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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Positive solutions of a fractional thermostat model

Juan J Nieto12* and Johnatan Pimentel1

Author Affiliations

1 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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Boundary Value Problems 2013, 2013:5  doi:10.1186/1687-2770-2013-5

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/5


Received:17 October 2012
Accepted:29 December 2012
Published:16 January 2013

© 2013 Nieto and Pimentel; 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

We study the existence of positive solutions of a nonlinear fractional heat equation with nonlocal boundary conditions depending on a positive parameter. Our results extend the second-order thermostat model to the non-integer case. We base our analysis on the known Guo-Krasnosel’skii fixed point theorem on cones.

1 Introduction

Fractional calculus has been studied for centuries mainly as a pure theoretical mathematical discipline, but recently, there has been a lot of interest in its practical applications. In current research, fractional differential equations have arisen in mathematical models of systems and processes in various fields such as aerodynamics, acoustics, mechanics, electromagnetism, signal processing, control theory, robotics, population dynamics, finance, etc.[1-3]. For some recent results in fractional differential equations, see [4-12] and the references therein.

Infante and Webb [13] studied the nonlocal boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M1">View MathML</a>

which models a thermostat insulated at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M2">View MathML</a> with the controller at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M3">View MathML</a> adding or discharging heat depending on the temperature detected by the sensor at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M4">View MathML</a>. Using fixed point index theory and some results on their work on Hammerstein integral equations [14,15], they obtained results on the existence of positive solutions of the boundary value problem. In particular, they have shown that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M5">View MathML</a>, then positive solutions exist under suitable conditions on f. This type of boundary value problem was earlier investigated by Guidotti and Merino [16] for the linear case with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M6">View MathML</a> where they have shown a loss of positivity as β decreases. In the present paper, we consider the following fractional analog of the thermostat model:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M7">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M9">View MathML</a> denotes the Caputo fractional derivative of order α and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M10">View MathML</a> subject to the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M11">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M13">View MathML</a> are given constants.

We point out that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M14">View MathML</a>, we recover the second-order problem of [13]. We use the properties of the corresponding Green’s function and the Guo-Krasnosel’skii fixed point theorem to show the existence of positive solutions of (1)-(2) under the condition that the nonlinearity f is either sublinear or superlinear.

2 Preliminaries

Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [1-3].

Definition 2.1 The Riemann-Liouville fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M16">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M17">View MathML</a>

provided the integral exists.

Definition 2.2 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M16">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M21">View MathML</a> denotes the integer part of the real number α.

Definition 2.3 The Caputo derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M23">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M24">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M21">View MathML</a> denotes the integer part of the real number α.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M27">View MathML</a>.

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M28">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M29">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M28">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M31">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M32">View MathML</a>.

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M33">View MathML</a>.

Remark 2.1 In addition to the above properties, the Caputo derivative of a power function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M35">View MathML</a>, is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38">View MathML</a>.

Lemma 2.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M15">View MathML</a>, the general solution of the fractional differential equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M40">View MathML</a>is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M41">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M43">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38">View MathML</a>).

Lemma 2.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M46">View MathML</a>

(3)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M43">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M38">View MathML</a>).

We start by solving an auxiliary problem to get an expression for the Green’s function of boundary value problem (1)-(2).

Lemma 2.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M51">View MathML</a>. A function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M52">View MathML</a>is a solution of the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M53">View MathML</a>

if and only if it satisfies the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M54">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a>is the Green’s function (depending onα) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M56">View MathML</a>

(4)

and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M58">View MathML</a>is defined as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M59">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M61">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M62">View MathML</a>.

Proof Using (3) we have, for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M64">View MathML</a>

(5)

In view of Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M65">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M66">View MathML</a>, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M67">View MathML</a>.

It also follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M68">View MathML</a>

Using the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M69">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M70">View MathML</a>

Finally, substituting the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M72">View MathML</a> in (5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> is given by (4). This completes the proof. □

Remark 2.2 We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M75">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M76">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M77">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> given by (4) is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M79">View MathML</a>.

Remark 2.3 By taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M14">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M81">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> in this case coincides with the one obtained in [13] for the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M83">View MathML</a>

Remark 2.4 We observe that for each fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M85">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M87">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M88">View MathML</a> and deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> is a decreasing function of t. It then follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M90">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M91">View MathML</a>

Consequently, by looking at the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> with respect to s, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M93">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M94">View MathML</a>

To establish the existence of positive solutions of problem (1)-(2), we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> satisfies the following property introduced by Lan and Webb in [17]:

(A) There exist a measurable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M96">View MathML</a>, a subinterval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M97">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M98">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M100">View MathML</a>

Lemma 2.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M102">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M103">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a>satisfies property (A).

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M107">View MathML</a>. We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M108">View MathML</a>, and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M109">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M110">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M111">View MathML</a>

(6)

 □

Lemma 2.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M112">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M113">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M103">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a>satisfies property (A).

