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A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions

Khaled M Furati

Author Affiliations

Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Boundary Value Problems 2012, 2012:58  doi:10.1186/1687-2770-2012-58

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/58


Received:12 February 2012
Accepted:17 May 2012
Published:17 May 2012

© 2012 Furati; 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

A Cauchy-type nonlinear problem for a class of fractional differential equations with sequential derivatives is considered in the space of weighted continuous functions. Some properties and composition identities are derived. The equivalence with the associated integral equation is established. An existence and uniqueness result of the global continuous solution is proved.

AMS Subject Classification: 26A33; 34A08; 34A34; 34A12; 45J08.

Keywords:
fractional derivatives; Riemann-Liouville fractional derivative; sequential fractional derivative; fractional differential equation

1 Introduction

We consider a Cauchy-type problem associated with the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M3">View MathML</a> are the Riemann-Liouville fractional derivatives.

In recent years there has been a considerable interest in the theory and applications of fractional differential equations. As for the theory, the investigations include the existence and uniqueness of solutions, asymptotic behavior, stability, etc. See for example the books [1-3] and the articles [4-10] and the references therein.

As for the applications, fractional models provide a tool for capturing and understanding complex phenomena in many fields. See for example the surveys in [1,11] and the collection of applications in [12].

Some recent applications include control systems [13,14], viscoelasticity [15-18], and nanotechnology [19]. Also fractional models are used to model a vibrating string [20], and anomalous transport [21], anomalous diffusion [22-24].

Another field of applications is in random walk and stochastic processes [25-27] and its applications in financial modeling [28-30]. Other physical and engineering processes are given in [31,32]

In a series of articles, [33-35], Glushak studied the uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator. He also proposed a criterion for the uniform correctness of unbounded operator.

In this article we prove an existence and uniqueness result for a nonlinear Cauchy-type problem associated with the Equation (1) in the space of weighted continuous functions.

We start with some preliminaries in Section 2. In Section 3 we define the sequential derivative and develop some properties and composition identities. In Section 4 we set up the Cauchy-type problem and establish the equivalence with the associated integral equation. Finally, in Section 5 we prove the existence and uniqueness of the solution.

2 Preliminaries

In this section we present some definitions, lemmas, properties and notation which we use later. For more details please see [1].

Let -∞ < a < b < ∞. Let C[a, b] denote the spaces of continuous functions on [a, b]. We denote by L(a, b) the spaces of Lebesgue integrable functions on (a, b). Let CL(a, b) = L(a, b) ⋂ C(a, b].

We introduce the weighted spaces of continuous functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M4">View MathML</a>

(2)

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M5">View MathML</a>

(3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M6">View MathML</a>

(4)

In the case f is not defined at x = a or γ < 0 we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M7">View MathML</a> The spaces Cγ[a, b] satisfy the following properties.

C0[a,b] = C[a,b].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M8">View MathML</a>.

Cγ[a,b] ⊂CL(a,b),γ < 1.

f Cγ [a, b] if and only if f C(a, b) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M9">View MathML</a> exists and is finite.

The left-sided Riemann-Liouville fractional integrals and derivatives are defined as follows.

Definition 1 Let f L(a,b). The integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M10">View MathML</a>

(5)

is called the left-sided Riemann-Liouville fractional integral of order α of the function f.

Definition 2 The expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M11">View MathML</a>

(6)

is called the left-sided Riemann-Liouville fractional derivative of order α of f provided the right-hand side exists.

For power functions we have the following formulas.

Lemma 3 For x > a we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M12">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M13">View MathML</a>

(8)

Next we present some mapping properties of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14">View MathML</a>.

Lemma 4 For α > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14">View MathML</a>maps L(a, b) into L(a, b).

The proof of Lemma 4 is given in [36]. The following lemma is proved in [37].

Lemma 5 For α > 0, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M15">View MathML</a>maps C[a, b] into C[a, b].

The following lemma is proved in [38].

Lemma 6 Let α ≥ 0. If f CL(a, b) then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M16">View MathML</a>.

The mapping properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M15">View MathML</a> in the spaces Cγ[a, b], 0 ≤ α γ < 1, are given in [1], Lemma 2.8 which is proved in [39] in Russian. For completeness we present here a more general result for α > 0 and γ < 1. First we prove the necessity condition at the left end.

