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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Existence results for fractional integral inclusions via nonlinear alternative for contractive maps

Ahmed Alsaedi1, Sotiris K Ntouyas2, Bashir Ahmad1* and Hamed H Alsulami1

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 2014, 2014:25  doi:10.1186/1687-2770-2014-25


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/25


Received:18 November 2013
Accepted:10 January 2014
Published:30 January 2014

© 2014 Alsaedi 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, a new existence result is obtained for a fractional multivalued problem with fractional integral boundary conditions by applying a (Krasnoselskii type) fixed-point result for multivalued maps due to Petryshyn and Fitzpatric [Trans. Am. Math. Soc. 194:1-25, 1974]. The case for lower semi-continuous multivalued maps is also discussed. An example for the illustration of our main result is presented.

MSC: 34A60, 34A08.

Keywords:
fractional differential inclusions; nonlocal boundary conditions; fixed-point theorems

1 Introduction

The theory of fractional differential equations and inclusions has developed into an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [1-4]. The nonlocal behavior exhibited by a fractional-order differential operator makes it distinct from the integer-order differential operator. It means that the future state of a dynamical system or process involving fractional derivatives depends on its current state as well its past states. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of several materials and processes. This characteristic of fractional calculus has contributed to its popularity and has convinced many researchers of the need to shift their focus from classical integer-order models to fractional-order models. There has been a great surge in developing new theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see [5-18] and the references cited therein.

In this paper, we consider the following boundary value problem of fractional differential inclusions with fractional integral boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M5">View MathML</a> denotes the Riemann-Liouville fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M7">View MathML</a> are multivalued maps, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M8">View MathML</a> is the family of all nonempty subsets of ℝ and A, B are real constants.

We establish two new existence results for the problem (1.1). The first result relies on a nonlinear alternative for contractive maps, while in the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semi-continuous multivalued maps with nonempty closed and decomposable values.

The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel and Section 3 deals with the main results.

2 Preliminaries

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

Definition 2.1 The Riemann-Liouville derivative of fractional order q for a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M9">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M11">View MathML</a> denotes the integer part of the real number q.

Definition 2.2 The Riemann-Liouville fractional integral of order q for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M9">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M13">View MathML</a>

provided the integral exists.

Observe that the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M14">View MathML</a> transforms the problem (1.1) to the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M15">View MathML</a>

(2.1)

To define the solutions of the problem (1.1), we need the following lemma. Though the proof of this lemma involves standard arguments, we trace its proof for the convenience of the reader.

Lemma 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M16">View MathML</a>, the unique solution of the linear fractional boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M17">View MathML</a>

(2.2)

is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M18">View MathML</a>

Proof It is well known [3] that the solution of fractional differential equation in (2.2) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M19">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M20">View MathML</a> are arbitrary constants. Using the boundary conditions in (2.2), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M22">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M23">View MathML</a>

Substituting these values in (2.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M24">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M25">View MathML</a> in view of the given values of the parameters involved in the expression. This completes the proof. □

Thus, the solution of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M26">View MathML</a> subject to the boundary conditions given by (1.1) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M27">View MathML</a>

where we have used the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M28">View MathML</a> in the integral of the last term. Using the relation for the Beta function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M29">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M30">View MathML</a>

we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M31">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M32">View MathML</a> denote the Banach space of all continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M33">View MathML</a> endowed with the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M34">View MathML</a>.

To establish the main results of this paper, we use the following form of the nonlinear alternative for contractive maps [[19], Corollary 3.8].

Theorem 2.4LetXbe a Banach space, andDa bounded neighborhood of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M35">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M36">View MathML</a> (here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M37">View MathML</a>denotes the family of all nonempty, compact and convex subsets ofX) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M38">View MathML</a>two multivalued operators satisfying

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M39">View MathML</a>is contraction, and

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M40">View MathML</a>is upper semi-continuous (u.s.c. for shortly) and compact.

Then, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M41">View MathML</a>, either

(i) Hhas a fixed point in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M42">View MathML</a>or

(ii) there is a point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M44">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M45">View MathML</a>.

