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

Open Access Research

Existence and multiplicity of positive bound states for Schrödinger equations

Sun Sheng, Fanglei Wang* and Tianqing An

Author Affiliations

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

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Boundary Value Problems 2013, 2013:271  doi:10.1186/1687-2770-2013-271


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/271


Received:5 September 2013
Accepted:19 November 2013
Published:12 December 2013

© 2013 Sheng et al.; licensee Springer.

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

Abstract

In this paper, we study the existence and multiplicity of positive bound states of non-autonomous systems of nonlinear Schrödinger equations. The proof is based on the fixed point theorems in a cone.

MSC: 34B15, 35J20.

Keywords:
nonlinear Schrödinger systems; positive solutions; fixed point theorems

1 Introduction

Because of the important background in nonlinear optics and other fields, many authors pay more attention to the study of different types of vector nonlinear Schrödinger equations, we refer the readers to [1-5]. Most of these results have been proven using critical point theory, variational approaches or a fixed point theorem. More recently, Chu has applied another topological approach, a nonlinear alternative principle of Leray-Schauder, to establish some new existence results for the following Schrödinger equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M2">View MathML</a> is nonnegative almost everywhere, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M3">View MathML</a>. The author considered two different cases. One is the singular case, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M4">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M5">View MathML</a>

The other is the regular case, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M6">View MathML</a>. However, the references [4-6] are not concerned with multiplicity of positive solutions for the scalar Schrödinger equation or system.

Motivated by the study of solitary wave solutions, in this paper we mainly aim to study the existence and multiplicity of positive bound states of the more general system of non-autonomous Schrödinger equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M7">View MathML</a>

(1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M8">View MathML</a>

The methods used here are the Krasnoselskii fixed point theorem and the Leggett-Williams fixed point theorem together with a compactness criterion due to Zima.

We organize the paper as follows. In Section 2, we give some preliminaries; in Section 3, we discuss the existence and multiplicity of positive solutions for (1).

2 Preliminaries

For convenience, we assume that the following conditions hold throughout this paper.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M9">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10">View MathML</a> satisfies the following property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M11">View MathML</a>

(H2) The support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M12">View MathML</a> denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M13">View MathML</a> is a nonempty compact set, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M14">View MathML</a>

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M15">View MathML</a> is continuous, and the following notations are introduced:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M16">View MathML</a>

Since (H1) holds, then for the homogeneous problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M17">View MathML</a>

the associated Green’s function is expressed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M18">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20">View MathML</a> are solutions such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M22">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20">View MathML</a> can be chosen as positive increasing and positive decreasing functions, respectively. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M20">View MathML</a> intersect at a unique point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M27">View MathML</a>. Therefore, we can define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M28">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M29">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M30">View MathML</a> denotes the space of bounded continuous functions.

Lemma 2.1[4]

For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31">View MathML</a>, Green’s function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M32">View MathML</a>satisfies the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M33">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M34">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M35">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M34">View MathML</a>;

(iii) Given a nonempty compact subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M37">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M38">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M39">View MathML</a>.

Lemma 2.2[4]

Assume that (H1) holds and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M40">View MathML</a>. Then the unique solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M41">View MathML</a>

belongs to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M42">View MathML</a>, and the solution can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M43">View MathML</a>

The proof of our main results is based on the following fixed points, which can be found in [7].

Lemma 2.3LetEbe a Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M44">View MathML</a>be a cone inE. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M46">View MathML</a>are open subsets ofEwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M48">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M49">View MathML</a>be a completely continuous operator such that either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53">View MathML</a>.

ThenThas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M58">View MathML</a>.

Let E be a real Banach space and P be a cone in E. A map α is said to be a nonnegative continuous concave functional on P if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M59">View MathML</a>

is continuous and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M60">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M62">View MathML</a>.

For numbers a, b such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M63">View MathML</a>, letting α be a nonnegative continuous concave functional on P, we define the following convex sets:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M64">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M65">View MathML</a>

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M66">View MathML</a>be completely continuous andαbe a nonnegative continuous concave functional onPsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M67">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M68">View MathML</a>. Suppose that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M69">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M70">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M71">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M72">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M73">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M74">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M71">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M76">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M77">View MathML</a>.

ThenThas at least three fixed points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M80">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M81">View MathML</a>

In addition, the following compactness criterion proved by Zima in [8] is also used in our proof.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M82">View MathML</a>. Let us assume that the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M83">View MathML</a>are equicontinuous in each compact interval ofRand that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M84">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M85">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M86">View MathML</a>verifies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M87">View MathML</a>

Then Ω is relatively compact.

