Open Access Research

Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space

Ruijuan Liu12*, Chunhai Kou3 and Ran Jin1

Author Affiliations

1 College of Information Science and Technology, Donghua University, Shanghai, 201620, China

2 College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, China

3 Department of Applied mathematics, Donghua University, Shanghai, 201620, China

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Boundary Value Problems 2013, 2013:79  doi:10.1186/1687-2770-2013-79


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/79


Received:25 December 2012
Accepted:18 March 2013
Published:8 April 2013

© 2013 Liu et al.; licensee Springer

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

Abstract

In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators.

MSC: 26A33, 34B15.

Keywords:
fractional order; integro-differential equation; measure of noncompactness; fixed point index; boundary value problem

1 Introduction

Fractional differential equations (FDEs) have been of great interest for the last three decades [1-11]. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity [12], electrochemistry [13], control, porous media [14], etc. Therefore, the theory of FDEs has been developed very quickly. Many qualitative theories of FDEs have been obtained. Many important results can be found in [15-19] and references cited therein.

In this paper, we shall use the fixed point index theory of completely continuous operators to investigate the multiple positive solutions of a boundary value problem for a class of α order nonlinear integro-differential equations in a Banach space.

Let E be a real Banach space, P be a cone in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M1">View MathML</a> denote the interior points of P. A partial ordering in E is introduced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M2">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M3">View MathML</a>. P is said to be normal if there exists a positive constant N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M4">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M5">View MathML</a>, where θ denotes the zero element of E, and the smallest constant N is called the normal constant of P. P is called solid if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M1">View MathML</a> is nonempty. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M8">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M9">View MathML</a>. If P is solid and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M10">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M11">View MathML</a>. For details on cone theory, see [1].

For the application in the sequel, we first state the following lemmas and definitions which can be found in [1,10,20].

Lemma 1.1LetPbe a cone in a real Banach spaceE, and let Ω be a nonempty bounded open convex subset ofP. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M12">View MathML</a>is completely continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M13">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M14">View MathML</a>denotes the closure of Ω inP. Then the fixed point index

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M15">View MathML</a>

Lemma 1.2LetPbe a cone in a real Banach spaceE, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M16">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M18">View MathML</a>) are nonempty bounded open convex subsets ofPand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M19">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M12">View MathML</a>is a strict set contraction and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M13">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M22">View MathML</a>

Lemma 1.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M23">View MathML</a>is bounded and equicontinuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M24">View MathML</a>is continuous onI, and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M25">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M27">View MathML</a>.

Definition 1.1 The fractional integral of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M29">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M30">View MathML</a>

provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M31">View MathML</a>.

Definition 1.2 The fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28">View MathML</a> of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M29">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M34">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M35">View MathML</a>, provided the right-hand side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M36">View MathML</a>.

Lemma 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M28">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M38">View MathML</a>

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M41">View MathML</a>.

In this article, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M43">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a> is a Banach space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M45">View MathML</a>

Consider the boundary value problem (BVP) for a fractional nonlinear integro-differential equation of mixed type in E:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M46">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M47">View MathML</a> is the standard Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M50">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M51">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M53">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M54">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M58">View MathML</a> denotes the set of all nonnegative real numbers.

2 Several lemmas

To establish the existence of multiple positive solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a> of (1), let us list the following assumptions.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M63">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M64">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a>).

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M68">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M69">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M71">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M72">View MathML</a>

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M74">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M75">View MathML</a>) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M77">View MathML</a>

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M79">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80">View MathML</a>) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M83">View MathML</a> is relatively compact in E, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M84">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85">View MathML</a>) P is normal and solid, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M89">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M90">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M92">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M97">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M92">View MathML</a>.

Remark 2.1 It is clear that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80">View MathML</a>) is satisfied automatically when E is finite dimensional.

Remark 2.2 It is clear that assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>) is weaker than assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85">View MathML</a>).

We shall reduce BVP (1) to an integral equation in E. To this end, we first consider the operator A defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M104">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M105">View MathML</a>.

In our main results, we make use of the following lemmas.

Lemma 2.1Let assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>) be satisfied, then the operatorsTandSdefined by (2) are bounded linear operators from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M109">View MathML</a>

(4)

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M110">View MathML</a>

(5)

Proof Inequalities (4) follow from two simple inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M111">View MathML</a>

and (5) is obvious. □

Lemma 2.2Let assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>) be satisfied, then the operatorAdefined by (3) is a continuous operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>.

Proof

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M117">View MathML</a>

where λ is defined in the operator A.

