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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 solutions for a system of fractional boundary value problems

Johnny Henderson1 and Rodica Luca2*

Author Affiliations

1 Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USA

2 Department of Mathematics, Gh. Asachi Technical University, Iasi, 700506, Romania

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Boundary Value Problems 2014, 2014:60  doi:10.1186/1687-2770-2014-60


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/60


Received:8 January 2014
Accepted:11 March 2014
Published:20 March 2014

© 2014 Henderson and Luca; 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 credited.

Abstract

We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated.

MSC: 34A08, 45G15.

Keywords:
Riemann-Liouville fractional differential equation; integral boundary conditions; positive solutions

1 Introduction

We consider the system of nonlinear ordinary fractional differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M1">View MathML</a>

with the integral boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M6">View MathML</a> denote the Riemann-Liouville derivatives of orders α and β, respectively, and the integrals from (BC) are Riemann-Stieltjes integrals.

Under sufficient conditions on functions f and g, which can be nonsingular or singular in the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8">View MathML</a>, we study the existence and multiplicity of positive solutions of problem (S)-(BC). We use the Guo-Krasnosel’skii fixed point theorem (see [1]) and some theorems from the fixed point index theory (from [2] and [3]). By a positive solution of problem (S)-(BC) we mean a pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9">View MathML</a> satisfying (S) and (BC) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M14">View MathML</a>. The system (S) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M16">View MathML</a> and the boundary conditions (BC) where H and K are scale functions (that is, multi-point boundary conditions) has been investigated in [4] (the nonsingular case) and [5] (the singular case). In [6], the authors give sufficient conditions for λ, μ, f, and g such that the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M17">View MathML</a>

with the boundary conditions (BC) with H and K scale functions, has positive solutions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M21">View MathML</a>).

Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics, and so on (see [7-13]).

In Section 2, we present the necessary definitions and properties from the fractional calculus theory and some auxiliary results dealing with a nonlocal boundary value problem for fractional differential equations. In Section 3, we give some existence and multiplicity results for positive solutions with respect to a cone for our problem (S)-(BC), where f and g are nonsingular functions. The case when f and g are singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8">View MathML</a> is studied in Section 4. Finally, in Section 5, we present two examples which illustrate our main results.

2 Preliminaries and auxiliary results

We present here the definitions, some lemmas from the theory of fractional calculus and some auxiliary results that will be used to prove our main theorems.

Definition 2.1 The (left-sided) fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M25">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M26">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M27">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M28">View MathML</a> is the Euler gamma function defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24">View MathML</a>.

Definition 2.2 The Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M31">View MathML</a> for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M32">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M34">View MathML</a>, provided that the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M27">View MathML</a>.

The notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M36">View MathML</a> stands for the largest integer not greater than α. We also denote the Riemann-Liouville fractional derivative of f by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M37">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M38">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M39">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M40">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M41">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M42">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M40">View MathML</a>.

Lemma 2.1 ([10])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M45">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M46">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M47">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M48">View MathML</a>; that is, nis the smallest integer greater than or equal toα. Then the solutions of the fractional differential equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M50">View MathML</a>, are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M51">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M52">View MathML</a>are arbitrary real constants.

Lemma 2.2 ([6,10])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M24">View MathML</a>, nbe the smallest integer greater than or equal toα (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M54">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M55">View MathML</a>. The solutions of the fractional equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M50">View MathML</a>, are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M58">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M52">View MathML</a>are arbitrary real constants.

We consider now the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M60">View MathML</a>

(1)

with the integral boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M61">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M63">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64">View MathML</a> is a function of the bounded variation.