Definition 2.5 A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M46">View MathML</a> is said to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47">View MathML</a>-Carathéodory if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M48">View MathML</a> is measurable for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M49">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M50">View MathML</a> is upper semi-continuous for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51">View MathML</a>, and

(iii) for each real number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M52">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M54">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M55">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M56">View MathML</a>.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M57">View MathML</a>

Lemma 2.6 (Lasota and Opial [20])

LetXbe a Banach space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M58">View MathML</a>be an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47">View MathML</a>Carathéodory multivalued map and let Θ be a linear continuous mapping from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M60">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M61">View MathML</a>. Then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M62">View MathML</a>

is a closed graph operator in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M63">View MathML</a>.

3 Existence results

Before presenting the main results, we define the solutions of the boundary value problem (1.1).

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M64">View MathML</a> is said to be a solution of the problem (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M66">View MathML</a> and there exist functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M68">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M69">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M70">View MathML</a>

(3.1)

In the sequel, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M71">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M72">View MathML</a>

(3.3)

Theorem 3.2Assume that

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M46">View MathML</a>is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47">View MathML</a>Carathéodory multivalued map;

(H2) there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M75">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M76">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M78">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M79">View MathML</a>is given by (3.2);

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M80">View MathML</a>is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M47">View MathML</a>Carathéodory multivalued map;

(H4) there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M82">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M83">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51">View MathML</a>and a nondecreasing function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M85">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M86">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M87">View MathML</a>;

(H5) there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M88">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M89">View MathML</a>

(3.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M91">View MathML</a>are given by (3.2) and (3.3), respectively, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M92">View MathML</a>.

Then the problem (1.1) has a solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a>.

Proof To transform the problem (1.1) to a fixed-point problem, let us define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M94">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M95">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M97">View MathML</a>, where Q is given by (3.1).

We study the integral inclusion in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M98">View MathML</a> of all continuous real valued functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a> with supremum norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M100">View MathML</a>. Define two multivalued maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M101">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M102">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M67">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M104">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M68">View MathML</a>.

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M106">View MathML</a>. We shall show that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> satisfy all the conditions of Theorem 2.4 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a>. For the sake of clarity, we split the proof into a sequence of steps and claims.

Step 1. We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107">View MathML</a>is a multivalued contraction on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M111">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M113">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M114">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M115">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M116">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M117">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M119">View MathML</a>. Thus the multivalued operator U is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M120">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M121">View MathML</a>

has nonempty values and is measurable. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M122">View MathML</a> be a measurable selection for U (which exists by Kuratowski-Ryll-Nardzewski’s selection theorem [21,22]). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M124">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a>.

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M126">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M127">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M128">View MathML</a>

Taking the supremum over the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M129">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M130">View MathML</a>

(3.5)

Combining the inequality (3.5) with the corresponding one obtained by interchanging the roles of x and y, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M131">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M112">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107">View MathML</a> is a multivalued contraction as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M134">View MathML</a>

Step 2. We shall show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a>is u.s.c. and compact. It is well known [[23], Proposition 1.2] that if an operator is completely continuous and has a closed graph, then it is u.s.c. Therefore we will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> is completely continuous and has a closed graph. This step involves several claims.

Claim I<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a>maps bounded sets into bounded sets in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M111">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M139">View MathML</a> be a bounded set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M140">View MathML</a>.

Now for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M141">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M143">View MathML</a>

Then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M145">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M146">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> is bounded.

Claim II<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a>maps bounded sets into equicontinuous sets.

As in the proof of Claim I, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M149">View MathML</a> be a bounded set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M141">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M151">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M153">View MathML</a>

Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M154">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M155">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M156">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/2014/1/25/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M157">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M158">View MathML</a>. Therefore it follows by the Arzelá-Ascoli theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M159">View MathML</a> is completely continuous.

Claim IIINext we prove that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a>has a closed graph.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M163">View MathML</a>. Then we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M164">View MathML</a>. Associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M165">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M166">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M168">View MathML</a>

Thus it suffices to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M169">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M171">View MathML</a>

Let us consider the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M172">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M173">View MathML</a>

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M174">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M175">View MathML</a>. Thus, it follows by Lemma 2.6 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M176">View MathML</a> is a closed graph operator. Further, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M177">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M161">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M179">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M169">View MathML</a>.

