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Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces

Yujun Cui1* and Jingxian Sun2

Author Affiliations

1 Department of Mathematics, Shandong University of Science and Technology, Qingdao, 266590, P.R. China

2 Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu, 221116, P.R. China

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Boundary Value Problems 2012, 2012:107  doi:10.1186/1687-2770-2012-107


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/107


Received:2 June 2012
Accepted:24 September 2012
Published:9 October 2012

© 2012 Cui and Sun; licensee Springer

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

Abstract

This paper deals with the positive solutions of a fourth-order boundary value problem in Banach spaces. By using the fixed-point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to a fourth-order boundary value problem in Banach spaces are obtained. An example illustrating the main results is given.

MSC: 34B15.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive operator; boundary value problem; positive solution; fixed-point theorem; measure of noncompactness

1 Introduction

In this paper, we consider the existence of multiple positive solutions for the fourth-order ordinary differential equation boundary value problem in a Banach space E

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M3">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M4">View MathML</a>, θ is the zero element of E. This problem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem in a scalar space has been studied by many authors using Schauder’s fixed-point theorem and the Leray-Schauder degree theory (see [1-5] and references therein). On the other hand, the theory of ordinary differential equations (ODE) in abstract spaces has become an important branch of mathematics in last thirty years because of its application in partial differential equations and ODEs in appropriately infinite dimensional spaces (see, for example, [6-8]). For an abstract space, it is here worth mentioning that Guo and Lakshmikantham [9] discussed the multiple solutions of two-point boundary value problems of ordinary differential equations in a Banach space. Recently, Liu [10] obtained the sufficient condition for multiple positive solutions to fourth-order singular boundary value problems in an abstract space. In [11], by using the fixed-point index theory in a cone for a strict-set-contraction operator, the authors have studied the existence of multiple positive solutions for the singular boundary value problems with an integral boundary condition.

However, the above works in a Banach space were carried out under the assumption that the second-order derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M5">View MathML</a> is not involved explicitly in the nonlinear term f. This is because the presence of second-order derivatives in the nonlinear function f will make the study extremely difficult. As a result, the goal of this paper is to fill up the gap, that is, to investigate the existence of solutions for fourth-order boundary value problems of (1.1) in which the nonlinear function f contains second-order derivatives, i.e., f depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M5">View MathML</a>.

The main features of this paper are as follows. First, we discuss the existence results in an abstract space E, not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M7">View MathML</a>. Secondly, we will consider the nonlinear term which is more extensive than the nonlinear term of [10,11]. Finally, the technique for dealing with fourth-order BVP is completely different from [10,11]. Hence, we improve and generalize the results of [10,11] to some degree, and so, it is interesting and important to study the existence of positive solutions of BVP (1.1). The arguments are based upon the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive operator and the fixed-point theorem in a cone for a strict-set-contraction operator.

The paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, various conditions on the existence of positive solutions to BVP (1.1) are discussed. In Section 4, we give an example to demonstrate our result.

2 Preliminaries

Let the real Banach space E with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M9">View MathML</a> be partially ordered by a cone P of E, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M10">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M11">View MathML</a>. P is said to be normal if there exists a positive constant N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M12">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M13">View MathML</a>. We consider problem (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a>. Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a> is a Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M17">View MathML</a> is a cone of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a>. In the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M19">View MathML</a> is called a solution of problem (1.1) if it satisfies (1.1). x is a positive solution of (1.1) if, in addition, x is nonnegative and nontrivial, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M21">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a>.

For a bounded set V in a Banach space, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M23">View MathML</a> the Kuratowski measure of noncompactness (see [6-8] for further understanding). In this paper, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M24">View MathML</a> the Kuratowski measure of noncompactness of a bounded set in E and in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a>.

Lemma 2.1[6]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M26">View MathML</a>andDbe a bounded set, fbe uniformly continuous and bounded from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M27">View MathML</a>intoE, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M28">View MathML</a>

The key tool in our approach is the following fixed-point theorem of strict-set-contractions:

Theorem 2.1[8]

LetPbe a cone of a real Banach spaceEand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M29">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M30">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M31">View MathML</a>is a strict set contraction such that one of the following two conditions is satisfied:

(i) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M34">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M35">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M37">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M38">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M34">View MathML</a>andfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M37">View MathML</a>;

then the operatorAhas at least one fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M44">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M45">View MathML</a>.

The following concept is due to Krasnosel’skii [12,13], with a slightly more general definition in [12].

Definition 2.1 We say that a bounded linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M46">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive on a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M48">View MathML</a> if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M49">View MathML</a> such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M50">View MathML</a>, there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M52">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M53">View MathML</a>

Lemma 2.2LetTbe<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive on a coneK. IfTis completely continuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M55">View MathML</a>, the spectral radius ofT, is the unique positive eigenvalue ofTwith its eigenfunction in K. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M56">View MathML</a>holds with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M57">View MathML</a>, then for an arbitrary non-zero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M58">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M59">View MathML</a>) the elementsuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M60">View MathML</a>are incomparable

In the following, the closed balls in spaces E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a> are denoted, respectively, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64">View MathML</a>).

