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Existence results for fractional differential inclusions with three-point fractional integral boundary conditions

Xi Fu

Author Affiliations

Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang, 312000, P.R. China

Advances in Difference Equations 2013, 2013:304  doi:10.1186/1687-1847-2013-304

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/304


Received:15 May 2013
Accepted:16 October 2013
Published:8 November 2013

© 2013 Fu; licensee Springer.

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

Abstract

This paper is concerned with fractional differential inclusions with three-point fractional integral boundary conditions. We consider the fractional differential inclusions under both convexity and nonconvexity conditions on the multivalued term. Some new existence results are obtained by using standard fixed point theorems. Two examples are given to illustrate the main results.

MSC: 34A60, 26A33, 34B15.

Keywords:
fractional differential inclusions; boundary value problems; existence results; multivalued maps

1 Introduction

Fractional differential equations have recently gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many physical phenomena [1-3]. For some recent developments on the existence results of fractional differential equations, we can refer, for instance, to [4-17] and the references therein.

Differential inclusions arise in the mathematical modeling of certain problems in economics, optimal control, etc. and are widely studied by many authors, see [18,19] and the references therein. For some recent works on differential inclusions of fractional order, we refer the reader to the references [4,5,20-29].

Motivated by the above papers, in this article, we study a new class of fractional boundary value problems, i.e., the following fractional differential inclusions with three-point fractional integral boundary conditions:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M2">View MathML</a> denotes the Caputo fractional derivative of order p, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M3">View MathML</a> the Riemann-Liouville fractional integral of order q, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M4">View MathML</a> is a multifunction and a, b, c are real constants with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M5">View MathML</a>.

We remark that when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M7">View MathML</a> and third variable of the function F in (1) vanishes, problem (1) reduces to a three-point fractional integral boundary value problem (see [17] with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M8">View MathML</a> a given continuous function).

The rest of this paper is organized as follows. In Section 2 we present the notations, definitions and give some preliminary results that we need in the sequel, Section 3 is dedicated to the existence results of problem (1), in the final Section 4, two examples are given to illustrate the main results.

2 Preliminaries

In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M9">View MathML</a> be a normed space. We use the notations: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M12">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M13">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M14">View MathML</a> and so on.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M15">View MathML</a>, the Pompeiu-Hausdorff distance of A, B is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M16">View MathML</a>

A multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M17">View MathML</a> is convex (closed) valued if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18">View MathML</a> is convex (closed) for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19">View MathML</a>. F is said to be completely continuous if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M20">View MathML</a> is relatively compact for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M21">View MathML</a>. F is called upper semicontinuous on X if, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19">View MathML</a>, the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18">View MathML</a> is a nonempty closed subset of X, and for every open set O of X containing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M18">View MathML</a>, there exists an open neighborhood U of x such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M25">View MathML</a>. Equivalently, F is upper semicontinuous if the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M26">View MathML</a> is open for any open set O of X. F is called lower semicontinuous if the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M27">View MathML</a> is open for each open set O in X. If a multivalued map F is completely continuous with nonempty compact values, then F is upper semicontinuous if and only if F has a closed graph, i.e., if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M29">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M30">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M31">View MathML</a>[30].

A multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M32">View MathML</a> is said to be measurable if, for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M19">View MathML</a>, the function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M34">View MathML</a> is a measurable function.

Definition 2.1 A multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M35">View MathML</a> is called

(1) γ-Lipschitz if there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M36">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M37">View MathML</a>

(2) a contraction if it is γ-Lipschitz with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M38">View MathML</a>.

Definition 2.2 A multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M39">View MathML</a> is said to be Carathéodory if:

(1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M40">View MathML</a> is measurable for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M41">View MathML</a>;

(2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M42">View MathML</a> is upper semicontinuous for a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M43">View MathML</a>.

Further, a Carathéodory function F is said to be <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M44">View MathML</a>-Carathéodory if

(3) for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M45">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M47">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M48">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M49">View MathML</a> and a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>.

The following lemmas will be used in the sequel.

Lemma 2.1 (see [31])

LetXbe a Banach space. Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M51">View MathML</a>be an<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M44">View MathML</a>-Carathéodory multivalued map andPbe a linear continuous map from<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M53">View MathML</a>to<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M54">View MathML</a>, then the operator

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M55">View MathML</a>

is a closed graph operator in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M56">View MathML</a>.

Here the set of selections

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M57">View MathML</a>

Definition 2.3 ([32])

The Riemann-Liouville fractional integral of order q for a function f is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M58">View MathML</a>

provided the integral exists.

Definition 2.4 ([32])

For at least n-times differentiable function f, the Caputo derivative of order q is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M59">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M60">View MathML</a> denotes the integer part of the real number q.

Lemma 2.2 ([16])

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M61">View MathML</a>, then the differential equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M62">View MathML</a>

has solutions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M63">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M64">View MathML</a>

here<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M65">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M66">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M67">View MathML</a>.

