Open Access Research

On fractional differential inclusions with anti-periodic type 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, 2013:82  doi:10.1186/1687-2770-2013-82


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/82


Received:4 October 2012
Accepted:12 March 2013
Published:10 April 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

This paper investigates the existence of solutions for fractional differential inclusions of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M1">View MathML</a> with anti-periodic type integral boundary conditions by means of some standard fixed point theorems for inclusions. Our results include the cases when the multivalued map involved in the problem has convex as well as non-convex values. The paper concludes with an illustrative example.

MSC: 34A60, 34A08.

Keywords:
fractional differential inclusions; anti-periodic; integral boundary conditions; fixed point theorems

1 Introduction

The topic of fractional differential equations and inclusions has recently emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc.[1-5]. An important characteristic of a fractional-order differential operator, in contrast to its integer-order counterpart, is its nonlocal nature. This feature of fractional-order operators (equations) is regarded as one of the key factors for the popularity of the subject. As a matter of fact, the use of fractional-order operators in the mathematical modeling of several real world processes gives rise to more realistic models as these operators are capable of describing memory and hereditary properties. For some recent results on fractional differential equations, see [6-22] and the references cited therein, whereas some recent work dealing with fractional differential inclusions can be found in [23-28].

In this paper, we study a boundary value problem of fractional differential inclusions with anti-periodic type integral boundary conditions given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M3">View MathML</a> denotes the Caputo derivative of fractional order q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M4">View MathML</a> denotes jth derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M5">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M7">View MathML</a> is a multivalued map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M8">View MathML</a> is the family of all subsets of ℝ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M9">View MathML</a> are given continuous functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M10">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M11">View MathML</a>).

The present work is motivated by a recent paper [22], where the authors considered (1.1) with F as a single-valued map. The existence of solutions for problem (1.1) has been discussed for the cases when the right-hand side is convex as well as non-convex valued. The first result is based on the nonlinear alternative of Leray-Schauder type, whereas the second result is established by combining 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. In the third result, we use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Though the methods used are well known, their exposition in the framework of problem (1.1) is new. We recall some preliminary facts about fractional calculus and multivalued maps in Section 2, while the main results are presented in Section 3.

2 Preliminaries

2.1 Fractional calculus

Let us recall some basic definitions of fractional calculus [1-3].

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M12">View MathML</a> be an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M13">View MathML</a>-times absolutely continuous function. Then the Caputo derivative of fractional order ν for h is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M14">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M15">View MathML</a> denotes the integer part of the real number ν.

Definition 2.2 The Riemann-Liouville fractional integral of order ν is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M16">View MathML</a>

provided the integral exists.

Definition 2.3 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M17">View MathML</a> is called a solution of problem (1.1) if there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M18">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M19">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M21">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M24">View MathML</a>.

In the sequel, the following lemma plays a pivotal role.

Lemma 2.4 ([22])

For a given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M25">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M26">View MathML</a>, the unique solution of the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M28">View MathML</a>subject to the boundary conditions of (1.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M29">View MathML</a>

(2.1)

where

2.2 Basic concepts of multivalued analysis

Let us begin this section with some basic concepts of multi-valued maps [29,30].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a> denote a normed space equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M32">View MathML</a>. A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M33">View MathML</a> is

• convex (closed) valued if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M34">View MathML</a> is convex (closed) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M35">View MathML</a>;

• bounded on bounded sets if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M36">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a> for all bounded sets B in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M39">View MathML</a>;

• upper semi-continuous (u.s.c.) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a> if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M41">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M42">View MathML</a> is a nonempty closed subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a>, and if for each open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M44">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a> containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M42">View MathML</a>, there exists an open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M47">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M48">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M49">View MathML</a>;

• completely continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M50">View MathML</a> is relatively compact for every bounded set B in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a>.

Remark 2.5 If the multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> is completely continuous with nonempty compact values, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> is u.s.c. if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> has a closed graph, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M57">View MathML</a> imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M58">View MathML</a>.

Definition 2.6 The multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> has a fixed point if there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M35">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M61">View MathML</a>. The fixed point set of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> is denoted by FixG.

