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Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions

Bashir Ahmad1* and Sotiris K Ntouyas2

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 451 10 Ioannina, Greece

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Boundary Value Problems 2012, 2012:55  doi:10.1186/1687-2770-2012-55

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/55


Received:30 December 2011
Accepted:9 May 2012
Published:9 May 2012

© 2012 Ahmad and Ntouyas; 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 article studies a new class of nonlocal boundary value problems of nonlinear differential equations and inclusions of fractional order with strip conditions. We extend the idea of four-point nonlocal boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M1">View MathML</a> to nonlocal strip conditions of the form: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M3">View MathML</a>. These strip conditions may be regarded as six-point boundary conditions. Some new existence and uniqueness results are obtained for this class of nonlocal problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed.

MSC 2000: 26A33; 34A12; 34A40.

Keywords:
fractional differential equations; fractional differential inclusions; nonlocal boundary conditions; fixed point theorems; Leray-Schauder degree

1 Introduction

The subject of fractional calculus has recently evolved as an interesting and popular field of research. A variety of results on initial and boundary value problems of fractional order can easily be found in the recent literature on the topic. These results involve the theoretical development as well as the methods of solution for the fractional-order problems. It is mainly due to the extensive application of fractional calculus in the mathematical modeling of physical, engineering, and biological phenomena. For some recent results on the topic, see [1-19] and the references therein.

In this article, we discuss the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations and inclusions of order q ∈ (1, 2] with nonlocal strip conditions. As a first problem, we consider the following boundary value problem of fractional differential equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M4">View MathML</a>

(1.1)

where cDq denotes the Caputo fractional derivative of order q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M5">View MathML</a> is a given continuous function and σ, η are appropriately chosen real numbers.

The boundary conditions in the problem (1.1) can be regarded as six-point nonlocal boundary conditions, which reduces to the typical integral boundary conditions in the limit α, γ → 0, β, δ → 1. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. For a detailed description of the integral boundary conditions, we refer the reader to the articles [20,21] and references therein. Regarding the application of the strip conditions of fixed size, we know that such conditions appear in the mathematical modeling of real world problems, for example, see [22,23].

As a second problem, we study a two-strip boundary value problem of fractional differential inclusions given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M6">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M7">View MathML</a> is a multivalued map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M8">View MathML</a> is the family of all subsets of ℝ.

We establish existence results for the problem (1.2), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. 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 semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

The methods used are standard, however their exposition in the framework of problems (1.1) and (1.2) is new.

2 Linear problem

Let us recall some basic definitions of fractional calculus [24-26].

Definition 2.1 For at least n-times continuously differentiable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M9">View MathML</a>, the Caputo derivative of fractional order q is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M10">View MathML</a>

where [q] denotes the integer part of the real number q.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M11">View MathML</a>

provided the integral exists.

By a solution of (1.1), we mean a continuous function x(t) which satisfies the equation cDqx(t) = f (t, x(t)), 0 < t < 1, together with the boundary conditions of (1.1).

To define a fixed point problem associated with (1.1), we need the following lemma, which deals with the linear variant of problem (1.1).

Lemma 2.3 For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M12">View MathML</a>, the solution of the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M13">View MathML</a>

(2.1)

subject to the boundary conditions in (1.1) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M14">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M15">View MathML</a>

Proof. It is well known that the solution of (2.1) can be written as [24]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M16">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M17">View MathML</a> are constants. Applying the boundary conditions given in (1.1), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M18">View MathML</a>

Solving these equations simultaneously, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M19">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M20">View MathML</a>

Substituting the values of c0 and c1 in (2.3), we obtain the solution (2.2). □

3 Existence results for single-valued case

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M21">View MathML</a> denotes the Banach space of all continuous functions from [0, 1] → ℝ endowed with the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M22">View MathML</a> .

In view of Lemma 2.3, we define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M23">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M24">View MathML</a>

(3.1)

Observe that the problem (1.1) has solutions if and only if the operator equation Fx = x has fixed points.

