modified differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M218">View MathML</a>

(22)

with boundary conditions (2). Similarly to Theorem 3.2, it suffices to show that problem (22)-(2) has at least three solutions. To this end, define the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M219">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M220">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221">View MathML</a> is completely continuous. By using the degree theory, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221">View MathML</a> has at least three fixed points which coincide with the solutions of (22)-(2).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M223">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M225">View MathML</a> are defined as above, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M226">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M227">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M228">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M229">View MathML</a>

and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M230">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M231">View MathML</a>.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M232">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M235">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M236">View MathML</a>

Now since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200">View MathML</a> are strict lower and upper solutions, there is no solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M239">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M240">View MathML</a>

Next we will show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M241">View MathML</a>

For this, we define another mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M242">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M243">View MathML</a>

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M244">View MathML</a> is similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M215">View MathML</a> except changing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199">View MathML</a>. Similar to the proof of Theorem 3.2, we find that u is a fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M248">View MathML</a> only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M249">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M250">View MathML</a>. From the Schäuder fixed point theorem and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M251">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M252">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M253">View MathML</a>

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M254">View MathML</a>, and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M255">View MathML</a>

Finally, using the properties of the degree, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M221">View MathML</a> has at least three fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M258">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M259">View MathML</a>. □

5 An example

Example Consider the second-order differential equation with Sturm-Liouville boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M260">View MathML</a>

(23)

Clearly, BVP (23) is a particular case of problem (1)-(2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M261">View MathML</a>

We let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M262">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M263">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M267">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M268">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M269">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M199">View MathML</a> are strict lower solutions of problem (23).

Now we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M272">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M274">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M275">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M276">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M206">View MathML</a> are strict upper solutions of problem (23). Further, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M279">View MathML</a>

Moreover, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M280">View MathML</a>, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M281">View MathML</a> is bounded. Finally, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M283">View MathML</a>. Hence, all conditions in Theorem 4.1 are satisfied and therefore problem (23) has at least three solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11101385) and by the Fundamental Research Funds for the Central Universities.

References

  1. Graef, JR, Kong, L, Mihós, FM: Higher order ϕ-Laplacian BVP with generalized Sturm-Liouville boundary conditions. Differ. Equ. Dyn. Syst.. 18(4), 373–383 (2010). Publisher Full Text OpenURL

  2. Graef, JR, Kong, L, Mihós, FM, Fialho, J: On the lower and upper solution method for higher order functional boundary value problems. Appl. Anal. Discrete Math.. 5, 133–146 (2011). Publisher Full Text OpenURL

  3. Grossinho, MR, Minhós, FM: Existence result for some third order separated boundary value problems. Nonlinear Anal.. 47, 2407–2418 (2001). Publisher Full Text OpenURL

  4. Bai, Z: Positive solutions of some nonlocal fourth-order boundary value problem. Appl. Math. Comput.. 215, 4191–4197 (2010). Publisher Full Text OpenURL

  5. Du, Z, Liu, W, Lin, X: Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations. J. Math. Anal. Appl.. 335, 1207–1218 (2007). Publisher Full Text OpenURL

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

  7. Countryman, M, Kannan, R: Nonlinear boundary value problems on semi-infinite intervals. Comput. Math. Appl.. 28, 59–75 (1994). Publisher Full Text OpenURL

  8. Philos, CG: A boundary value problem on the half-line for higher-order nonlinear differential equations. Proc. R. Soc. Edinb.. 139A, 1017–1035 (2009)

  9. Umamahesearam, S, Venkata Rama, M: Multipoint focal boundary value problems on infinite interval. J. Appl. Math. Stoch. Anal.. 5, 283–290 (1992). Publisher Full Text OpenURL

  10. Agarwal, RP, O’Regan, D: Nonlinear boundary value problems on the semi-infinite interval: an upper and lower solution approach. Mathematika. 49, 129–140 (2002). Publisher Full Text OpenURL

  11. Eloe, PW, Kaufmann, ER, Tisdell, CC: Multiple solutions of a boundary value problem on an unbounded domain. Dyn. Syst. Appl.. 15(1), 53–63 (2006)

  12. Lian, H, Wang, P, Ge, W: Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals. Nonlinear Anal.. 70, 2627–2633 (2009). Publisher Full Text OpenURL

  13. Lian, H, Zhao, J: Existence of unbounded solutions for a third-order boundary value problem on infinite intervals. Discrete Dyn. Nat. Soc. (2012). Publisher Full Text OpenURL

  14. Yan, B, O’Regan, D, Agarwal, RP: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. J. Comput. Appl. Math.. 197, 365–386 (2006). Publisher Full Text OpenURL

  15. Zhao, Y, Chen, H, Xu, C: Existence of multiple solutions for three-point boundary-value problems on infinite intervals in Banach spaces. Electron. J. Differ. Equ.. 44, 1–11 (2012)

  16. Ehme, J, Eloe, PW, Henderson, J: Upper and lower solution methods for fully nonlinear boundary value problems. J. Differ. Equ.. 180, 51–64 (2002). Publisher Full Text OpenURL

  17. Agarwal, RP, O’Regan, D: Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory. Stud. Appl. Math.. 111, 339–358 (2003). Publisher Full Text OpenURL

  18. Baxley, JV: Existence and uniqueness for nonlinear boundary value problems on infinite interval. J. Math. Anal. Appl.. 147, 122–133 (1990). Publisher Full Text OpenURL

  19. Chen, S, Zhang, Y: Singular boundary value problems on a half-line. J. Math. Anal. Appl.. 195, 449–468 (1995). Publisher Full Text OpenURL

  20. Gross, OA: The boundary value problem on an infinite interval, existence, uniqueness and asymptotic behavior of bounded solutions to a class of nonlinear second order differential equations. J. Math. Anal. Appl.. 7, 100–109 (1963). Publisher Full Text OpenURL

  21. Jiang, D, Agarwal, RP: A uniqueness and existence theorem for a singular third-order boundary value problem on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/100/mathml/M38">View MathML</a>. Appl. Math. Lett.. 15, 445–451 (2002). Publisher Full Text OpenURL

  22. Liu, Y: Boundary value problems for second order differential equations on infinite intervals. Appl. Math. Comput.. 135, 211–216 (2002)

  23. Ma, R: Existence of positive solution for second-order boundary value problems on infinite intervals. Appl. Math. Lett.. 16, 33–39 (2003). Publisher Full Text OpenURL

  24. Bai, C, Li, C: Unbounded upper and lower solution method for third-order boundary-value problems on the half-line. Electron. J. Differ. Equ.. 119, 1–12 (2009)