Definition 2.7 A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M63">View MathML</a> with nonempty compact convex values is said to be measurable if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M64">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M65">View MathML</a>

is measurable.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M66">View MathML</a> denote the Banach space of all continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a> into ℝ with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M68">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M69">View MathML</a> be the Banach space of measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M70">View MathML</a> which are Lebesgue integrable and normed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M71">View MathML</a>

Definition 2.8 A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M72">View MathML</a> is called Carathéodory if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M73">View MathML</a> is measurable for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M75">View MathML</a> is upper semicontinuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M76">View MathML</a>. A Carathéodory function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M52">View MathML</a> is said to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M78">View MathML</a>-Carathéodory if, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M79">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M80">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M81">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M82">View MathML</a> and for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M84">View MathML</a>, we define the set of selections of F by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M85">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M86">View MathML</a> denote a nonempty closed subset of a Banach space E, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M87">View MathML</a> be a multivalued operator with nonempty closed values. The map G is lower semi-continuous (l.s.c.) if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M88">View MathML</a> is open for any open set B in E. Let A be a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M89">View MathML</a>. A is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M90">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/2013/1/82/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M91">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M92">View MathML</a> is Lebesgue measurable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M94">View MathML</a> is Borel measurable in ℝ. A subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M95">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M96">View MathML</a> is decomposable if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M97">View MathML</a> and measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M98">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M99">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M100">View MathML</a> stands for the characteristic function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M92">View MathML</a>.

Definition 2.9 Let Y be a separable metric space. A multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M102">View MathML</a> has the property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M103">View MathML</a> be a multivalued map with nonempty compact values. Define a multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M104">View MathML</a> associated with F as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M105">View MathML</a>

which is called the Nemytskii operator associated with F.

Definition 2.10 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M106">View MathML</a> be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semi-continuous and has nonempty closed and decomposable values.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M107">View MathML</a> be a metric space induced from the normed space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M108">View MathML</a>. Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M109">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M110">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M112">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M113">View MathML</a> is a metric space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M114">View MathML</a> is a generalized metric space (see [31]), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M115">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M116">View MathML</a>.

Definition 2.11 A multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M117">View MathML</a> is called γ-Lipschitz if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M119">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M120">View MathML</a> and is a contraction if and only if it is γ-Lipschitz with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M121">View MathML</a>.

3 Existence results

3.1 The Carathéodory case

We recall the following lemmas to prove the existence of solutions for problem (1.1) when the multivalued map F in (1.1) is of Carathéodory type.

Lemma 3.1 (Nonlinear alternative for Kakutani maps) [32]

LetEbe a Banach space, letCbe a closed convex subset ofE, letUbe an open subset ofC, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M122">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M123">View MathML</a>is an upper semicontinuous compact map; here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M124">View MathML</a>denotes the family of nonempty, compact convex subsets ofC. Then either

(i) Fhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M125">View MathML</a>, or

(ii) there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M126">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M128">View MathML</a>.

Lemma 3.2 ([33])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a>be a Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M130">View MathML</a>denote a family of nonempty, compact and convex subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M31">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M132">View MathML</a>be an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M78">View MathML</a>-Carathéodory multivalued map, and let Θ be a linear continuous mapping from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M134">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M135">View MathML</a>. Then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M136">View MathML</a>

is a closed graph operator in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M137">View MathML</a>.

Theorem 3.3Suppose that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M138">View MathML</a>is Carathéodory and has nonempty compact and convex values;

(H2) there exists a continuous nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M139">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M140">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M141">View MathML</a>

(H3) there exist continuous nondecreasing functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M142">View MathML</a>and functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M143">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M144">View MathML</a>

(H4) there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M145">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M146">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M147">View MathML</a>

Then the boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a>.

Proof Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M149">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M150">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M151">View MathML</a>. We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a> satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a>is convex for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M154">View MathML</a>. This step is obvious since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M155">View MathML</a> is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a>maps bounded sets (balls) into bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M157">View MathML</a>. For a positive number ρ, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M158">View MathML</a> be a bounded ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M159">View MathML</a>. Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M163">View MathML</a>

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M28">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M165">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M166">View MathML</a>

Now we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a>maps bounded sets into equicontinuous sets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M157">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M169">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M172">View MathML</a>, we obtain

Obviously, the right-hand side of the above inequality tends to zero independently of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M161">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M175">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a> satisfies the above three assumptions, it follows by the Ascoli-Arzelá theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M177">View MathML</a> is completely continuous.