For the forthcoming analysis, we need the following assumptions:

(A1) |f (t, x) - f (t, y)| ≤ L|x - y|, ∀t ∈ [0, 1], L > 0, x, y ∈ ℝ;

(A2) |f (t, x)| ≤ μ(t), ∀(t, x) ∈ [0, 1] × ℝ, and μ C([0, 1], ℝ+).

For convenience, let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M25">View MathML</a>

(3.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M26">View MathML</a>

Theorem 3.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M27">View MathML</a>is a jointly continuous function and satisfies the assumption (A1) with L < 1/Λ, where Λ is given by (3.2). Then the boundary value problem (1.1) has a unique solution.

Proof. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M28">View MathML</a> and choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M29">View MathML</a>, we show that FBr Br, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M193">View MathML</a>. For x Br, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M30">View MathML</a>

Now, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M31">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M32">View MathML</a>

where Λ is given by (3.2). Observe that Λ depends only on the parameters involved in the problem. As L < 1/Λ, therefore F is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). □

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii's fixed point theorem [27].

Theorem 3.2 (Krasnoselskii's fixed point theorem). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (i) Ax + By M whenever x, y M; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists z M such that z = Az + Bz.

Theorem 3.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M33">View MathML</a>be a jointly continuous function satisfying the assumptions (A1) and (A2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M34">View MathML</a>

(3.3)

Then the boundary value problem (1.1) has at least one solution on [0, 1].

Proof. Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M35">View MathML</a>, we fix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M36">View MathML</a>

and consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M37">View MathML</a>. We define the operators and on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M40">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M42">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M43">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M44">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M45">View MathML</a> It follows from the assumption (A1) together with (3.3) that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M48">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M49">View MathML</a>

Now we prove the compactness of the operator .

In view of (A1), we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M51">View MathML</a>, and consequently we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M52">View MathML</a>

which is independent of x. Thus, is equicontinuous. Hence, by the Arzelá-Ascoli Theorem, is compact on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M55">View MathML</a> Thus all the assumptions of Theorem 3.2 are satisfied. So the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on [0, 1]. □

Our next existence result is based on Leray-Schauder degree theory.

Theorem 3.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M56">View MathML</a>. Assume that there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M57">View MathML</a>, where Λ is given by (3.2) and M > 0 such that |f(t, x)| κ|x|+M for all t ∈ [0, 1], x C[0, 1]. Then the boundary value problem (1.1) has at least one solution.

Proof. Consider the fixed point problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M58">View MathML</a>

(3.4)

where F is defined by (3.1). In view of the fixed point problem (3.4), we just need to prove the existence of at least one solution x C[0, 1] satisfying (3.4). Define a suitable ball BR C[0, 1] with radius R > 0 as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M59">View MathML</a>

where R will be fixed later. Then, it is sufficient to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M60">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M61">View MathML</a>

(3.5)

Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M62">View MathML</a>

Then, by the Arzelá-Ascoli Theorem, hλ (x) = x - H (λ, x) = x - λFx is completely continuous. If (3.5) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M63">View MathML</a>

where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, h1(t) = x - λFx = 0 for at least one x BR. In order to prove (3.5), we assume that x = λFx, λ ∈ [0, 1]. Then for x ∂BR and t ∈ [0, 1] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M64">View MathML</a>

which, on taking norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M65">View MathML</a> and solving for ‖x‖, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M66">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M67">View MathML</a>, (3.5) holds. This completes the proof. □

Example 3.5 Consider the following strip fractional boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M68">View MathML</a>

(3.6)

Here, q = 3/2, σ = 1, η = 1, α = 1/3, β = 1/2, γ = 2/3, δ = 3/4 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M69">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M70">View MathML</a>, therefore, (A1) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M71">View MathML</a>. Further, Δ1 = 65/72, Δ2 = 535/288, Δ = 4945/5184, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M72">View MathML</a>

Clearly, LΛ = 0.282191 < 1. Thus, by the conclusion of Theorem 3.1, the boundary value problem (3.6) has a unique solution on [0, 1].

