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On positive solutions for nonhomogeneous m-point boundary value problems with two parameters

Fanglei Wang1* and Yukun An2

Author Affiliations

1 College of Science, Hohai University, Nanjing, 210098, P.R. China

2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

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Boundary Value Problems 2012, 2012:87  doi:10.1186/1687-2770-2012-87


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/87


Received:14 May 2012
Accepted:27 July 2012
Published:6 August 2012

© 2012 Wang and An; 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 is concerned with the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the fixed-point theorem, the upper-lower solutions method, and the fixed-point index.

MSC: 34B10, 34B18.

Keywords:
nonhomogeneous BVP; positive solutions; upper-lower solutions; fixed-point theorem; fixed point index

1 Introduction

Many authors have studied the existence, nonexistence, and multiplicity of positive solutions for multipoint boundary value problems by using the fixed-point theorem, the fixed point index theory, and the lower and upper solutions method. We refer the readers to the references [1-4]. Recently, Hao, Liu and Wu [5] studied the existence, nonexistence, and multiplicity of positive solutions for the following nonhomogeneous boundary value problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M4">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M7">View MathML</a> may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M8">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M9">View MathML</a>. They showed that there exists a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M10">View MathML</a> such that the problem has at least two positive solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M11">View MathML</a>, at least one positive solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M12">View MathML</a> and no solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M13">View MathML</a> by using the Krasnosel’skii-Guo fixed-point theorem, the upper-lower solutions method, and the topological degree theory.

Inspired by the above references, the purpose of this paper is to study the following more general nonhomogeneous boundary value problems:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M14">View MathML</a>

(1)

where λ, μ are positive parameters, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M5">View MathML</a>. The main result of the present paper is summarized as follows.

Theorem 1.1Assume the following conditions hold:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17">View MathML</a>are nonnegative parameters;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M18">View MathML</a>is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M19">View MathML</a>does not vanish identically on any subinterval of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M20">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M21">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M22">View MathML</a>is given in Sect. 2;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M23">View MathML</a>is nondecreasing with respect tou, respectively, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M24">View MathML</a>

And either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M25">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M26">View MathML</a>;

(H4) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M27">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M28">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M29">View MathML</a>, respectively, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M30">View MathML</a>;

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M32">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M33">View MathML</a>, then there exists a bounded and continuous curve Γ separating<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34">View MathML</a>into two disjoint subsets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36">View MathML</a>such that (1) has at least two positive solutions for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37">View MathML</a>, one positive solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M38">View MathML</a>, and no solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M39">View MathML</a>. Moreover, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M40">View MathML</a>be the parametric representation of Γ, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M41">View MathML</a>

Then on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M42">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M43">View MathML</a>is continuous and nonincreasing, that is, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M44">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M45">View MathML</a>.

For the proof of Theorem 1.1, we also need the following lemmas.

Lemma 1.2[6]

LetEbe a Banach space, Ka cone inEand Ω bounded open inE. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M47">View MathML</a>be condensing. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M48">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M49">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M50">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M51">View MathML</a>

Lemma 1.3[6]

LetEbe a Banach space andKa cone inE. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M52">View MathML</a>, define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M53">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M54">View MathML</a>is a compact map such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M55">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M57">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M56">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M59">View MathML</a>

2 Preliminaries

Lemma 2.1[5]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M60">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M61">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M62">View MathML</a>, then the Green function for the homogeneous BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M63">View MathML</a>

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M64">View MathML</a>

Moreover, the Green function satisfies the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M65">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M66">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M67">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M68">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M69">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M70">View MathML</a>.

Lemma 2.2Assume that (H1)-(H5) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M60">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M72">View MathML</a>is a solution of (1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M73">View MathML</a>satisfies the following nonlinear integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M74">View MathML</a>

Proof Integrating both sides of (1) from 0 to t twice and applying the boundary conditions, then we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M75">View MathML</a>

Furthermore, by Lemma 2.1, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M76">View MathML</a>

 □

Let E denote the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M77">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M78">View MathML</a>. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79">View MathML</a> is said to be a solution of (1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M80">View MathML</a> satisfies (1). Moreover, from Lemma 2.2, it is clear to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79">View MathML</a> is a solution of (1) is equivalent to the fixed point of the operator T defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M82">View MathML</a>

In addition, define a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M83">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M84">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M85">View MathML</a>. Then we have

Lemma 2.3If (H1)-(H3) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M86">View MathML</a>is completely continuous.

The proof procedure of Lemma 2.3 is standard, so we omit it.

Now, we will establish the classical lower and upper solutions method for our problem. As usual, we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M87">View MathML</a> is a lower solution for (1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M88">View MathML</a>

Similarly, we define the upper solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M89">View MathML</a> of the problem (1):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M90">View MathML</a>

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M89">View MathML</a>be lower and upper solutions, respectively, of (1) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M93">View MathML</a>. Then (1) has a nonnegative solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M95">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M96">View MathML</a>.

Proof Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M97">View MathML</a>

It is clear to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M98">View MathML</a> is a bounded, convex and closed subset in Banach space E. Now we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M99">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M100">View MathML</a>, from (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M101">View MathML</a>

On the other hand, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M102">View MathML</a>

From above inequalities, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M99">View MathML</a>.

Therefore, by Schauder’s fixed theorem, the operator T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M100">View MathML</a>, which is the solution of (1). □

3 Proof of Theorem 1.1

Lemma 3.1Assume (H1)-(H5) hold and Σ be a compact subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M106">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M107">View MathML</a>and all possible positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79">View MathML</a>of (1) at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a>, one has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M110">View MathML</a>.