For convenience, let us list the following assumptions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M68">View MathML</a>, f is bounded and uniformly continuous in t on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M69">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M70">View MathML</a>, and there exist two nonnegative constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M72">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M73">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M74">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>) There are three positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M78">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M79">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M81">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M84">View MathML</a> denotes the dual cone of P) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M85">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86">View MathML</a>, two nonnegative constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M88">View MathML</a> and a real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M89">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M90">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M92">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M85">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86">View MathML</a> and two nonnegative constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M98">View MathML</a> and a real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M99">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M100">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M101">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>) There are three positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M105">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M106">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M108">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109">View MathML</a>) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M110">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M111">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M112">View MathML</a>) There is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M83">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M115">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M86">View MathML</a> and a real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M110">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M118">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M120">View MathML</a>.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M121">View MathML</a> be the Green’s function of the linear problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M122">View MathML</a> together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M123">View MathML</a>, which can be explicitly given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M124">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M121">View MathML</a> have the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M126">View MathML</a>

(2.1)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M127">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M128">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M129">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M130">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M131">View MathML</a>

(2.2)

Using the above transformation and (2.2), BVP (1.1) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M132">View MathML</a>

(2.3)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M133">View MathML</a>

(2.4)

From (2.3) and (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M134">View MathML</a>

Now, define an operator A on Q by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M135">View MathML</a>

(2.5)

The following Lemma 2.3 can be easily obtained.

Lemma 2.3Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>) holds. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M137">View MathML</a>and

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M138">View MathML</a>is continuous and bounded;

(ii) BVP (1.1) has a solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M139">View MathML</a>if and only ifAhas a fixed point in Q.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M140">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M141">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M144">View MathML</a>.

Lemma 2.4Suppose that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>) holds. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M147">View MathML</a>is a strict set contraction.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M149">View MathML</a>, by the expression of S, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M150">View MathML</a>

and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M151">View MathML</a> is continuous and bounded. By the uniformly continuous f and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>), and Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M153">View MathML</a>

Since f is uniformly continuous and bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M154">View MathML</a>, we see from (2.5) that A is continuous and bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M155">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M156">View MathML</a>, according to (2.5), it is easy to show that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M157">View MathML</a> are uniformly bounded and equicontinuous, and so in [9] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M158">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M159">View MathML</a>

Using the obvious formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M160">View MathML</a>

and observing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M161">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M162">View MathML</a>

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M164">View MathML</a>.

From the fact of [9], we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M165">View MathML</a>

(2.7)

It follows from (2.6) and (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M166">View MathML</a>

and consequently, A is a strict set contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M155">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M168">View MathML</a>. □

3 Main results

Theorem 3.1Let a conePbe normal and condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>) be satisfied. If (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>) or (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>) are satisfied, then BVP (1.1) has at least one positive solution.

Proof Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M174">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M175">View MathML</a> is a cone of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M177">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M178">View MathML</a>, by (2.1), we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M179">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M180">View MathML</a>

We first assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>) are satisfied. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M183">View MathML</a>

In the following, we prove that W is bounded.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M184">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M185">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a>. And so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M188">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M189">View MathML</a>, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M191">View MathML</a>

(3.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M192">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M193">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M194">View MathML</a> is a bounded linear operator. From (3.1), one deduces that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M195">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M196">View MathML</a> is the first eigenvalue of T, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>), the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M199">View MathML</a>. Therefore, by [14], the inverse operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M200">View MathML</a> exists and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M201">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M202">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M203">View MathML</a>. So, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M149">View MathML</a> and W is bounded.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M206">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M207">View MathML</a>

(3.2)

Next, we are going to verify that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M208">View MathML</a>,

(3.3)

If this is false, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M210">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M211">View MathML</a>. This together with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M213">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M192">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M215">View MathML</a>, then the above inequality can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M216">View MathML</a>

(3.4)

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M217">View MathML</a>

In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M218">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M219">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a>, and consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M221">View MathML</a> in contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M222">View MathML</a>. Now, notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M223">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M225">View MathML</a>, then by Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M226">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M227">View MathML</a>. This together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M228">View MathML</a> and (3.4) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M229">View MathML</a>

which is a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M230">View MathML</a>. So, (3.3) holds.

By Lemma 2.4, A is a strict set contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M231">View MathML</a>. Observing (3.2) and (3.3) and using Theorem 2.1, we see that A has a fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M232">View MathML</a>.

Next, in the case that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>) are satisfied, by the method as in establishing (3.3), we can assert from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M236">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M237">View MathML</a>

(3.5)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M238">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M239">View MathML</a> is a completely continuous linear <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M242">View MathML</a> in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M243">View MathML</a>. In addition, the spectral radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M245">View MathML</a> is the positive eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M246">View MathML</a> corresponding to its first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M247">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M248">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M249">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M250">View MathML</a> is a completely continuous linear <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M253">View MathML</a>. Thus, the spectral radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255">View MathML</a> has a positive eigenfunction corresponding to its first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M256">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M257">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M258">View MathML</a>) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M260">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M261">View MathML</a>). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M263">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M264">View MathML</a>

By [12], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M265">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M266">View MathML</a>, by Gelfand’s formula, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M267">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M268">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M261">View MathML</a>.