Lemma 2.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64">View MathML</a>is a function of bounded variation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67">View MathML</a>, then the solution of problem (1)-(2) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M68">View MathML</a>

(3)

Proof By Lemma 2.2, the solutions of equation (1) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M69">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M70">View MathML</a>. By using the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M71">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M72">View MathML</a>. Then we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M73">View MathML</a>

Now, by condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M74">View MathML</a>, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M75">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M76">View MathML</a>

So, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M77">View MathML</a>

Therefore, we get the expression (3) for the solution of problem (1)-(2). □

Lemma 2.4Under the assumptions of Lemma 2.3, the Green’s function for the boundary value problem (1)-(2) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M78">View MathML</a>

(4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M79">View MathML</a>

(5)

Proof By Lemma 2.3 and relation (3), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82">View MathML</a> are given in (5) and (4), respectively. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12">View MathML</a>. □

Lemma 2.5 ([6])

The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81">View MathML</a>given by (5) has the properties:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M86">View MathML</a>is a continuous function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M87">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88">View MathML</a>.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M89">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88">View MathML</a>.

(c) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M92">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M94">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M95">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M96">View MathML</a>), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M97">View MathML</a>

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M99">View MathML</a>.

Lemma 2.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64">View MathML</a>is a nondecreasing function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101">View MathML</a>, then the Green’s function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82">View MathML</a>of the problem (1)-(2) is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M103">View MathML</a>and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M104">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M107">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12">View MathML</a>, then the unique solutionuof problem (1)-(2) satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M10">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

Proof By using the assumptions of this lemma, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M104">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M88">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M113">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>. □

Lemma 2.7Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64">View MathML</a>is a nondecreasing function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101">View MathML</a>. Then the Green’s function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82">View MathML</a>of the problem (1)-(2) satisfies the inequalities:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M119">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M120">View MathML</a>

(b) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M122">View MathML</a>

Proof The first inequality (a) is evident. For part (b), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M125">View MathML</a>, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M126">View MathML</a>

Therefore, we obtain the inequalities (b) of this lemma. □

Lemma 2.8Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M64">View MathML</a>is a nondecreasing function and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M107">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>. Then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>of problem (1)-(2) satisfies the inequality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M135">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M138">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M139">View MathML</a>

Then we deduce the conclusion of this lemma. □

We can also formulate similar results as Lemmas 2.3-2.8 above for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M140">View MathML</a>

(6)

with the integral boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M141">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M144">View MathML</a> is a nondecreasing function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M145">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M150">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151">View MathML</a> the corresponding constants and functions for the problem (6)-(7) defined in a similar manner as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M82">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157">View MathML</a>, respectively.

3 The nonsingular case

In this section, we investigate the existence and multiplicity of positive solutions for our problem (S)-(BC) under various assumptions on nonsingular functions f and g.

We present the basic assumptions that we shall use in the sequel.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M158">View MathML</a> are nondecreasing functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M160">View MathML</a>.

(H2) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M161">View MathML</a> are continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M162">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

A pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9">View MathML</a> is a solution for our problem (S)-(BC) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M9">View MathML</a> is a solution for the nonlinear integral system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M166">View MathML</a>

We consider the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M167">View MathML</a> with supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M168">View MathML</a> and define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M169">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M170">View MathML</a>.

We also define the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M171">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M172">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M174">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M175">View MathML</a>

Under the assumptions (H1) and (H2), using also Lemma 2.6, it is easy to see that , ℬ, and are completely continuous from P to P. Thus the existence and multiplicity of positive solutions of the system (S)-(BC) are equivalent to the existence and multiplicity of fixed points of the operator .

Theorem 3.1Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:

(H3) There exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M179">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M181">View MathML</a>

(H4) There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M182">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M183">View MathML</a>

then the problem (S)-(BC) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

Proof Because the proof of the theorem is similar to that of Theorem 3.1 from [4], we will sketch some parts of it. From assumption (i) of (H3), we deduce that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M186">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M187">View MathML</a>

(8)

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M188">View MathML</a>, by using (8), Lemma 2.6, and Lemma 2.7, we obtain after some computations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M189">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M190">View MathML</a>.