Lemma 2.3For any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M68">View MathML</a>, the unique solution of the three-point boundary value problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M69">View MathML</a>

(2)

is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M70">View MathML</a>

Proof For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M71">View MathML</a> and some constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M72">View MathML</a>, the general solution of the equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M73">View MathML</a> can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M74">View MathML</a>

(3)

From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M75">View MathML</a>, it follows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M76">View MathML</a>. Using the integral boundary conditions of (2), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M77">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M78">View MathML</a>

Substituting the values of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M79">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M80">View MathML</a>, we obtain the result. This completes the proof. □

Let us define what we mean by a solution of problem (1).

Definition 2.5 A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M81">View MathML</a> is a solution of problem (1) if it satisfies the boundary conditions in (1) and there exists a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M82">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M83">View MathML</a> a.e. on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M85">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M86">View MathML</a> be the space of all continuous functions defined on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>. Define the space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M88">View MathML</a> endowed with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M89">View MathML</a>. Obviously, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M90">View MathML</a> is a Banach space.

Theorem 2.1 (Nonlinear alternative of Leray-Schauder type)

LetXbe a Banach space, Cbe a closed convex subset ofX, Ube an open subset ofCwith<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M91">View MathML</a>. Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M92">View MathML</a>is an upper semicontinuous compact map. Then either (1) Fhas a fixed point in<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M93">View MathML</a>, or (2) there are<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M94">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M96">View MathML</a>.

Theorem 2.2 (Covitz and Nadler)

Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M97">View MathML</a>be a complete metric space. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M35">View MathML</a>is a contraction, thenFhas a fixed point.

3 Existence results

In this section, three existence results of problem (1) are presented. The first one concerns the convex valued case, and the others are related to the nonconvex valued case.

Now let us begin with the convex valued case.

Theorem 3.1Suppose that the following (H1), (H2) and (H3) are satisfied.

(H1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M99">View MathML</a>is a Carathéodory multivalued map.

(H2) There exist<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M100">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M101">View MathML</a>continuous, nondecreasing such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M102">View MathML</a>

for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M103">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>.

(H3) There exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M105">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M106">View MathML</a>

(4)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M107">View MathML</a>

Then boundary value problem (1) has at least one solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>.

Proof Consider the multivalued operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M109">View MathML</a> defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M110">View MathML</a>

(5)

with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M111">View MathML</a>

Clearly, by Lemma 2.3, we know that the fixed points of N are solutions of problem (1). From (H1) and (H2), we have, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a>, that the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113">View MathML</a> is nonempty [31]. Next we will show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof is given in the following five steps.

Step 1: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114">View MathML</a>is convex valued. Since F is convex valued, we know that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113">View MathML</a> is convex and therefore it is obvious that for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114">View MathML</a> is convex.

Step 2: Nmaps bounded sets into bounded sets in. Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M119">View MathML</a>

be a bounded subset of . We need to prove that there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M45">View MathML</a> such that for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122">View MathML</a>, one has <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M123">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124">View MathML</a>, then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M128">View MathML</a>

By simple calculations, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M129">View MathML</a>

Similarly, we can obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M130">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M131">View MathML</a>

Hence, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M132">View MathML</a>

Step 3: Nmaps bounded sets into equicontinuous sets in. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M134">View MathML</a> be as in Step 2 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M135">View MathML</a>. Then, for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M139">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>. Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M141">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M142">View MathML</a>

we deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M143">View MathML</a>

independently of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124">View MathML</a>.

Step 4: Nhas a closed graph. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M147">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M148">View MathML</a>, we need to show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M149">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M147">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M151">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M152">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>. We must prove that there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M155">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>.

Now, let us consider the continuous linear operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M157">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M158">View MathML</a>

and denote

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M159">View MathML</a>

Clearly, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M160">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M161">View MathML</a>

By the definition of P, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M162">View MathML</a>

It follows from Lemma 2.1 that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M163">View MathML</a> is a closed graph operator. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M28">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M165">View MathML</a>

for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M154">View MathML</a>. This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M149">View MathML</a>.

Step 5: A priori bounds for solutions. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M168">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95">View MathML</a>. Then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M171">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>. By a similar discussion as in Step 2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M173">View MathML</a>

Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M174">View MathML</a>

By the assumption of (H3), there exists M such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M175">View MathML</a>. Let us set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M176">View MathML</a>

As a consequence of Steps 1-4, together with the Arzela-Ascoli theorem, we can obtain that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M177">View MathML</a> is an upper semicontinuous and completely continuous map. From the choice of U, there is no <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M94">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M168">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M95">View MathML</a>. Hence, by Theorem 2.1, we deduce that N has a fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M181">View MathML</a> which is a solution of problem (1). This is the end of the proof. □

Next we study the case when F is not necessarily convex valued.