Lemma 7 Let α ≥ 0 and γ < 1. If f Cγ[a, b] then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M17">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M18">View MathML</a>.

Proof. Note that from Lemma 3 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M19">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M20">View MathML</a>

Now, given ϵ > 0 there exists δ > 0 such that x - a < δ implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M21">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M22">View MathML</a>

This yields the limit (9).

Next we present the mapping properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M14">View MathML</a> in the spaces Cγ[a, b].

Lemma 8 Let α > 0 and γ < 1. If f Cγ[a, b] then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M23">View MathML</a>and for x ∈ (a, b] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M24">View MathML</a>

(10)

Proof. From Lemmas 6 and 7 we have Iαf C(a, b) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M25">View MathML</a> exists and is finite. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M26">View MathML</a>. Now for x ∈ (a, b] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M27">View MathML</a>

The relation (10) follows by applying Lemma 3.

Consequently, from Lemma 8 we have the following property.

Lemma 9 Let α > 0, γ < 1, and r ∈ ℝ. If f Cγ[a, b] then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M28">View MathML</a>. In particular, if γ + r - α < 1 then Iαf CL(a, b).

Later, the following observation is important.

Lemma 10 Let α > 0 and r < α. If f CL(a, b) then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M29">View MathML</a>.

Proof. When r ≤ 0 the result follows clearly from Lemma 6. When r > 0 it follows from Lemma 6 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M30">View MathML</a> and we only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M31">View MathML</a>.

For any x ∈ (a, b] we have the following inequality.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M32">View MathML</a>

Or,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M33">View MathML</a>

From Lemma 4 the right-hand side is in L(a, b) and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M34">View MathML</a>. This completes the proof.

The following lemma follows by direct calculations using Dirichlet formula, [36].

Lemma 11 Let α ≥ 0, β ≥ 0, and f CL(a, b). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M35">View MathML</a>

(11)

for all x ∈ (a, b].

Lemma 11 leads to the left inverse operator.

Lemma 12 Let α > 0 and f CL(a, b). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M36">View MathML</a>

(12)

for all x ∈ (a, b].

Now we present a version of the fundamental theorem of fractional calculus.

Lemma 13 Let 0 < α < 1. If f C(a, b) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M37">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M38">View MathML</a>exists and is finite, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M39">View MathML</a>

(13)

for all x ∈ (a, b].

Proof. From Lemma 12 we have for all x ∈ (a, b] the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M40">View MathML</a>

which we can write as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M41">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M42">View MathML</a>

(14)

for some constant c. Since Lemma 6 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M43">View MathML</a>, we also have f CL(a, b). Also, if we apply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M44">View MathML</a> to both sides of (14) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M45">View MathML</a>

Taking the limit yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M46">View MathML</a> and (13) is obtained.

In the proof of our existence and uniqueness result we will use the following results.

Lemma 14 Let γ ∈ ℝ, a < c < b, g Cγ[a, c], g C[c, b] and g is continuous at c. Then g Cγ[a, b].

Theorem 15 ([1], Banach Fixed Point Theorem) Let (U, d) be a nonempty complete metric space. Let T : U U be a map such that for every u, v U, the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M47">View MathML</a>

holds. Then the operator T has a unique fixed point u* ∈ U.

3 Sequential derivative

In this section we define the sequential derivative and integral that we consider and develop some of their properties. In particular, we derive the composition identities.

Definition 16 Let α > 0, β > 0, r ∈ ℝ. Let f CL(a, b). Define the sequential integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M48">View MathML</a>and the sequential derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M49">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M50">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M51">View MathML</a>

(16)

if the right-hand sides exist.

From Lemma 3 we have the following formula for the power function.

Lemma 17 Let α > 0, β > 0, r ∈ ℝ. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M52">View MathML</a>

then for x > a,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M53">View MathML</a>

(17)

Moreover, from Lemmas 3 and 17 we have the following vanishing derivatives.

Lemma 18

(a) Let α > 0, 0 < β < 1, r ∈ ℝ. Then for x > a,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M54">View MathML</a>

(18)

(b) Let 0 < α < 1 and β > 0. Let r ∈ ℝ be such that r < α + β. Then for x > a,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M55">View MathML</a>

(19)

Lemma 19 (Left inverse) Let α > 0, β > 0, and r ∈ ℝ. If f CL(a, b) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M56">View MathML</a>then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M57">View MathML</a>

(20)

for all x ∈ (a, b].

