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Existence results for nonlinear boundary value problems with integral boundary conditions on an infinite interval

Fulya Yoruk* and Nuket Aykut Hamal

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Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey

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Citation and License

Boundary Value Problems 2012, 2012:127  doi:10.1186/1687-2770-2012-127


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/127


Received:29 June 2012
Accepted:22 October 2012
Published:5 November 2012

© 2012 Yoruk and Hamal; 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, by using fixed point theorems in a cone, the existence of one positive solution and three positive solutions for nonlinear boundary value problems with integral boundary conditions on an infinite interval are established.

MSC: 34B10, 39A10, 34B18, 45G10.

Keywords:
positive solutions; fixed point theorems; integral boundary conditions; infinite interval

1 Introduction

Consider the following boundary value problem with integral boundary conditions on the half-line of an infinite interval of the form

(1.1)

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M3">View MathML</a>, f may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M5">View MathML</a> are continuous, nondecreasing functions and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6">View MathML</a>, z in a bounded set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M8">View MathML</a> are bounded; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M9">View MathML</a> is a continuous function with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M11">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M16">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M18">View MathML</a> in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M19">View MathML</a>.

Boundary value problems on an infinite interval appear often in applied mathematics and physics. There are many papers concerning the existence of solutions on the half-line for boundary value problems; see [1-5] and the references therein.

At the same time, boundary value problems with integral boundary conditions are of great importance and are an interesting class of problems. They constitute two, three, multi-point, and nonlocal boundary value problems as special cases. For an overview of the literature on integral boundary value problems, see [6-11] and the references therein.

Yan Sun et al.[4] studied the existence of positive solutions for singular boundary value problems on the half-line for the following Sturm-Liouville boundary value problem:

where μ is a positive parameter; f is a continuous, non-negative function and may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M26">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M27">View MathML</a>. Wang et al.[5] investigated the existence theorems for the boundary value problem given by

where f is a continuous, non-negative function and may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M30">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M12">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M14">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M34">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M27">View MathML</a>. Also, Feng [11] considered the following boundary value problem with integral boundary conditions on a finite interval:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M37">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M40">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M41">View MathML</a> are symmetric functions; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M42">View MathML</a> is continuous. The author obtained the existence of symmetric positive solutions by using the fixed point index theory in cones.

Motivated by the above works, we consider the existence of one and three positive solutions for the BVP (1.1), (1.2). However, to our knowledge, although various existence theorems are obtained for Sturm-Liouville boundary value problems with homogeneous boundary conditions, problems with nonhomogeneous boundary conditions, especially integral boundary conditions on an infinite interval have rarely been considered. Therefore, our boundary conditions are more general.

The rest of the paper is organized as follows. In Section 2, we present some necessary lemmas that will be used to prove our main results. In Section 3, we apply the Schauder fixed point theorem to get the existence of at least one positive solution for the nonlinear boundary value problem (1.1) and (1.2). In Section 4, we use the Leggett-Williams fixed point theorem [12] to get the existence of at least three positive solutions for the nonlinear boundary value problem (1.1) and (1.2).

2 Preliminaries

In this section, we will employ several lemmas to prove the main results in this paper. These lemmas are based on the following BVP for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M43">View MathML</a>:

(2.1)

(2.2)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47">View MathML</a> to be the solutions of the corresponding homogeneous equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M48">View MathML</a>

(2.3)

under the initial conditions,

(2.4)

Using the initial conditions (2.4), we can deduce, from equation (2.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46">View MathML</a>, the following equations:

(2.5)

(2.6)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> be the Green function for (2.1), (2.2) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M55">View MathML</a>

(2.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M46">View MathML</a> are given in (2.5) and (2.6) respectively.

Lemma 2.1Suppose the conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M59">View MathML</a>hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M60">View MathML</a>, the BVP (2.1), (2.2) has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M61">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a>is given by (2.7).

Furthermore, it is easy to prove the following properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a>:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M65">View MathML</a>.

(2) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> is continuously differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M68">View MathML</a> except <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M69">View MathML</a>.

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M70">View MathML</a>.

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M71">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M72">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M73">View MathML</a>

(2.8)

(5) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> satisfies the corresponding homogeneous BVP (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M76">View MathML</a> in the BVP (2.1)) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M68">View MathML</a> except <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M69">View MathML</a>.