Proof We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M116">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M117">View MathML</a>. Following the arguments in the previous lemma, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M118">View MathML</a>

Also, by taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M119">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M120">View MathML</a>

 □

Lemma 2.7If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M121">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a>changes sign on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M123">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a>satisfies property (A).

Proof We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M116">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M126">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M127">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M128">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M129">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M130">View MathML</a>

For the main results, we use the known Guo-Krasnosel’skii fixed point theorem [18]. □

Theorem 2.1LetEbe a Banach space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M131">View MathML</a>be a cone. Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M133">View MathML</a>are open bounded subsets ofEsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M134">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M135">View MathML</a>be a completely continuous operator such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M137">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M139">View MathML</a>; or

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M137">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M139">View MathML</a>.

Then the operatorPhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M144">View MathML</a>.

3 Main results

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M145">View MathML</a>

We now state the main result of this paper.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M146">View MathML</a>. Assume that one of the following conditions is satisfied:

(i) (Sublinear case) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148">View MathML</a>.

(ii) (Superlinear case) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M101">View MathML</a>, then problem (1)-(2) admits at least one positive solution.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M152">View MathML</a>. Assume that one of the following conditions is satisfied:

(i) (Sublinear case) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148">View MathML</a>.

(ii) (Superlinear case) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M157">View MathML</a>, then problem (1)-(2) admits a solution which is positive on an interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M158">View MathML</a>.

Proof of Theorem 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M159">View MathML</a> be the Banach space of all continuous real-valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M76">View MathML</a> endowed with the usual supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M161">View MathML</a>.

We define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M162">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M163">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> is defined by (4).

It is clear from Lemma 2.4 that the fixed points of the operator T coincide with the solutions of problem (1)-(2).

We now define the cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M165">View MathML</a>

where λ is given by (6).

First, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M166">View MathML</a>.

It follows from the continuity and the non-negativity of the functions G and f on their domains of definition that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M169">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170">View MathML</a>.

For a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170">View MathML</a>, the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M55">View MathML</a> satisfies property (A) leads to the following inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M174">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M175">View MathML</a>.

We now show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M176">View MathML</a> is completely continuous.

In view of the continuity of the functions G and f, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M177">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M178">View MathML</a> be bounded, that is, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M179">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M180">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M181">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M182">View MathML</a>

Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M183">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M184">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M170">View MathML</a>. That is, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M186">View MathML</a> is bounded.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M188">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M189">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M190">View MathML</a>

Clearly, the right-hand side of the above inequalities tends to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M191">View MathML</a> and therefore the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M186">View MathML</a> is equicontinuous. It follows from the Arzela-Ascoli theorem that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M177">View MathML</a> is completely continuous.

We now consider the two cases.

(i) Sublinear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148">View MathML</a>).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M197">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M198">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M199">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M200">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M201">View MathML</a>

(7)

We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M203">View MathML</a>, then we have the following inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M204">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M205">View MathML</a>. Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M207">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M208">View MathML</a> is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M209">View MathML</a>, we can define the function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M210">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M211">View MathML</a> is non-decreasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M212">View MathML</a> and since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148">View MathML</a>, we have (see [19])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M214">View MathML</a>

Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M215">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M216">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M217">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M218">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M219">View MathML</a>

(8)

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M220">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M222">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M223">View MathML</a>

Hence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M225">View MathML</a>.

Thus, by the first part of the Guo-Krasnosel’skii fixed point theorem, we conclude that (1)-(2) has at least one positive solution.

(ii) Superlinear case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150">View MathML</a>).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M228">View MathML</a> be given as in (8).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M149">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M230">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M231">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M232">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M167">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M234">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M235">View MathML</a>

If we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M236">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M237">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M238">View MathML</a>.

Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M150">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M240">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M198">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M242">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M200">View MathML</a> is as in (7).

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M244">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M245">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M247">View MathML</a> imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M248">View MathML</a>

and so we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M249">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M136">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M251">View MathML</a>. We conclude by the second part of the Guo-Krasnosel’skii fixed point theorem that (1)-(2) has at least one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M252">View MathML</a>. □

Remark 3.1

To prove Theorem 3.2, we use the cone

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M253">View MathML</a>

where b and λ are defined in Lemma 2.6 for the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M254">View MathML</a>, and in Lemma 2.7 for the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M255">View MathML</a>. We skip the rest of the proof as it is similar to the proof of Theorem 3.1.

Example 3.1

Consider the fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M256">View MathML</a>

(9)

which is problem (1)-(2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M260">View MathML</a>.

First, we note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M261">View MathML</a> is not a solution of (9).

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M148">View MathML</a>, and we also have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M264">View MathML</a>.

We take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M265">View MathML</a>

and consider the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/5/mathml/M266">View MathML</a>.

By the first part of Theorem 3.1, we conclude that the boundary value problem (9) has a positive solution in the cone P.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors, JJN and JP, contributed equally and read and approved the final version of the manuscript.

Acknowledgements

Dedicated to Professor Jean Mawhin for his 70th anniversary.

The research has been partially supported by Ministerio de Economía y Competitividad, and FEDER, project MTM2010-15314.

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