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

Open Access Research

New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions

Bashir Ahmad1*, Sotiris K Ntouyas2 and Ahmed Alsaedi1

Author Affiliations

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

2 Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece

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Boundary Value Problems 2013, 2013:275  doi:10.1186/1687-2770-2013-275

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/275


Received:30 September 2013
Accepted:25 November 2013
Published:13 December 2013

© 2013 Ahmad et al.; licensee Springer.

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

Abstract

In this paper, we study the existence of solutions for a fractional boundary value problem involving Hadamard-type fractional differential inclusions and integral boundary conditions. Our results include the cases for convex as well as non-convex valued maps and are based on standard fixed point theorems for multivalued maps. Some illustrative examples are also presented.

MSC: 34A60, 34A08.

Keywords:
Hadamard fractional derivative; integral boundary conditions; fixed point theorems

1 Introduction

The theory of fractional differential equations and inclusions has received much attention over the past years and has become an important field of investigation due to its extensive applications in numerous branches of physics, economics and engineering sciences [1-4]. Fractional differential equations and inclusions provide appropriate models for describing real world problems, which cannot be described using classical integer order differential equations. Some recent contributions to the subject can be seen in [5-21] and references cited therein.

It has been noticed that most of the work on the topic is based on Riemann-Liouville and Caputo-type fractional differential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [22], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains a logarithmic function of arbitrary exponent. Details and properties of the Hadamard fractional derivative and integral can be found in [1,23-27].

In this paper, we study the following boundary value problem of Hadamard-type fractional differential inclusions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M2">View MathML</a> is the Hadamard fractional derivative of order α, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M3">View MathML</a> is the Hadamard fractional integral of order β and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M4">View MathML</a> is a multivalued map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M5">View MathML</a> is the family of all nonempty subsets of ℝ.

We aim to establish a variety of results for inclusion problem (1.1) by considering the given multivalued map to be convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while the third result is obtained by using the fixed point theorem for contractive multivalued maps due to Covitz and Nadler.

We emphasize that the main idea of the present research is to introduce Hadamard-type fractional differential inclusions supplemented with Hadamard-type integral boundary conditions and develop some existence results for the problem at hand. It is imperative to note that our results are absolutely new in the context of Hadamard-type integral boundary value problems and provide a new avenue to the researchers working on fractional boundary value problems.

The paper is organized as follows. In Section 2, we solve a linear Hadamard-type integro-differential boundary value problem and recall some preliminary concepts of multivalued analysis that we need in the sequel. Section 3 contains the main results for problem (1.1). In Section 4, some illustrative examples are discussed.

2 Preliminaries

This section is devoted to the basic concepts of Hadamard-type fractional calculus and multivalued analysis. We also establish an auxiliary lemma to define the solutions for the given problem.

2.1 Fractional calculus

Definition 2.1[1]

The Hadamard derivative of fractional order q for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M6">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M8">View MathML</a> denotes the integer part of the real number q and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M9">View MathML</a>.

Definition 2.2[1]

The Hadamard fractional integral of order q for a function g is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M10">View MathML</a>

provided the integral exists.

Lemma 2.3 (Auxiliary lemma)

For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M12">View MathML</a>, the unique solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M13">View MathML</a>

(2.1)

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M14">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M15">View MathML</a>

(2.3)

Proof As argued in [1], the solution of the Hadamard differential equation in (2.1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M16">View MathML</a>

(2.4)

Using the given boundary conditions, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M17">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M18">View MathML</a>

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M19">View MathML</a>

(2.5)

Substituting the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M21">View MathML</a> in (2.4), we obtain (2.2). This completes the proof. □

2.2 Basic concepts of multivalued analysis

Here we outline some basic definitions and results for multivalued maps [28,29].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M22">View MathML</a> denote a Banach space of continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a> into ℝ with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M24">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M25">View MathML</a> be the Banach space of measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M26">View MathML</a> which are Lebesgue integrable and normed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M27">View MathML</a>.