For c given in (H3), we define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M191">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M192">View MathML</a>. From our assumptions and Lemma 2.8, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M193">View MathML</a>, we can easily show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M195">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M197">View MathML</a>.

We now consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M200">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>. We define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M202">View MathML</a>

We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M203">View MathML</a> and M is a bounded subset of X. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M205">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>. From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M208">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M209">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M210">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M211">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M203">View MathML</a>, and from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M213">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M214">View MathML</a>

(10)

From (ii) of assumption (H3), we conclude that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M215">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M216">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M217">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M219">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M124">View MathML</a>, by using Lemma 2.7 and the relations (9) and (11), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M222">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M223">View MathML</a>.

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M224">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M225">View MathML</a>

(12)

Now from relations (10) and (12), one obtains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M226">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M204">View MathML</a>, that is, M is a bounded subset of X.

Besides, there exists a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M228">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M229">View MathML</a>

From [2], we deduce that the fixed point index of the operator over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M231">View MathML</a> with respect to P is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M232">View MathML</a>

(13)

Next, from assumption (H4), we conclude that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M234">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M235">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M238">View MathML</a>. Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M241">View MathML</a>

(15)

Therefore, by (14) and (15), we deduce that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M244">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M245">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M246">View MathML</a>. From [2], we conclude that the fixed point index of the operator over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M248">View MathML</a> with respect to P is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M249">View MathML</a>

(16)

Combining (13) and (16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M250">View MathML</a>

We deduce that has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M252">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M253">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M254">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M255">View MathML</a> is a solution of (S)-(BC). In addition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M256">View MathML</a>. Indeed, if we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M257">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M12">View MathML</a>, then by using (H2) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M259">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M261">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M263">View MathML</a>. The proof of Theorem 3.1 is completed. □

Using similar arguments as those used in the proofs of Theorem 3.2 and Theorem 3.3 in [4], we also obtain the following results for our problem (S)-(BC).

Theorem 3.2Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:

(H5) There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M264">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M265">View MathML</a>

(H6) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M267">View MathML</a>

then the problem (S)-(BC) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

Theorem 3.3Assume that (H1)-(H3), and (H6) hold. If the functionsfandgalso satisfy the condition:

(H7) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M271">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M272">View MathML</a>are nondecreasing with respect tou, and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M273">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M274">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M277">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151">View MathML</a>are defined in Section 2, then the problem (S)-(BC) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

4 The singular case

In this section, we investigate the existence of positive solutions for our problem (S)-(BC) under various assumptions on functions f and g which may be singular at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M7">View MathML</a> and/or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M8">View MathML</a>.

The basic assumptions used here are the following.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M285">View MathML</a>) ≡ (H1).

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M287">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M290">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M294">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M295">View MathML</a>

We consider the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M167">View MathML</a> with supremum norm and define the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M169">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M170">View MathML</a>. We also define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M299">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M300">View MathML</a>

Lemma 4.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M301">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M302">View MathML</a>) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M303">View MathML</a>is completely continuous.

Proof We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M305">View MathML</a>. Using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M302">View MathML</a>), we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M307">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M308">View MathML</a>. By Lemma 2.6 and the corresponding lemma for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M150">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310">View MathML</a> maps P into P.

We shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310">View MathML</a> maps bounded sets into relatively compact sets. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M312">View MathML</a> is an arbitrary bounded set. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M313">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M314">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315">View MathML</a>. By using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>) and Lemma 2.7, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M317">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M319">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M320">View MathML</a>. In what follows, we shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321">View MathML</a> is equicontinuous. By using Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M322">View MathML</a>

Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M323">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M324">View MathML</a>

So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M323">View MathML</a>, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M326">View MathML</a>

(17)

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M327">View MathML</a>

For the integral of the function h, by exchanging the order of integration, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M328">View MathML</a>

For the integral of the function μ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M329">View MathML</a>

(18)

We deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M330">View MathML</a>. Thus for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M331">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M332">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M315">View MathML</a>, by (17), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M334">View MathML</a>