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> has a closed graph (and therefore it has closed values). In consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> is compact valued.

Therefore the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M108">View MathML</a> satisfy all the conditions of Theorem 2.4. So the conclusion of Theorem 2.4 applies and either condition (i) or condition (ii) holds. We show that the conclusion (ii) is not possible. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M185">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M186">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M188">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M189">View MathML</a>

(3.6)

By hypothesis (H2), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M190">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M191">View MathML</a>

Hence for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M192">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M193">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M51">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M195">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M196">View MathML</a>

(3.7)

Now, if condition (ii) of Theorem 2.4 holds, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M198">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M199">View MathML</a>. Then x is a solution of (3.6) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M200">View MathML</a> and consequently, the inequality (3.7) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M201">View MathML</a>

which contradicts (3.4). Hence, has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a> by Theorem 2.4, which in fact is a solution of the problem (1.1). This completes the proof.  □

3.1 The lower semi-continuous case

This section is devoted to the study of the case that the maps in (1.1) are not necessarily convex-valued. We establish the existence result for the problem at hand by applying the nonlinear alternative of Leray-Schauder type and a selection theorem due to Bressan and Colombo [24] for lower semi-continuous maps with decomposable values. Before presenting this result, we revisit some basic concepts.

Let be a nonempty closed subset of a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M206">View MathML</a> be a multivalued operator with nonempty closed values. is lower semi-continuous (l.s.c.) if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M208">View MathML</a> is open for any open set in . Let be a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M212">View MathML</a>. is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M214">View MathML</a> measurable if belongs to the σ algebra generated by all sets of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M216">View MathML</a>, where is Lebesgue measurable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a> and is Borel measurable in ℝ. A subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M60">View MathML</a> is decomposable if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M222">View MathML</a> and measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M223">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M224">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M225">View MathML</a> stands for the characteristic function of .

Definition 3.3 Let Y be a separable metric space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M227">View MathML</a> be a multivalued operator. We say has a property (BC) if is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M230">View MathML</a> be a multivalued map with nonempty compact values. Define a multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M231">View MathML</a> associated with F as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M232">View MathML</a>

which is called the Nemytskii operator associated with F.

Definition 3.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M233">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.

Lemma 3.5 ([25])

LetYbe a separable metric space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M227">View MathML</a>be a multivalued operator satisfying the property (BC). Thenhas a continuous selection, that is, there exists a continuous function (single-valued) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M236">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M237">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M238">View MathML</a>.

Theorem 3.6Assume that (H2), (H4), (H5), and the following condition hold:

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M239">View MathML</a>are nonempty compact-valued multivalued maps such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M241">View MathML</a>are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M242">View MathML</a>measurable,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M243">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M244">View MathML</a>are lower semicontinuous for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M167">View MathML</a>;

Then the boundary value problem (1.1) has at least one solution on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M93">View MathML</a>.

Proof It follows from (H2), (H4), and (H6) that F and G are of l.s.c. type. Then from Lemma 3.5, there exist continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M247">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M249">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M250">View MathML</a>.

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M251">View MathML</a>

(3.8)

Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M252">View MathML</a> is a solution of (3.8), then x is a solution to the problem (1.1). Now, we define two multivalued operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M253">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M254">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M255">View MathML</a>

Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M256">View MathML</a> are continuous. Also the argument in Theorem 3.2 guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M258">View MathML</a> satisfy all the conditions of the nonlinear alternative for contractive maps in the single-valued setting [26] and hence the problem (3.8) has a solution. □

Example 3.7 Consider the following fractional boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M259">View MathML</a>

(3.9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M260">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M261">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M262">View MathML</a>. Using the given data, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M263">View MathML</a>

Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M78">View MathML</a>, and by the condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M265">View MathML</a>

it is found that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M266">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/25/mathml/M267">View MathML</a>. Thus, all the assumptions of Theorem 3.2 are satisfied. Hence, the conclusion of Theorem 3.2 applies to the problem (3.9).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Authors’ information

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

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant no. 3-130/1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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