For a normed space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M28">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M32">View MathML</a>. A multi-valued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M33">View MathML</a>:

(i) is convex (closed) valued if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M34">View MathML</a> is convex (closed) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35">View MathML</a>;

(ii) is bounded on bounded sets if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M36">View MathML</a> is bounded in X for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M37">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M38">View MathML</a>);

(iii) is called upper semicontinuous (u.s.c.) on X if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M39">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M40">View MathML</a> is a nonempty closed subset of X and if for each open set N of X containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M40">View MathML</a>, there exists an open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M42">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M44">View MathML</a>;

(iv) G is lower semicontinuous (l.s.c.) if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M45">View MathML</a> is open for any open set B in E;

(v) is said to be completely continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M46">View MathML</a> is relatively compact for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M47">View MathML</a>;

(vi) is said to be measurable if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M48">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M49">View MathML</a>

is measurable;

(vii) has a fixed point if there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M51">View MathML</a>. The fixed point set of the multivalued operator G will be denoted by FixG.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M52">View MathML</a>, define the set of selections of F by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M53">View MathML</a>

(2.6)

We define the graph of G to be the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M54">View MathML</a> and recall two results for closed graphs and upper-semicontinuity.

Lemma 2.4 [[28], Proposition 1.2]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M55">View MathML</a>is u.s.c., then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M56">View MathML</a>is a closed subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M57">View MathML</a>; i.e., for every sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M58">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M59">View MathML</a>, if when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M62">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M63">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M64">View MathML</a>. Conversely, ifGis completely continuous and has a closed graph, then it is upper semicontinuous.

Lemma 2.5[30]

LetXbe a separable Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M65">View MathML</a>be measurable with respect totfor each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M35">View MathML</a>and u.s.c. with respect toxfor almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M67">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M68">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M69">View MathML</a>, and let Θ be a linear continuous mapping from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M70">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M71">View MathML</a>. Then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M72">View MathML</a>

is a closed graph operator in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M73">View MathML</a>.

3 Existence results

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M74">View MathML</a> is called a solution of problem (1.1) if there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M75">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M76">View MathML</a>, a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M78">View MathML</a>, a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M81">View MathML</a>.

3.1 The upper semicontinuous case

Our first main result for Carathéodory case is established via the nonlinear alternative of Leray-Schauder for multivalued maps.

Lemma 3.2 (Nonlinear alternative for Kakutani maps [31])

LetEbe a Banach space, Cbe a closed convex subset ofE, Ube an open subset ofCand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M82">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M83">View MathML</a>is an upper semicontinuous compact map. Then either

(i) Fhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M84">View MathML</a>, or

(ii) there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M85">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M87">View MathML</a>.

Theorem 3.3Assume that:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M88">View MathML</a>is Carathéodory, i.e.,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M89">View MathML</a>is measurable for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M90">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M91">View MathML</a>is u.s.c. for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M67">View MathML</a>;

(H2) there exist a continuous nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M93">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M94">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M95">View MathML</a>

(H3) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M96">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M97">View MathML</a>

where Ω is given by (2.3).

Then problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a>.

Proof In view of Lemma 2.3, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M99">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M100">View MathML</a>

(3.1)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M101">View MathML</a> (defined by (2.6)). Observe that the fixed points of the operator ℱ correspond to the solutions of problem (1.1). We will show that ℱ satisfies the assumptions of the Leray-Schauder nonlinear alternative (Lemma 3.2). The proof consists of several steps.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M102">View MathML</a>is convex for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M103">View MathML</a>.

This step is obvious since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M104">View MathML</a> is convex (F has convex values), and therefore we omit the proof.

Step 2. ℱ maps bounded sets (balls) into bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M105">View MathML</a>.