3 Main results

From now on, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M88">View MathML</a> is a nonempty compact set. Let E denote the Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M91">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M92">View MathML</a>. Define a cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M44">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M94">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M96">View MathML</a> and the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31">View MathML</a>, are defined by property (iii) of Lemma 2.1. Since M is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31">View MathML</a>. Moreover, from (iii) of Lemma 2.1 it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M101">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M31">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M103">View MathML</a> be a map with components <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M104">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M105">View MathML</a>

A fixed point of T is a solution of (1) which belongs to .

Lemma 3.1Assume that (H1)-(H3) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M107">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M108">View MathML</a>is completely continuous.

Proof The continuity is trivial. Since M is compact, there exists a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M109">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M110">View MathML</a> is attained. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M111">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M112">View MathML</a>

namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M113">View MathML</a>

Therefore, it is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M107">View MathML</a>.

Finally, we prove that each component of T is compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M115">View MathML</a> be a bounded set, then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M116">View MathML</a> which is uniformly bounded for its element. Since the derivative is bounded in compacts, the functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M117">View MathML</a> are equicontinuous on each compact interval. On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M119">View MathML</a>

 □

Theorem 3.2Assume that (H1)-(H3) hold. In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121">View MathML</a>) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M122">View MathML</a>

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M124">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125">View MathML</a>, then (1) has at least one positive solution.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125">View MathML</a>, then (1) has at least one positive solution.

Proof (a) On the one hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M131">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M132">View MathML</a> is sufficiently small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M133">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M134">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M136">View MathML</a>

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M137">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M138">View MathML</a>

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M124">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M140">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M141">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M142">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M143">View MathML</a> is sufficiently large such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M144">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M145">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M146">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M148">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M149">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M150">View MathML</a>

Now by Lemma 2.3, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M151">View MathML</a>, namely, (1) has a positive solution.

(b) On the one hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M155">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M143">View MathML</a> is sufficiently large such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M157">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M134">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M160">View MathML</a>

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M161">View MathML</a>

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M141">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M164">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M132">View MathML</a> is sufficiently small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M166">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M145">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M146">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M148">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M171">View MathML</a>

Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M172">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M173">View MathML</a>

Now, by Lemma 2.3, T has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M151">View MathML</a>, namely, (1) has a positive solution. □

Corollary 3.3Assume that (H1)-(H3) hold. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121">View MathML</a>) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M177">View MathML</a>

In addition, the following conditions hold.

(H4) If there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M178">View MathML</a>such that for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M180">View MathML</a>

whereρsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M181">View MathML</a>

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M126">View MathML</a>.

Then (1) has at least two positive solutions.

Corollary 3.4Assume that (H1)-(H3) hold. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121">View MathML</a>) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M186">View MathML</a>

In addition, the following conditions hold.

(H6) If there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M187">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M188">View MathML</a>

whereϑsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M189">View MathML</a>

(H7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M191">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M192">View MathML</a>.

Then (1) has at least two positive solutions.

Theorem 3.5Assume that (H1)-(H3) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M121">View MathML</a>) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M195">View MathML</a>

In addition, there exist numbersa, canddwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M196">View MathML</a>such that the following conditions are satisfied:

(H8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M197">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M198">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M199">View MathML</a>;

(H9) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M125">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M201">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M202">View MathML</a>;

(H10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M203">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M198">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M205">View MathML</a>.

Then (1) has at least three positive solutions.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M206">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M207">View MathML</a>

then it is easy to know that α is a nonnegative continuous concave functional on K with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M208">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M209">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M210">View MathML</a>. First, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M211">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M212">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M213">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M214">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M215">View MathML</a>.

In a similar way, we also can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M216">View MathML</a>. Then (ii) of Lemma 2.4 holds.

Next, we shall show that (i) of Lemma 2.4 is satisfied. It is clearly seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M217">View MathML</a>. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M219">View MathML</a>, it is easy to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M220">View MathML</a>

Then, by (H9), we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M221">View MathML</a>

Finally, we verify that (iii) of Lemma 2.4 is satisfied. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M222">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M223">View MathML</a>, then we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M224">View MathML</a>

From the above, the hypotheses of Leggett-Williams theorem are satisfied. Hence (1) has at least three positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M227">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M229">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M230">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/271/mathml/M231">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SS drafted the manuscript. FW and TA gave some suggestions to improve the manuscript. All authors typed, read and approved the final manuscript.

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