By virtue of assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>), there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M120">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M121">View MathML</a>

(6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M122">View MathML</a>

(7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M123">View MathML</a>

It follows from (6) and (7) that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M126">View MathML</a>

(8)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127">View MathML</a>, we have, by (8) and Lemma 2.1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M128">View MathML</a>

(9)

which implies the convergence of the infinite integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M129">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M130">View MathML</a>

(10)

Thus, we have, by (3), (9) and (10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M131">View MathML</a>

(11)

It follows from (11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M132">View MathML</a>

(12)

Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M133">View MathML</a>.

Finally, we show that A is continuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M135">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M136">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M138">View MathML</a>. By (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M139">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M140">View MathML</a>

(14)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M141">View MathML</a>

(15)

and by (9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M142">View MathML</a>

(16)

It follows from (15) and (16) and the dominated convergence theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M143">View MathML</a>

(17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M144">View MathML</a>

(18)

It follows from (14), (17) and (18) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M145">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M136">View MathML</a>), and the continuity of A is proved. □

Lemma 2.3Let assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>) be satisfied, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150">View MathML</a>is a solution of BVP (1) if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150">View MathML</a>is a solution of the following integral equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M152">View MathML</a>

(19)

i.e., uis a fixed point of the operatorAdefined by (3) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150">View MathML</a> is a solution of BVP (1), then by applying Lemma 1.4 we reduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M155">View MathML</a> to an equivalent integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M156">View MathML</a>

(20)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M159">View MathML</a>. (20) can be rewritten

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M160">View MathML</a>

(21)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M161">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M162">View MathML</a>

(22)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M163">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M164">View MathML</a>

(23)

Now, substituting (22) and (23) into (21), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M165">View MathML</a> satisfies integral equation (19).

Conversely, if u is a solution of (19), the direct differentiation of (19) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M166">View MathML</a>

(24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M167">View MathML</a>

(25)

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M150">View MathML</a>, and by (19), (24) and (25), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M165">View MathML</a> satisfies BVP (1). □

Lemma 2.4Integral equation (19) can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M170">View MathML</a>

(26)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M171">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M172">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M173">View MathML</a>

(27)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M174">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M175">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M176">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M177">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M178">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M179">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M180">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M182">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M183">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M184">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M185">View MathML</a>

By simple calculation, we can prove the rest of the lemma. □

Lemma 2.5Let assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>) be satisfied, and letUbe a bounded subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M190">View MathML</a>is equicontinuous on any finite subinterval ofJ, and for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M193">View MathML</a>

uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M197">View MathML</a>, by using (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M198">View MathML</a>

(28)

This, together with (9) and (10), implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M199">View MathML</a> are equicontinuous on any finite subinterval of J.

Now, we are going to prove that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exists sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a>, which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M202">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195">View MathML</a>.

Together with (28), we need only to show that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exists sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M207">View MathML</a>

It follows from (10) that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exists a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M209">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M210">View MathML</a>

(29)

and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M211">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M212">View MathML</a>

(30)

On the other hand, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M215">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M216">View MathML</a>

Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M219">View MathML</a>

(31)

Therefore, from (29), (30) and (31) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M220">View MathML</a>

Consequently, the proof is complete. □

Lemma 2.6Let assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M70">View MathML</a>) be satisfied, and letUbe a bounded subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M225">View MathML</a>

Proof By Lemma 2.2, we know AU is a bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M227">View MathML</a>

First, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M228">View MathML</a>.

In fact, by Lemma 2.5, we know that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M231">View MathML</a>

(32)

uniformly with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M195">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M234">View MathML</a> is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235">View MathML</a>, by Lemma 1.3, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M236">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M237">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M238">View MathML</a> is the restriction of AU on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M240">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M241">View MathML</a>

satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M242">View MathML</a>

(33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M243">View MathML</a> denote the diameters of bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>.

At the same time, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M245">View MathML</a>, by (32) and (33), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M246">View MathML</a>

(34)

It follows from (33) and (34) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M247">View MathML</a>

Then, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M248">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M249">View MathML</a>

On the other hand, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M191">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M252">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M253">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M252">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M257">View MathML</a>

(35)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M258">View MathML</a> together with (35), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M259">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M260">View MathML</a>

Because ε is arbitrary, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M261">View MathML</a>

Consequently, the proof is complete. □

3 Main results

In this section, we give and prove our main results.

Theorem 3.1Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85">View MathML</a>) be satisfied. Then BVP (1) has at least two positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M264">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>.

Proof By Lemma 2.2 and Lemma 2.4, the operator A defined by (3) is continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>, and by Lemma 2.3, we need only to show that A has two positive fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M264">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>.