In the following, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M270">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M271">View MathML</a> be the positive eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M272">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M273">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M274">View MathML</a>

(3.6)

satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M275">View MathML</a>. Without loss of generality, by standard argument, we may suppose by the Arzela-Ascoli theorem and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M268">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M277">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M278">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M279">View MathML</a> and by (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M280">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M281">View MathML</a>. This together with Lemma 2.2 guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M282">View MathML</a>.

By the above argument, it is easy to see that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M283">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M284">View MathML</a>

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M285">View MathML</a>

(3.7)

Now, we assert that

(3.8)

If this is not true, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M287">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M288">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M289">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M290">View MathML</a>. Moreover, by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M175">View MathML</a>, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M292">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M294">View MathML</a>, which implies by (3.7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M295">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M296">View MathML</a>

So, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M298">View MathML</a>

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M299">View MathML</a>

In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M300">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M301">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a>, and consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M303">View MathML</a> in contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M304">View MathML</a>. Now, notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M1">View MathML</a>-positive operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M252">View MathML</a>. Then by Lemma 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M308">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M227">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M310">View MathML</a> is the positive eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M255">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M312">View MathML</a>. This together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M313">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M314">View MathML</a>

which is a contradiction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M315">View MathML</a>. So, (3.8) holds.

By Lemma 2.4, A is a strict set contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M316">View MathML</a>. Observing (3.5) and (3.8) and using Theorem 2.1, we see that A has a fixed point on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M317">View MathML</a>. This together with Lemma 2.3 implies that BVP (1.1) has at least one positive solution. □

Theorem 3.2Let a conePbe normal. Suppose that conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109">View MathML</a>) are satisfied. Then BVP (1.1) has at least two positive solutions.

Proof We can take the same <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M322">View MathML</a> as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M323">View MathML</a>. And we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M325">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M326">View MathML</a> such that

(3.9)

(3.10)

On the other hand, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M329">View MathML</a>

(3.11)

In fact, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M330">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M331">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M332">View MathML</a>, then observing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M334">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M335">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M336">View MathML</a>

(3.12)

where, by virtue of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M338">View MathML</a>

(3.13)

It follows from (3.12) and (3.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M339">View MathML</a>

a contradiction. Thus (3.11) is true.

By Lemma 2.4, A is a strict set contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M340">View MathML</a>, and also on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M341">View MathML</a>. Observing (3.9), (3.10), (3.11) and applying, respectively, Theorem 2.1 to A, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M342">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M343">View MathML</a>, we assert that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M345">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M347">View MathML</a> and, by Lemma 2.3 and (3.11), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M349">View MathML</a> are positive solutions of BVP (1.1). □

Theorem 3.3Let a conePbe normal. Suppose that conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M75">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M102">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M112">View MathML</a>) are satisfied. Then BVP (1.1) has at least two positive solutions.

Proof We can take the same <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M322">View MathML</a> as in Theorem 3.1. As in the proof of Theorem 3.1, we can also obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M323">View MathML</a>. And we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M356">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M357">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M358">View MathML</a> such that

(3.14)

(3.15)

On the other hand, it is easy to see that

(3.16)

In fact, if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M362">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M363">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M364">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M365">View MathML</a>

Observing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M366">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M367">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M368">View MathML</a>

which is a contradiction. Hence, (3.16) holds.

By Lemma 2.4, A is a strict set contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M369">View MathML</a> and also on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M370">View MathML</a>. Observing (3.14), (3.15), (3.16) and applying, respectively, Theorem 2.1 to A, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M371">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M372">View MathML</a>, we assert that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M373">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M374">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M347">View MathML</a> and, by Lemma 2.3 and (3.16), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M349">View MathML</a> are positive solutions of BVP (1.1). □

4 One example

Now, we consider an example to illustrate our results.

Example 4.1 Consider the following boundary value problem of the finite system of scalar differential equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M379">View MathML</a>

(4.1)

where

(4.2)

(4.3)

Claim (4.1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M382">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M383">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M384">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M385">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M386">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M387">View MathML</a>. Then P is a normal cone in E, and the normal constant is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M388">View MathML</a>. System (4.1) can be regarded as a boundary value problem of (1.1) in E, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M389">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M390">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M391">View MathML</a>

Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M392">View MathML</a> is continuous. In this case, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M67">View MathML</a>) is automatically satisfied. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M394">View MathML</a> is identical zero for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M396">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M397">View MathML</a>, so we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M398">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M399">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M400">View MathML</a>

Noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M401">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M402">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M404">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M405">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M406">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M80">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M408">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M409">View MathML</a>. So, the conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M82">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M93">View MathML</a>) are satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M413">View MathML</a>.

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M414">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M416">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M417">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M418">View MathML</a>

So, condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/107/mathml/M109">View MathML</a>) is satisfied. Thus, our conclusion follows from Theorem 3.2. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179), NSF (BS2010SF023, BS2012SF022) of Shandong Province.

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