Example 3.6 Consider the following boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M73">View MathML</a>

(3.7)

Here,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M74">View MathML</a>

Clearly M = 1 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M75">View MathML</a>

Thus, all the conditions of Theorem 3.4 are satisfied and consequently the problem (3.7) has at least one solution.

4 Existence results for multi-valued case

4.1 Preliminaries

Let us recall some basic definitions on multi-valued maps [28,29].

For a normed space (X, ‖.‖), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M78">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M79">View MathML</a>. A multi-valued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M80">View MathML</a>is convex (closed) valued if G(x) is convex (closed) for all x X. The map G is bounded on bounded sets if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M81">View MathML</a> is bounded in X for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M82">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M83">View MathML</a>. G is called upper semi-continuous (u.s.c.) on X if for each x0 X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M84">View MathML</a> of x0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M85">View MathML</a>. G is said to be completely continuous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M86">View MathML</a> is relatively compact for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M87">View MathML</a>. If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., xn x*, yn y*, yn G(xn) imply y* G(x*). G has a fixed point if there is x X such that x G(x). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M88">View MathML</a> is said to be measurable if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M89">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M90">View MathML</a>

is measurable.

Let C([0, 1]) denotes a Banach space of continuous functions from [0, 1] into ℝ with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M91">View MathML</a>. Let L1([0, 1], ℝ) be the Banach space of measurable functions x : [0, 1] → ℝ which are Lebesgue integrable and normed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M92">View MathML</a>.

Definition 4.1 A multivalued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M93">View MathML</a>is said to be Carathéodory if

(i) t F (t, x) is measurable for each x ∈ ℝ;

(ii) x F (t, x) is upper semicontinuous for almost all t ∈ [0, 1];

Further a Carathéodory function F is called L1-Carathéodory if

(iii) for each α > 0, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M94">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M95">View MathML</a>

for all xα and for a. e. t ∈ [0, 1].

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M96">View MathML</a>, define the set of selections of F by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M97">View MathML</a>

Let X be a nonempty closed subset of a Banach space E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M98">View MathML</a> be a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {y X : G(y) ∩ B ≠ ∅} is open for any open set B in E. Let A be a subset of [0, 1] × ℝ. A is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M99">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/2012/1/55/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M100">View MathML</a>, where is Lebesgue measurable in [0, 1] and is Borel measurable in ℝ. A subset of L1([0, 1], ℝ) is decomposable if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M104">View MathML</a> and measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M105">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M106">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M107">View MathML</a> stands for the characteristic function of .

Definition 4.2 Let Y be a separable metric space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M109">View MathML</a>be a multivalued operator. We say N has a 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/2012/1/55/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M110">View MathML</a> be a multivalued map with nonempty compact values. Define a multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M111">View MathML</a> associated with F as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M112">View MathML</a>

which is called the Nemytskii operator associated with F.

Definition 4.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M113">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 is lower semi-continuous and has nonempty closed and decomposable values.

Let (X, d) be a metric space induced from the normed space (X; ‖.‖). Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M115">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M116">View MathML</a>

where d(A, b) = infaA d(a; b) and d(a, B) = infbB d(a; b). Then (Pb,cl(X), Hd) is a metric space and (Pcl(X), Hd) is a generalized metric space (see [30]).

Definition 4.4 A multivalued operator N : X Pcl(X) is called:

(a) γ-Lipschitz if and only if there exists γ > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M117">View MathML</a>

(b) a contraction if and only if it is γ-Lipschitz with γ < 1.

The following lemmas will be used in the sequel.

Lemma 4.5 (Nonlinear alternative for Kakutani maps) [31]. Let E be a Banach space, C is a closed convex subset of E, U is an open subset of C and 0 ∈ U. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M118">View MathML</a>is a upper semicontinuous compact map; here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M119">View MathML</a>denotes the family of nonempty, compact convex subsets of C. Then either

(i) F has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M120">View MathML</a>, or

(ii) there is a u ∂U and λ ∈ (0, 1) with u λF(u).