For a positive number ρ, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M106">View MathML</a> be a bounded ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M107">View MathML</a>. Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M109">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M101">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M111">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M112">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M113">View MathML</a>

Step 3. ℱ maps bounded sets into equicontinuous sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M105">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M115">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M117">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M118">View MathML</a> is a bounded set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M107">View MathML</a> as in Step 2. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M120">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M121">View MathML</a>

Obviously the right-hand side of the above inequality tends to zero independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M122">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M123">View MathML</a>. In view of Steps 1-3, the Arzelá-Ascoli theorem applies and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M124">View MathML</a> is completely continuous.

By Lemma 2.4, ℱ will be upper semicontinuous (u.s.c.) if we prove that it has a closed graph since ℱ is already shown to be completely continuous.

Step 4. ℱ has a closed graph.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M127">View MathML</a>. Then we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M128">View MathML</a>. Associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M129">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M130">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M131">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M132">View MathML</a>

Thus we have to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M133">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M131">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M135">View MathML</a>

Let us consider the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M136">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M137">View MathML</a>

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M138">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M139">View MathML</a>.

Thus, it follows by Lemma 2.5 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M140">View MathML</a> is a closed graph operator. Further, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M141">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M61">View MathML</a>, therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M143">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M133">View MathML</a>.

Step 5. We show that there exists an open set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M145">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M146">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M148">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M149">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M151">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M152">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M154">View MathML</a>

Using the computations of the second step above, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M155">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M156">View MathML</a>

In view of (H3), there exists M such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M157">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M158">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M159">View MathML</a> is upper semicontinuous and completely continuous. From the choice of U, there is no <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M160">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M161">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M86">View MathML</a>. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that ℱ has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M163">View MathML</a> which is a solution of problem (1.1). This completes the proof. □

3.2 The lower semicontinuous case

In what follows, we consider the case when F is not necessarily convex valued and obtain the existence result by combining the nonlinear alternative of Leray-Schauder type with the selection theorem due to Bressan and Colombo [32] for lower semicontinuous maps with decomposable values.

Definition 3.4 Let A be a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M164">View MathML</a>. A is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M165">View MathML</a> measurable if A belongs to the σ-algebra generated by all sets of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M166">View MathML</a>, where is Lebesgue measurable in I and is Borel measurable in ℝ.

Definition 3.5 A subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M170">View MathML</a> is decomposable if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M171">View MathML</a> and measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M172">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M173">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M174">View MathML</a> stands for the characteristic function of .

Lemma 3.6[32]

LetYbe a separable metric space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M176">View MathML</a>be a lower semicontinuous (l.s.c.) multivalued operator with nonempty closed and decomposable values. ThenNhas a continuous selection, that is, there exists a continuous function (single-valued) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M177">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M178">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M179">View MathML</a>.

Theorem 3.7Assume that (H2), (H3) and the following condition holds:

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M180">View MathML</a>is a nonempty compact-valued multivalued map such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M181">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M182">View MathML</a>measurable,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M91">View MathML</a>is lower semicontinuous for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>.

Then problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a>.

Proof It follows from (H2) and (H4) that F is of l.s.c. type [33]. Then, by Lemma 3.6, there exists a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M186">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M187">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M188">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M189">View MathML</a> is the Nemytskii operator associated with F, defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M190">View MathML</a>

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M191">View MathML</a>

(3.2)

Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M192">View MathML</a> is a solution of problem (3.2), then x is a solution to problem (1.1). In order to transform problem (3.2) into a fixed point problem, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M193">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M194">View MathML</a>

It can easily be shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M193">View MathML</a> is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.3. So we omit it. This completes the proof. □

3.3 The Lipschitz case

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M196">View MathML</a> be a metric space induced from the normed space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M197">View MathML</a>. Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M198">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M199">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M201">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M202">View MathML</a> is a metric space (see [34]).