(6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M79">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M80">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M81">View MathML</a>

(7) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M66">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M84">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M85">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M86">View MathML</a>.

It is convenient to list the following conditions which are to be used in our theorems:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M87">View MathML</a> and also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M89">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M90">View MathML</a>; and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6">View MathML</a>, x, y in a bounded set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M92">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M93">View MathML</a> is continuous and may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4">View MathML</a>; and also, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M95">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M96">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M97">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M5">View MathML</a> are continuous, nondecreasing functions, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M6">View MathML</a>, z in a bounded set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M8">View MathML</a> are bounded.

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M102">View MathML</a> is a continuous function with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M103">View MathML</a>.

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M105">View MathML</a>.

Consider the Banach space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M106">View MathML</a>

with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M107">View MathML</a>.

From the above assumptions, we can define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M109">View MathML</a>

(2.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> is given by (2.7).

Lemma 2.2 ([13])

Letbe defined as before and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M111">View MathML</a>. ThenMis relatively compact inif the following conditions hold:

(a) Mis uniformly bounded in ℬ;

(b) The functions belonging toMare equicontinuous on any compact interval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M112">View MathML</a>;

(c) The functions fromMare equiconvergent, that is, given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M113">View MathML</a>, there corresponds a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M114">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M115">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M117">View MathML</a>.

Definition 2.1 An operator is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.

3 Existence of at least one positive solution

In this section, we will apply the following Schauder fixed point theorem to get an existence of one positive solution.

Theorem 3.1 (Schauder fixed point theorem)

Letbe a Banach space andSbe a nonempty bounded, convex, and closed subset of ℬ. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M118">View MathML</a>is a completely continuous operator. If the operatorAleaves the setSinvariant, i.e., if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M119">View MathML</a>, thenAhas at least one fixed point inS.

For convenience, let us set

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M121">View MathML</a>

Theorem 3.2Assume conditions (H1)-(H4) are satisfied. In addition, let there exist a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M122">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M123">View MathML</a>

wherecis defined by (2.8).

Then the BVP (1.1), (1.2) has at least one solutionzwith

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M124">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108">View MathML</a> be the operator defined by (2.9). We claim that A is a completely continuous operator. To justify this, we first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108">View MathML</a> is well defined. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M127">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M129">View MathML</a> and from conditions (H1) and (H2), we have

and

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M133">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M134">View MathML</a>

(3.1)

Hence, by the Lebesgue dominated convergence theorem and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M54">View MathML</a> is continuous on t, we have

(3.2)

Also, by (H1) and (H2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M137">View MathML</a>

(3.3)

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M138">View MathML</a>.

We can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M139">View MathML</a>. Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M140">View MathML</a>

In addition, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M141">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M142">View MathML</a>.

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108">View MathML</a> is well defined.

Next, for any positive integer m, we denote the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M144">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M145">View MathML</a>

(3.4)

and prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M144">View MathML</a> is completely continuous for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M147">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149">View MathML</a>. We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M150">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151">View MathML</a> in ℬ. We know that

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M153">View MathML</a> is a real number such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M154">View MathML</a>, N is a natural number set, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M155">View MathML</a>.

Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156">View MathML</a>, there exists a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M157">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M158">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M159">View MathML</a>

(3.5)

From the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151">View MathML</a>, we can see that for the above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156">View MathML</a>, there exists a sufficiently large natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M163">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M164">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M165">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M166">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M167">View MathML</a>

On the other hand, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M168">View MathML</a>, for the above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M170">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M173">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M175">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M176">View MathML</a>

(3.6)

From the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151">View MathML</a>, there exists a natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M179">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M180">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M183">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M185">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M186">View MathML</a>

(3.7)

In addition to this, by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M188">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M189">View MathML</a>, for the above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M156">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M170">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M172">View MathML</a>, such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M174">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M195">View MathML</a>

(3.8)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M151">View MathML</a>, there exists a natural number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M198">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M199">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M182">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M184">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M203">View MathML</a>

(3.9)

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M204">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M205">View MathML</a>

Similarly, we can see that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M148">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M208">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M149">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M210">View MathML</a> is a continuous operator for each natural number m.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M211">View MathML</a> to be a bounded, convex, and closed set by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M212">View MathML</a>

We must show that there exists a positive constant R such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M213">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M214">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M215">View MathML</a>. Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M217">View MathML</a>. Since f, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M219">View MathML</a> are positive functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>. Furthermore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M223">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M224">View MathML</a>

(3.11)

Inequalities (3.10) and (3.11) yield that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M225">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M226">View MathML</a> is uniformly bounded. Using the similar proof as (3.2) and (3.3), we can obtain that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M213">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M229">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M230">View MathML</a> is equicontinuous. It follows from

(3.12)

and

(3.13)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M230">View MathML</a> is equiconvergent. Hence, by Lemma 2.2 and the above discussion, we conclude that for each natural number m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M234">View MathML</a> is completely continuous.