(19)

From (18), (19), and the absolute continuity of the integral function, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321">View MathML</a> is equicontinuous. By the Ascoli-Arzelà theorem, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M321">View MathML</a> is relatively compact. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M337">View MathML</a> is a compact operator. Besides, we can easily show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310">View MathML</a> is continuous on P. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M303">View MathML</a> is completely continuous. □

Theorem 4.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>) hold. If the functionsfandgalso satisfy the conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M342">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M343">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M344">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M345">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M347">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M348">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M350">View MathML</a>

then the problem (S)-(BC) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

Proof Because the proof of this theorem is similar to that of Theorem 3 in [5], we will sketch some parts of it. For c given in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346">View MathML</a>), we consider the cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M354">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M355">View MathML</a>. Under assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M358">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M342">View MathML</a>), we deduce that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M360">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M361">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M362">View MathML</a>

(20)

By using (20) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M364">View MathML</a>, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M365">View MathML</a>

By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M366">View MathML</a>, we can choose sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M367">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M368">View MathML</a>

(21)

From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M346">View MathML</a>), we deduce that there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M371">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M372">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M373">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M374">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M375">View MathML</a>. From the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M294">View MathML</a> and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M377">View MathML</a>, we conclude that there exists sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M378">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M379">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M380">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M381">View MathML</a>. Therefore for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M384">View MathML</a>

(23)

By (22), (23), Lemma 2.7, and Lemma 2.8, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M386">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M387">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M388">View MathML</a>

(24)

By (21), (24), and the Guo-Krasnosel’skii fixed point theorem, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M310">View MathML</a> has at least one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M390">View MathML</a>. Then our problem (S)-(BC) has at least one positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M391">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M392">View MathML</a>. The proof of Theorem 4.1 is completed. □

Using similar arguments as those used in the proof of Theorem 2 in [5] (see also [14] for a particular case of the problem studied in [5]), we also obtain the following result for our problem (S)-(BC).

Theorem 4.2Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M340">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M286">View MathML</a>) hold. If the functionsfandgalso satisfy the conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M395">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M396">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M397">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M398">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M399">View MathML</a>) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M400">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M401">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M403">View MathML</a>

then the problem (S)-(BC) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>.

5 Examples

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M406">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M407">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M408">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M409">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M410">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M411">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M413">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M414">View MathML</a>.

We consider the system of fractional differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M415">View MathML</a>

with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M416">View MathML</a>

Then we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M417">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M418">View MathML</a>. We also deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M419">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M420">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M421">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M260">View MathML</a>.

For the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M151">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M425">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M426">View MathML</a>

Example 1 We consider the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M427">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M429">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M430">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M431">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M432">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M434">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M435">View MathML</a>. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M272">View MathML</a> are nondecreasing with respect to u, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M20">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M439">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M91">View MathML</a> the assumptions (H3) and (H6) are satisfied; indeed we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M441">View MathML</a>

We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M442">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M443">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M444">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M445">View MathML</a>, then the assumption (H7) is satisfied. For example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M446">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M447">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M448">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M449">View MathML</a> (e.g.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M450">View MathML</a>), then the above inequality is satisfied. By Theorem 3.3, we deduce that the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M451">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M452">View MathML</a>) has at least two positive solutions.

Example 2 We consider the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M453">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M454">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M455">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M457">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M458">View MathML</a>

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M459">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M460">View MathML</a>.

In (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M395">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M462">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M463">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M397">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M465">View MathML</a>

In (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M399">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M467">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M468">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M401">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M349">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M471">View MathML</a>

For example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M472">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M473">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M474">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M475">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M476">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M477">View MathML</a>, the above conditions are satisfied. Then, by Theorem 4.2, we deduce that the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M451">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/60/mathml/M452">View MathML</a>) has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this paper. Both authors read and approved the final manuscript.

Acknowledgements

The work of R Luca was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.

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