Definition 3.8 A multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M203">View MathML</a> is called

(a) γ-Lipschitz if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M204">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M205">View MathML</a>

(b) a contraction if and only if it is γ-Lipschitz with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M206">View MathML</a>.

To show the existence of solutions for problem (1.1) with a nonconvex valued right-hand side, we need a fixed point theorem for multivalued maps due to Covitz and Nadler [35].

Lemma 3.9[35]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M196">View MathML</a>be a complete metric space. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M208">View MathML</a>is a contraction, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M209">View MathML</a>.

Theorem 3.10Assume that the following conditions hold:

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M210">View MathML</a>is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M211">View MathML</a>is measurable for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M90">View MathML</a>.

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M213">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M214">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M215">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M216">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M217">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>.

Then problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M23">View MathML</a>if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M220">View MathML</a>

Proof We transform problem (1.1) into a fixed point problem by means of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M221">View MathML</a> defined by (3.1) and show that the operator ℱ satisfies the assumptions of Lemma 3.9. The proof will be given in two steps.

Step 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M102">View MathML</a> is nonempty and closed for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M223">View MathML</a>.

Since the set-valued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M224">View MathML</a> is measurable with the measurable selection theorem (e.g., [[36], Theorem III.6]), it admits a measurable selection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M225">View MathML</a>. Moreover, by assumption (H6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M226">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M227">View MathML</a> and hence F is integrably bounded. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M228">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M229">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M103">View MathML</a>. Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M231">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M232">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M60">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M234">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M235">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M236">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M238">View MathML</a>

As F has compact values, we pass onto a subsequence (if necessary) to obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M239">View MathML</a> converges to g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M240">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M241">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M214">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M243">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M244">View MathML</a>.

Step 2. Next we show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M245">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M246">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M248">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M249">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M251">View MathML</a>

By (H6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M252">View MathML</a>

So, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M253">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M254">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M255">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M256">View MathML</a>

Since the multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M257">View MathML</a> is measurable (Proposition III.4 [36]), there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M258">View MathML</a> which is a measurable selection for U. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M259">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M261">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M153">View MathML</a>, let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M263">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M264">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M265">View MathML</a>

Analogously, interchanging the roles of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M266">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M267">View MathML</a>

Since ℱ is a contraction, it follows by Lemma 3.9 that ℱ has a fixed point x which is a solution of (1.1). This completes the proof. □

4 Examples

In this section we present some concrete examples to illustrate our results.

Let us consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M268">View MathML</a>

(4.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M271">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M272">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M273">View MathML</a>

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M274">View MathML</a> be a multivalued map given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M275">View MathML</a>

(4.2)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M276">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M277">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M279">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M280">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M281">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M282">View MathML</a>. Then, by Theorem 3.3, problem (4.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M283">View MathML</a> given by (4.2) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M284">View MathML</a>.

Example 4.2 Consider the multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M285">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M286">View MathML</a>

(4.3)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M287">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M288">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M289">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M290">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M291">View MathML</a>. By Theorem 3.10, problem (4.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M292">View MathML</a> given by (4.3) has at least one solution on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/275/mathml/M284">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, BA, SKN and AA contributed to each part of this work equally and read and approved the final version of the manuscript.

Authors’ information

All authors are members of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Acknowledgements

This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

References

  1. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

  2. Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge (2009)

  3. Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

  4. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon (1993)

  5. Benchohra, M, Hamani, S, Ntouyas, SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal.. 71, 2391–2396 (2009). Publisher Full Text OpenURL

  6. Ahmad, B, Ntouyas, SK: Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. Equ.. 2010, Article ID 71 (2010)

  7. Bai, ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal.. 72, 916–924 (2010). Publisher Full Text OpenURL

  8. Balachandran, K, Trujillo, JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal.. 72, 4587–4593 (2010). Publisher Full Text OpenURL

  9. Baleanu, D, Mustafa, OG: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl.. 59, 1835–1841 (2010). Publisher Full Text OpenURL