Finally, observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M235">View MathML</a>

(3.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M236">View MathML</a>

(3.15)

Hence, inequalities (3.14) and (3.15) imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M238">View MathML</a>. Then by the assumption (H4) and the absolute continuity of the integral, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M239">View MathML</a>

Therefore, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M108">View MathML</a> is completely continuous and maps the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M211">View MathML</a> into itself. Hence, the Schauder fixed point theorem can be applied to obtain a solution of the BVP (1.1), (1.2). The theorem is proved. □

Example 3.1 Consider the following boundary value problem:

(3.16)

(3.17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M251">View MathML</a>.

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M252">View MathML</a> is continuous and singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M4">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M255">View MathML</a>, it follows from a direct calculation that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M257">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M258">View MathML</a> such that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M259">View MathML</a>

Then by Theorem 3.2, the boundary value problem (3.16)-(3.17) has at least one positive solution.

4 Existence of at least three positive solutions

Definition 4.1 Let ℬ be a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M260">View MathML</a> be a cone in ℬ. By a concave nonnegative continuous functional φ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261">View MathML</a>, we mean a continuous functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M262">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M263">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M264">View MathML</a> being constants with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261">View MathML</a> and φ as above, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M266">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M267">View MathML</a>

Theorem 4.1 (Leggett-Williams fixed point theorem [12])

Letbe a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M260">View MathML</a>be a cone of ℬ, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M122">View MathML</a>be a constant. Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M270">View MathML</a>is a completely continuous operator andφis a nonnegative, continuous, concave functional on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M261">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M272">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M273">View MathML</a>. If there existr, L, andKwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M274">View MathML</a>such that the following conditions hold:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M275">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M276">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M277">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M278">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M279">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M276">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M281">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M282">View MathML</a>.

ThenAhas at least three positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M284">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M285">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M286">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M287">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M288">View MathML</a>

Theorem 4.2Assume that (H1)-(H4) are satisfied and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M289">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M290">View MathML</a>holds. Then the boundary value problem (1.1), (1.2) has at least three positive solutions if the following conditions hold:

(H5) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M291">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M292">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M295">View MathML</a>;

(H6) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M296">View MathML</a>and an interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M297">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M298">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M299">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M300">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M301">View MathML</a>;

(H7) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M303">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M304">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M305">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M307">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M308">View MathML</a>.

Proof The conditions of the Leggett-Williams fixed point theorem will be shown to be satisfied. Define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M309">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M310">View MathML</a> and the nonnegative, continuous, concave functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M311">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M312">View MathML</a>.

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M313">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M314">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M315">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M316">View MathML</a> and from (H7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M317">View MathML</a>

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M318">View MathML</a>

(4.1)

Therefore, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M319">View MathML</a>, and this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M320">View MathML</a>.

Now we show that condition (i) of Theorem 4.1 is satisfied. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M321">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M192">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M324">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M325">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M324">View MathML</a>, then by (H6) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M327">View MathML</a>

Therefore, condition (i) of Theorem 4.1 is satisfied.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M328">View MathML</a>, then by (H5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M329">View MathML</a>

In a similar way as (4.1), we can see that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M330">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M331">View MathML</a>

Hence, condition (ii) of Theorem 4.1 holds.

Finally, we show that condition (iii) of Theorem 4.1 is also satisfied. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M332">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M333">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M334">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M335">View MathML</a>

Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M337">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M338">View MathML</a>

Therefore, condition (iii) is also satisfied. Then the Leggett-Williams fixed point theorem implies that A has at least three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M340">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M341">View MathML</a> which are solutions to the problem (1.1)-(1.2). Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M342">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/127/mathml/M343">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors typed, read and approved the final manuscript.

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