Lemma 4.6 [32]Let X be a Banach space. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M121">View MathML</a>be an L1-Carathéodory multivalued map and let θ be a linear continuous mapping from L1([0, 1], X) to C([0, 1], X). Then the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M122">View MathML</a>

is a closed graph operator in C([0, 1], X) × C([0, 1], X).

Lemma 4.7 [33]Let Y be a separable metric space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M123">View MathML</a>be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M124">View MathML</a>such that g(x) ∈ N(x) for every x Y .

Lemma 4.8 [34]Let (X, d) be a complete metric space. If N : X Pcl(X) is a contraction, then FixN ≠ ∅.

Definition 4.9 A function x C2([0, 1], ℝ) is a solution of the problem (1.2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M125">View MathML</a>, and there exists a function f L1([0, 1], ℝ) such that

f(t) ∈ F (t, x(t)) a.e. on [0, 1] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M126">View MathML</a>

(4.1)

4.2 The Carathéodory case

Theorem 4.10 Assume that:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M127">View MathML</a>is Carathéodory and has nonempty compact and convex values;

(H2) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M128">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M129">View MathML</a>

(H3) there exists a constant M > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M130">View MathML</a>

Then the boundary value problem (1.2) has at least one solution on [0, 1].

Proof. Define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M131">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M132">View MathML</a>

for f SF,x. We will show that ΩF satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ΩF is convex for each x C([0, 1], ℝ). This step is obvious since SF,x is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that ΩF maps bounded sets (balls) into bounded sets in C([0, 1], ℝ). For a positive number ρ, let Bρ = {x C([0, 1], ℝ): ‖x‖ ≤ ρ} be a bounded ball in C([0, 1], ℝ). Then, for each h ∈ ΩF (x), x Bρ, there exists f SF,x such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M133">View MathML</a>

Then for t∈[0, 1] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M134">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M135">View MathML</a>

Now we show that ΩF maps bounded sets into ;equicontinuous sets of C([0, 1], ℝ).

Let t', t'' ∈ [0, 1] with t' < t'' and x Bρ. For each h ∈ ΩF(x), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M136">View MathML</a>

Obviously the right-hand side of the above inequality tends to zero independently of x Bρ as t'' - t' → 0. As ΩF satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M137">View MathML</a> is completely continuous.

In our next step, we show that ΩF has a closed graph. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M138">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M139">View MathML</a>. Then we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M140">View MathML</a>. Associated with hn ∈ ΩF(xn), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M141">View MathML</a> such that for each t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M142">View MathML</a>

Thus it suffices to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M143">View MathML</a> such that for each t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M144">View MathML</a>

Let us consider the linear operator θ: L1([0, 1], ℝ) → C([0, 1], ℝ) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M145">View MathML</a>

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M146">View MathML</a>

as n → ∞.

Thus, it follows by Lemma 4.6 that θ ο SF is a closed graph operator. Further, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M147">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M148">View MathML</a>, therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M149">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M150">View MathML</a>.

Finally, we show there exists an open set U C([0, 1], ℝ) with x ∉ ΩF(x) for any λ ∈ (0, 1) and all x ∂U. Let λ ∈ (0, 1) and x ∈ λΩF(x). Then there exists f L1([0, 1], ℝ) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M151">View MathML</a> such that, for t ∈ [0, 1], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M152">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M153">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M154">View MathML</a>

In view of (A10), there exists M such that ||x|| ≠ M. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M155">View MathML</a>

Note that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M156">View MathML</a> is upper semicontinuous and completely continuous. From the choice of U, there is no x ∈ ∂U such that x ∈ λΩF(x) for some λ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 4.5), we deduce that ΩF has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M157">View MathML</a> which is a solution of the problem (1.1). This completes the proof. □

Example 4.11 Consider the following strip fractional boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M158">View MathML</a>

(4.2)

Here, q = 3/2, σ = 1, η = 1, α = 1/3, β = 1/2, γ = 2/3, δ = 3/4, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M159">View MathML</a> is a multivalued map given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M160">View MathML</a>

For f F, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M161">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M162">View MathML</a>

with p(t) = 1, ψ(||x||) = 9.