  10. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math.. 109, 973–1033 (2010). Publisher Full Text OpenURL

  11. Ahmad, B: Existence of solutions for fractional differential equations of order with anti-periodic boundary conditions. J. Appl. Math. Comput.. 34, 385–391 (2010). Publisher Full Text OpenURL

  12. Agarwal, RP, Cuevas, C, Soto, H: Pseudo-almost periodic solutions of a class of semilinear fractional differential equations on type of periodicity and ergodicity to a class of fractional order differential equations. J. Appl. Math. Comput.. 37, 625–634 (2011). Publisher Full Text OpenURL

  13. Ford, NJ, Luisa Morgado, M: Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal.. 14, 554–567 (2011)

  14. Liu, X, Liu, Z: Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative. Abstr. Appl. Anal.. 2012, Article ID 423796 (2012)

  15. Ahmad, B, Ntouyas, SK: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal.. 15, 362–382 (2012)

  16. Debbouche, A, Baleanu, D, Agarwal, RP: Nonlocal nonlinear integrodifferential equations of fractional orders. Bound. Value Probl.. 2012, Article ID 78 (2012)

  17. Ahmad, B, Ntouyas, SK, Alsaedi, A: An existence result for fractional differential inclusions with nonlinear integral boundary conditions. J. Inequal. Appl.. 2013, Article ID 296 (2013)

  18. Chen, Y, Tang, X, He, X: Positive solutions of fractional differential inclusions at resonance. Mediterr. J. Math.. 10, 1207–1220 (2013). Publisher Full Text OpenURL

  19. Baleanu, D, Agarwal, RP, Mohammadi, H, Rezapour, S: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces. Bound. Value Probl.. 2013, Article ID 112 (2013)

  20. Nyamoradi, N, Baleanu, D, Agarwal, RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys.. 2013, Article ID 823961 (2013)

  21. Ahmad, B, Ntouyas, SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ.. 2013, Article ID 20 (2013)

  22. Hadamard, J: Essai sur l’etude des fonctions donnees par leur developpment de Taylor. J. Math. Pures Appl.. 8, 101–186 (1892)

  23. Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl.. 269, 387–400 (2002). Publisher Full Text OpenURL

  24. Butzer, PL, Kilbas, AA, Trujillo, JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl.. 269, 1–27 (2002). Publisher Full Text OpenURL

  25. Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl.. 270, 1–15 (2002). Publisher Full Text OpenURL

  26. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc.. 38, 1191–1204 (2001)

  27. Kilbas, AA, Trujillo, JJ: Hadamard-type integrals as G-transforms. Integral Transforms Spec. Funct.. 14, 413–427 (2003). Publisher Full Text OpenURL

  28. Deimling, K: Multivalued Differential Equations, de Gruyter, Berlin (1992)

  29. Hu, S, Papageorgiou, N: Handbook of Multivalued Analysis: Volume I: Theory, Kluwer Academic, Dordrecht (1997)

  30. Lasota, A, Opial, Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys.. 13, 781–786 (1965)

  31. Granas, A, Dugundji, J: Fixed Point Theory, Springer, New York (2003)

  32. Bressan, A, Colombo, G: Extensions and selections of maps with decomposable values. Stud. Math.. 90, 69–86 (1988)

  33. Frigon, M: Théorèmes d’existence de solutions d’inclusions différentielles. In: Granas A, Frigon M (eds.) Topological Methods in Differential Equations and Inclusions, pp. 51–87. Kluwer Academic, Dordrecht (1995)

  34. Kisielewicz, M: Differential Inclusions and Optimal Control, Kluwer Academic, Dordrecht (1991)

  35. Covitz, H, Nadler, SB Jr..: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math.. 8, 5–11 (1970). Publisher Full Text OpenURL

  36. Castaing, C, Valadier, M: Convex Analysis and Measurable Multifunctions, Springer, Berlin (1977)