Proof. Relation (20) follows directly by applying Lemma 12 twice.

From Lemmas 8 and 9 we have the following mapping property of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58">View MathML</a>.

Lemma 20 Let α > 0, β > 0, and r < 1 + α. Let 0 ≤ γ < min{1, 1 + α - r}. If f Cγ[a, b] then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M59">View MathML</a>and for x ∈ (a, b] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M60">View MathML</a>

(21)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M61">View MathML</a>

(22)

Lemma 20 implies the following.

Lemma 21 Let α > 0, β > 0, and r < 1 + α. Let 0 ≤ γ < min{1, 1 + α-r}. If r α + β, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58">View MathML</a>is bounded in Cγ[a, b] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M62">View MathML</a>

(23)

where k is given by (22).

Proof. Since γ + r - α - β γ, then from Lemma 20 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M63">View MathML</a>

The bound in (23) follows by multiply (21) by (x - a)γ and taking the maximum.

As a special case of Lemma 21, we have

Lemma 22 Let α > 0, β > 0, and r < min{α + 1, α + β}. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58">View MathML</a>, maps C[a, b] into C[a, b] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M64">View MathML</a>

(24)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M65">View MathML</a>

(25)

The following is an analogous result to the result for the Riemann-Liouville integral proved in [10].

Lemma 23 Let α > 0, β > 0, and r < α. Let f CL(a, c). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M66">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M67">View MathML</a>

Proof. Since r < α, Lemma 10 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M68">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M69">View MathML</a> is finite and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M70">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M71">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M72">View MathML</a>, the limit of the right-hand side vanishes and the proof is complete.

The following lemma relates the fractional derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73">View MathML</a> to the Riemann-Liouville derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M74">View MathML</a>.

Lemma 24 Let 0 < α < 1, β ≥ 0, and r ∈ ℝ. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M75">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M76">View MathML</a>then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M77">View MathML</a>exists and finite, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M78">View MathML</a>

(26)

for all x ∈ (a, b]. If in addition, r < α then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M79">View MathML</a>.

Proof. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M81">View MathML</a>. Thus we can apply Lemma 13 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M82">View MathML</a> and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M83">View MathML</a>

By multiplying both sides by (x - a)r we obtain (26). If r < α then Lemma 10 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M84">View MathML</a> and thus from (26) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85">View MathML</a>. This proves the result.

The Next lemma gives an analogous result to the fundamental theorem of calculus in terms of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M86">View MathML</a>.

Lemma 25 Let 0 < α < 1 and 0 < β < 1. Let y C(a, b) be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M87">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85">View MathML</a>. Then both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M88">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M89">View MathML</a>exist, y CL(a, b), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M90">View MathML</a>

(27)

for all x ∈ (a, b].

Proof. By applying Lemma 13 twice we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M91">View MathML</a>

4 Cauchy-type problem and equivalency

Consider the Cauchy-type problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M92">View MathML</a>

(28)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M93">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M94">View MathML</a>

(30)

where c0 and c1 are real numbers.

In this problem there are two conditions even when 0 < α + β < 1. The two initial conditions are based on the composition (27). The condition (29) is of one order less than that in the differential Equation (28) while the condition (30) is one order less than the equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M95">View MathML</a>.

In addition, from [1, Lemma 3.2], the condition (30) follows from the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M96">View MathML</a>

(31)

and if 0 < α - r < 1 then (29) follows from the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M97">View MathML</a>

(32)

Consequently, the results below hold under conditions of the type (31) and (32).

Now, Based on the composition in Lemma 24, in the next theorem we establish an equivalence with the following fractional integro-differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M98">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M99">View MathML</a>

(34)

Theorem 26 Let 0 < α < 1, β > 0 and r ∈ ℝ. Let f : (a, b] × ℝ → ℝ be a function such that f(.,y(.)) ∈ C1-α [a, b] for any y C1-α [a, b]. Then we have the following.

(a) If y C1-α[a, b] satisfies (33) and (34) then y(x) satisfies (28-30).