First, we shall prove A is compact.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M272">View MathML</a> be bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M273">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M274">View MathML</a>). From (9), we can choose a sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M192">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M277">View MathML</a>

(36)

It follows from Lemma 2.5 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M278">View MathML</a>

(37)

is equicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M235">View MathML</a>. Thus, by (3), (36) and (37), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M280">View MathML</a>

(38)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M284">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M285">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M286">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M287">View MathML</a>, we see that, by virtue of assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M66">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M289">View MathML</a>

(39)

It follows from (38) and (39) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M290">View MathML</a>

which implies, by virtue of the arbitrariness of ε, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M291">View MathML</a>

Using Lemma 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M292">View MathML</a>

Thus, we can conclude that AU is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M44">View MathML</a>, i.e., A is compact.

As in the proof of Lemma 2.2, (12) holds. Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M294">View MathML</a>

(40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86">View MathML</a> is given in assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85">View MathML</a>), and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M297">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M298">View MathML</a> and, by (12) and (40), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M299">View MathML</a>

(41)

By virtue of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M75">View MathML</a>), there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M301">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M302">View MathML</a>

(42)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M303">View MathML</a>

(43)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M304">View MathML</a>

Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M306">View MathML</a>, we have by (42)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M307">View MathML</a>

(44)

It follows from (3), (43) and (44) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M308">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M309">View MathML</a>

(45)

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M310">View MathML</a>

(46)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M311">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M312">View MathML</a>, and we have, by (45) and (46),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M313">View MathML</a>

(47)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M314">View MathML</a>, and we are going to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315">View MathML</a> is an open set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>. It is clear that we need only to show the following: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M317">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M318">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M320">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M321">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M323">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>. So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M325">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M326">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M327">View MathML</a>

(48)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M329">View MathML</a> is continuous on J, we can find an open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M330">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M331">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M332">View MathML</a>

which implies by virtue of (48) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M333">View MathML</a>

Since I is compact, there is a finite collection of such intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M334">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M335">View MathML</a>) which cover I, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M336">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M337">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M335">View MathML</a>). Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M339">View MathML</a>

(49)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M340">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M86">View MathML</a>, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M342">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M343">View MathML</a>

(50)

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M127">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M320">View MathML</a>, which implies by virtue of (49) and (50) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M346">View MathML</a>

Thus, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315">View MathML</a> is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>.

On the other hand, Lemma 2.4 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M85">View MathML</a>) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M350">View MathML</a>

(51)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M351">View MathML</a>

(52)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M352">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M353">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M315">View MathML</a> are nonempty bounded convex open subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>, we see that (41), (47) and (52) imply by virtue of Lemma 1.1 the fixed point indices

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M356">View MathML</a>

(53)

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M357">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M321">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M359">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M360">View MathML</a>

(54)

By (53), (54) and the additivity of the fixed point index (Lemma 1.2), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M361">View MathML</a>

(55)

Finally, (53), (54) and (55) imply that A has two fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M362">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M363">View MathML</a>. We have, by (51), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M265">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>. The proof is complete. □

Remark 3.1 Assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>) and the continuity of f imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M367">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M65">View MathML</a>. Hence, under the assumptions of the theorem, BVP (1) has the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M369">View MathML</a> besides two positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M371">View MathML</a>.

Theorem 3.2Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M60">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M80">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>) be satisfied. Then BVP (1) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M375">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M376">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>.

Proof By Lemma 2.2, Lemma 2.4 and the proof of Theorem 3.1, the operator A defined by (3) is completely continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a>, and by Lemma 2.3, we need only to show that A has one positive fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M380">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M381">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>.

As in the proof of Lemma 2.2, (12) holds. Choose R satisfying (40) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M383">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M94">View MathML</a> is given by assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>). It is clear that U is a nonempty bounded closed convex subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M387">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M388">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M194">View MathML</a>, by (40), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M390">View MathML</a>. On the other hand, as in the proof of Theorem 3.1, Lemma 2.4 and assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M93">View MathML</a>) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M392">View MathML</a>

(56)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M393">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M394">View MathML</a>. Thus, the Schauder fixed point theorem implies that A has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M395">View MathML</a>, and by (56) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M376">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/79/mathml/M266">View MathML</a>. The proof is complete. □

4 Conclusion

In this paper, the issue on the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space has been addressed for the first time. Taking advantage of the fixed point index theory of completely continuous operators, the existence conditions for such boundary value problems have been established.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RL completed the proof and wrote the initial draft. CK provided the problem and gave some suggestions for amendment. RL then finalized the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Natural Science Foundation of China under grant No. 11271248 and the Science and Technology Research Program of Zhejiang Province under grant No. 2011C21036.

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