Further, we see that (H3) is satisfied with M > 20.679031. Clearly, all the conditions of Theorem 4.10 are satisfied. So there exists at least one solution of the problem (4.2) on [0, 1].

4.3 The lower semicontinuous case

As a next result, we study 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 [35] for lower semi-continuous maps with decomposable values.

Theorem 4.12 Assume that (A10), (H2) and the following condition holds:

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M163">View MathML</a>is a nonempty compact-valued multivalued map such that

(a) (t, x) ↦ F (t, x) is Λ <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M99">View MathML</a>measurable,

(b) x F (t, x) is lower semicontinuous for each t ∈ [0, 1];

Then the boundary value problem (1.2) has at least one solution on [0, 1].

Proof. It follows from (H2) and (H4) that F is of l.s.c. type. Then from Lemma 4.7, there exists a continuous function f : C([0, 1], ℝ) → L1([0, 1], ℝ) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M192">View MathML</a> for all x C([0, 1], ℝ).

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M164">View MathML</a>

(4.3)

Observe that if x C2([0, 1], ℝ) is a solution of (4.3), then x is a solution to the problem (1.2). In order to transform the problem (4.3) into a fixed point problem, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M165">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M166">View MathML</a>

It can easily be shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M167">View MathML</a> is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof. □

4.4 The Lipschitz case

Now we prove the existence of solutions for the problem (1.2) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [34].

Theorem 4.13 Assume that the following conditions hold:

(H5) F : [0, 1] × ℝ → Pcp(ℝ) is such that F(·, x): [0, 1] → Pcp(ℝ) is measurable for each x ∈ ℝ.

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M168">View MathML</a>for almost all t ∈ [0, 1] and x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M169">View MathML</a>with m L1([0, 1], ℝ+) and d(0, F(t, 0)) ≤ m(t) for almost all t ∈ [0, 1].

Then the boundary value problem (1.2) has at least one solution on [0, 1] if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M170">View MathML</a>

Proof. Observe that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M171">View MathML</a> is nonempty for each x C([0, 1], ℝ) by the assumption (H5), so F has a measurable selection (see Theorem III.6 [36]). Now we show that the operator ΩF, defined in the beginning of proof of Theorem 4.10, satisfies the assumptions of Lemma 4.8. To show that ΩF(x) ∈ Pcl((C[0, 1], ℝ)) for each x C([0, 1], ℝ), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M172">View MathML</a> be such that un u (n → ∞) in C([0, 1], ℝ). Then u C([0, 1], ℝ) and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M173">View MathML</a> such that, for each t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M174">View MathML</a>

As F has compact values, we pass onto a subsequence to obtain that vn converges to v in L1([0, 1], ℝ). Thus, v SF,x and for each t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M175">View MathML</a>

Hence, u ∈ Ω(x).

Next, we show that there exists γ < 1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M176">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M177">View MathML</a> and h1 ∈ □(x). Then there exists v1(t) ∈ F(t, x(t)) such that, for each t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M178">View MathML</a>

By (H7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M179">View MathML</a>

So, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M180">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M181">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M182">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M183">View MathML</a>

Since the multivalued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M184">View MathML</a> is measurable [36, Proposition III.4], there exists a function v2(t) which is a measurable selection for U. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M185">View MathML</a> and for each t ∈ [0, 1], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M186">View MathML</a>

For each t ∈ [0, 1], let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M187">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M188">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M189">View MathML</a>

Analogously, interchanging the roles of x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M190">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/55/mathml/M191">View MathML</a>

Since ΩF is a contraction, it follows by Lemma 4.8 that □ has a fixed point x which is a solution of (1.2). This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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

Acknowledgements

The authors were grateful to the reviewers for their useful comments. The research of B. Ahmad was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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