Proof Suppose on the contrary that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M111">View MathML</a> of positive solutions of Eq. (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M112">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M113">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M114">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M115">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M116">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M117">View MathML</a>

(2)

Since Σ is compact, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M118">View MathML</a> has a convergent subsequence which we denote without loss of generality still by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M120">View MathML</a>

and at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M121">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M122">View MathML</a>.

Case (I). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M121">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M124">View MathML</a> for n sufficient large. Then by (H5), there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M126">View MathML</a>

where L satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M127">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M128">View MathML</a>, for n sufficient large, we

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M129">View MathML</a>

This is a contradiction.

Case (II). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M122">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M131">View MathML</a> for n sufficient large. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M32">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M134">View MathML</a>

where M satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M135">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M128">View MathML</a>, then for n sufficient large, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M137">View MathML</a>

This is a contradiction. □

Lemma 3.2Assume (H1)-(H4) hold. If (1) has a positive solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M138">View MathML</a>, then Eq. (1) has a positive solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M140">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M141">View MathML</a> be the solution of Eq. (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M138">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M141">View MathML</a> be the upper solution of (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M140">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M25">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M148">View MathML</a> is not a solution of (1), but it is the lower solution of (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a>. Therefore, by Lemma 2.4, we obtain the result. □

Lemma 3.3Assume (H1)-(H5) hold. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M150">View MathML</a>such that Eq. (1) has a positive solution for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M151">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152">View MathML</a> be the unique solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M153">View MathML</a>

(3)

It is clear to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152">View MathML</a> is a positive solution of (3). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M156">View MathML</a>, then by (H4), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M158">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M160">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M152">View MathML</a> is an upper solution of (3) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162">View MathML</a>. On the other hand, 0 is a lower solution of (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M163">View MathML</a>. By (H3), 0 is not a solution of (1). Hence, (1) has a positive solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162">View MathML</a>, Lemma 3.2 now implies the conclusion of Lemma 3.3. □

Define a set S by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M165">View MathML</a>

Then it follows from Lemma 3.3 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M167">View MathML</a> is a partially ordered set.

Lemma 3.4Assume (H1)-(H5) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M167">View MathML</a>is bounded above.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M79">View MathML</a> be a positive solution of (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M172">View MathML</a>

by (H4). Furthermore, we can obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M173">View MathML</a>

 □

Lemma 3.5Assume (H1)-(H5) hold. Then every chain inShas a unique supremum inS.

Lemma 3.6Assume (H1)-(H5) hold. Then there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M174">View MathML</a>such (1) has a positive solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M175">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M176">View MathML</a>, no solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M175">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M178">View MathML</a>. Similarly, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M179">View MathML</a>such that (1) has a positive solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M180">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M181">View MathML</a>, and no solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M180">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M183">View MathML</a>.

Lemma 3.7Assume (H1)-(H5) hold. Then there exists a continuous curve Γ separating<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34">View MathML</a>into two disjoint subsets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35">View MathML</a>is bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M36">View MathML</a>is unbounded, Eq. (1) has at least one solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M189">View MathML</a>, and no solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M39">View MathML</a>. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M43">View MathML</a>is nonincreasing, that is, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M192">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M193">View MathML</a>

Lemma 3.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M195">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M196">View MathML</a>is an upper solution of (1) at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M198">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M199">View MathML</a>is the positive solution of Eq. (1) corresponding to some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M200">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M201">View MathML</a>

Proof From (H4), there exists constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M202">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M203">View MathML</a>

Then by the uniform continuity of f and g on a compact set, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M204">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M205">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M207">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M208">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M209">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M210">View MathML</a>

From above inequalities, it is clear to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M211">View MathML</a>, is an upper solution of (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M207">View MathML</a>. □

Proof of Theorem 1.1 From above lemmas, we need only to show the existence of the second positive solution of (1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M37">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M217">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M199">View MathML</a> be the positive solution of (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M162">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M195">View MathML</a> given by Lemma 3.8 and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M221">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M222">View MathML</a>

Define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M223">View MathML</a>

Then D is bounded open set in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M224">View MathML</a>. The map T satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M225">View MathML</a> and is condensing, since it is completely continuous. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M226">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M227">View MathML</a> such that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M228">View MathML</a>. Then by (H) and Lemma 3.8, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M229">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M230">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M231">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M230">View MathML</a>, Lemma 1.2 now implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M234">View MathML</a>

Now for some fixed λ and μ, it follows from assumption (H4) that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M125">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M236">View MathML</a>

(4)

where L satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M237">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M238">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M239">View MathML</a> is given by Lemma 3.1 with Σ a compact set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M34">View MathML</a> containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M109">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M242">View MathML</a>

Then it follows from Lemma 3.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M243">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M244">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M245">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M246">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M247">View MathML</a> and it follows from Lemma 1.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M248">View MathML</a>

By the additivity of the fixed-point index,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M249">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M250">View MathML</a>

Hence, T has at least one fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M251">View MathML</a> and another one in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M252">View MathML</a>; this shows that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M35">View MathML</a>, (1) has at least two positive solution. □

Example Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M254">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M256">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/87/mathml/M257">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

In this paper, the author studies the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the upper-lower solutions method and fixed-point index. All authors typed, read, and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this paper. The first author is supported financially by the Fundamental Research Funds for the Central Universities.

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