Let A be a subset of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M182">View MathML</a>. A is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M183">View MathML</a> measurable if A belongs to the σ-algebra generated by all sets of the form <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M184">View MathML</a>, where J is Lebesgue measurable in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a> and D is a Borel set of ℝ. A subset A of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M186">View MathML</a> is decomposable if for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M187">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M188">View MathML</a> Lebesgue measurable, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M189">View MathML</a>, where χ stands for the characteristic function.

Theorem 3.2Let (H2) and (H3) hold and assume:

(H4) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M190">View MathML</a>is such that: (1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M191">View MathML</a>is<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M192">View MathML</a>measurable; (2) the map<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M193">View MathML</a>is lower semicontinuous for a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>.

Then problem (1) has at least one solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>.

Proof From (H2), (H4) and Lemma 4.4 of [27], the map

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M196">View MathML</a>

(6)

is lower semicontinuous and has nonempty closed and decomposable values. Then, from a selection theorem due to Bressan and Colombo [33], there exists a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M197">View MathML</a> such that for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M199">View MathML</a> a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M43">View MathML</a>. Now consider the problem

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M201">View MathML</a>

(7)

with the boundary conditions in (2). Note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a> is a solution of problem (7), then x is a solution to problem (1).

Problem (7) is then reformulated as a fixed point problem for the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M203">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M204">View MathML</a>

It can easily be shown that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M205">View MathML</a> is continuous and completely continuous and satisfies all conditions of the Leray-Schauder nonlinear alternative for single-valued maps [34]. By a discussion similar to the one in Theorem 3.1, Theorem 3.2 follows. □

Theorem 3.3We assume that:

(H5) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M190">View MathML</a>is such that: (1) the map<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M40">View MathML</a>is measurable for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M208">View MathML</a>; (2) there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M100">View MathML</a>such that for a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>and all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M211">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M212">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M213">View MathML</a>

(8)

then problem (1) has at least one solution on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>.

Proof From (H5), for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a>, the multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M216">View MathML</a> is measurable and closed valued. Hence it has measurable selection (Theorem 2.2.1 [30]) and the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M113">View MathML</a> is nonempty. Let N be defined in (5). We will show that N satisfies the requirements of Theorem 2.2.

Step 1: For each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M112">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M219">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M220">View MathML</a> be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M221">View MathML</a> in . Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M223">View MathML</a> and there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M224">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M225">View MathML</a>

By (H5), the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M226">View MathML</a> is integrable bounded. Since F has compact values, we may pass to a subsequence if necessary to get that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M226">View MathML</a> converges to v in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M186">View MathML</a>. Thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M127">View MathML</a> and for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M231">View MathML</a>

This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M124">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M114">View MathML</a> is closed.

Step 2: There exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M234">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M235">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M236">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M237">View MathML</a>, then there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M238">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M239">View MathML</a>

From (H5), we know that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M240">View MathML</a>

Hence, for a.e. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M242">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M243">View MathML</a>

(9)

Consider the multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M244">View MathML</a> given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M245">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M246">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M247">View MathML</a> are measurable, Theorem III.41 in [35] implies that V is measurable. It follows from (H5) that the map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M248">View MathML</a> is measurable. Hence, by (9) and Proposition 2.1.43 in [30], the multivalued map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M249">View MathML</a> with nonempty closed values is measurable. Therefore, we can find <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M250">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M251">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M252">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M253">View MathML</a>. Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M254">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M255">View MathML</a>

we obtain that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M256">View MathML</a>

Define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M257">View MathML</a>

By using an analogous relation obtained by interchanging the roles of x and y, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M258">View MathML</a>

Therefore from condition (8), Theorem 2.2 implies that N has a fixed point which is a solution of problem (1). This completes the proof. □

4 Examples

In this section, we give two examples to illustrate the results.

Example 1 Consider the following three-point fractional integral boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M259">View MathML</a>

(10)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M260">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M261">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M262">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M263">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M264">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M265">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M266">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M267">View MathML</a> is a multivalued map given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M268">View MathML</a>

In the context of this problem, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M269">View MathML</a>

It is clear that F is convex compact valued and is of Carathéodory type. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M270">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M271">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M272">View MathML</a>, we get that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M50">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M274">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M275">View MathML</a>

As for condition (4), since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M276">View MathML</a> (see O, Q in (H3)) is a constant, we can choose M large enough so that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M277">View MathML</a>

Thus, by the conclusion of Theorem 3.1, boundary value problem (10) has at least one solution on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>.

Example 2 Consider the following three-point fractional integral boundary value problem:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M279">View MathML</a>

(11)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M280">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M281">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M262">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M283">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M264">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M285">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M286">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M287">View MathML</a>

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M288">View MathML</a>.

From the data given above, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M289">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M290">View MathML</a>, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M291">View MathML</a>

Hence it follows from Theorem 3.3 that problem (11) has at least one solution on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/304/mathml/M87">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author carried out the proofs of the theorems and approved the final manuscript.

Acknowledgements

The author would like to express his thanks to the referees for their helpful suggestions.

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