In our next step, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a>has a closed graph. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M181">View MathML</a>. Then we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M182">View MathML</a>. Associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M180">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M184">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M186">View MathML</a>

Thus it suffices to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M187">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M189">View MathML</a>

Let us consider the continuous linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M190">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M191">View MathML</a>

Observe that

Thus, it follows by Lemma 3.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M193">View MathML</a> is a closed graph operator. Further, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M194">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M179">View MathML</a>, therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M196">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M187">View MathML</a>.

Finally, we show there exists an open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M198">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M199">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M201">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M203">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M204">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162">View MathML</a> such that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M207">View MathML</a>

and using the computations of the second step above, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M208">View MathML</a>

Consequently, we have

In view of (H4), there exists M such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M210">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M211">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M212">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/2013/1/82/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M214">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M127">View MathML</a>. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a> has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M217">View MathML</a> which is a solution of problem (1.1). This completes the proof. □

3.2 The lower semicontinuous case

This section deals with the case when F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo for lower semi-continuous maps with decomposable values.

Lemma 3.4 (Bressan and Colombo [34])

LetYbe a separable metric space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M218">View MathML</a>be a multivalued operator satisfying the property (BC). ThenNhas a continuous selection, that is, there exists a continuous function (single-valued) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M219">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M220">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M221">View MathML</a>.

Theorem 3.5Assume that (H2), (H3), (H4) and the following condition hold:

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M138">View MathML</a>is a nonempty compact-valued multivalued map such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M223">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M224">View MathML</a>measurable;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M225">View MathML</a>is lower semicontinuous for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>;

then the boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a>.

Proof It follows from (H2) and (H4) that F is of l.s.c. type. Then from Lemma 3.4, there exists a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M228">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M229">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M230">View MathML</a>.

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M231">View MathML</a>

(3.1)

Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M232">View MathML</a> is a solution of (3.1), then x is a solution to problem (1.1). In order to transform problem (3.1) into a fixed point problem, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M233">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M234">View MathML</a>

It can easily be shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M233">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

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

Lemma 3.6 ([35])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M236">View MathML</a>be a complete metric space. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M237">View MathML</a>is a contraction, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M238">View MathML</a>.

Theorem 3.7Assume that the following conditions hold:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M239">View MathML</a>is such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M240">View MathML</a>is measurable for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M74">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M242">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M243">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M244">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M245">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M246">View MathML</a>for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>;

(A3) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M24">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M250">View MathML</a>

Then the boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M20">View MathML</a>if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M252">View MathML</a>

Proof Observe that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M155">View MathML</a> is nonempty for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M230">View MathML</a> by the assumption (A1), so F has a measurable selection (see Theorem III.6 [36]). Now we show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a>, defined in the beginning of the proof of Theorem 3.3, satisfies the assumptions of Lemma 3.6. To show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M256">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M154">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M258">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M259">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M260">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M261">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M184">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M264">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/2013/1/82/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M265">View MathML</a> converges to v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M266">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M162">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M243">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M269">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M270">View MathML</a>.

Next we show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M271">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M272">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M273">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M274">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M275">View MathML</a> such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M277">View MathML</a>

By (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M278">View MathML</a>

So, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M279">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M280">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M281">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M282">View MathML</a>

Since the multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M283">View MathML</a> is measurable (Proposition III.4 [36]), there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M284">View MathML</a> which is a measurable selection for U. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M285">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M287">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M83">View MathML</a>, let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M289">View MathML</a>

Thus,

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M291">View MathML</a>

Analogously, interchanging the roles of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M292">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M293">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a> is a contraction, it follows by Lemma 3.6 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M152">View MathML</a> has a fixed point x which is a solution of (1.1). This completes the proof. □

Example 3.8

Consider the following boundary value problem of fractional differential inclusions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M296">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M297">View MathML</a> is a multivalued map given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M298">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M299">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M300">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M301">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M303">View MathML</a>. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M304">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M307">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M308">View MathML</a>.

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M309">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M316">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M319">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M320">View MathML</a>. In view of the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M321">View MathML</a>

we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M322">View MathML</a>. Thus, all the conditions of Theorem 3.3 are satisfied. So, there exists at least one solution of problem (3.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/82/mathml/M22">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.

Acknowledgements

The authors are grateful to the anonymous referees for their useful comments. The research of B. Ahmad and A. Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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