(b) If y C1-α[a, b] with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a> satisfy (28-30), then y(x) satisfies (33-34).

Proof.

For assertion (a), let y C1-α[a, b] satisfy (33-34). We multiply (33) by (x - a)r to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M100">View MathML</a>

(35)

Next we apply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M2">View MathML</a> to both sides of (35) to obtain (28). As for the initial condition, apply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M44">View MathML</a> to both sides of (35) and then take the limit to obtain (29).

For assertion (b), let y C1-α[a, b] satisfy (28-30). Since f(x, y(x)) ∈ C1-α[a, b], then from (28) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M101">View MathML</a>. Since also by hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a>, we can apply Lemma 24 and the formula (26) holds. By substituting the initial condition we obtain (33). This completes the proof.

The composition in Lemma 25 leads to the nonlinear integral equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M102">View MathML</a>

(36)

The following theorem establishes an equivalence with this equation.

Theorem 27 Let 0 < α < 1, 0 < β < 1 and r < α. Let f : (a, b] × ℝ → ℝ be a function such that f(.,y(.)) ∈ C1-β[a, b] for any y C1-β[a, b]. Then the following statements hold.

(a) If y C1-β[a, b] satisfies the integral Equation (36) then y(x) satisfies the Cauchy-type problem (28-30).

(b) If y C1-β[a, b] with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a>satisfies the Cauchy-type problem (28-30), then y(x) satisfies the integral Equation (36).

Proof. (a). Let y C1-β[a, b] satisfy the integral Equation (36). By hypothesis we have f C1-β[a, b]. Moreover, from Lemma 9 and the hypothesis r < α, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M103">View MathML</a>

Thus the hypothesis of Lemmas 18 and 19 are satisfied. Applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M73">View MathML</a> to both sides of (36) and using Lemmas 18 and 19 yields (28) as follows.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M104">View MathML</a>

Next, applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M105">View MathML</a> to both sides of (36) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M106">View MathML</a>

(37)

Since r < α, taking the limit we obtain the initial condition (30).

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M107">View MathML</a> to both sides of (36) and using Lemmas 3, 11, and 12 yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M108">View MathML</a>

Again, taking the limit we obtain the initial condition (29).

(b). Let y C1-β[a, b] satisfy (28-30). Since f(x, y(x)) ∈ CL(a, b) then from (28), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M87">View MathML</a>. Since r < α then from Lemma 24 we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M85">View MathML</a>. Thus we can apply Lemma 25 and the formula (27) holds. By using the initial conditions we obtain (36). This completes the proof.

In the next section we use this equivalence to prove the existence and uniqueness of solutions.

5 Existence and uniqueness of the solution of the Cauchy-type problem

In this section we prove an existence and uniqueness result for the Cauchy-type problem (28-30) using the integral Equation (36). For this purpose we introduce the following lemma.

Lemma 28 Let 0 < r < α < 1, 0 < β < 1, then the fractional differentiation operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M58">View MathML</a>is bounded in C1-β[a, b] and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M109">View MathML</a>

(38)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M110">View MathML</a>

(39)

Proof. Clearly from the hypothesis we have r < α + β and 0 < 1 - β < min{1, 1 + α - r}. Thus the result follows by taking γ = 1 - β in Lemma 21.

Theorem 29 Let 0 < r < α < 1, 0 ≤ β < 1. Let f : (a, b] × ℝ → ℝ be a function such that f(.,y(.)) ∈ C1-β[a, b] for any y C1-β[a, b] and the condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M111">View MathML</a>

(40)

is satisfied for all x ∈ (a, b] and for all y1, y2 ∈ ℝ.

Then the Cauchy-type problem (28-30) has a solution y C1-β[a, b]. Furthermore, if for this solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a>, then this solution is unique.

Proof.

According to Theorem 27(a), we can consider the existence of an C1-β[a, b] solution for the integral Equation (36). This equation holds in any interval (a, x1] ⊂ (a, b], a < x1 < b. Choose x1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M112">View MathML</a>

where K is given by (39). We rewrite the integral equation in the form y(x) = Ty(x), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M113">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M114">View MathML</a>

Since r < α then v0 C1-β[a, b]. Thus, it follows from Lemma 28 that if y C1-β[a, x1] then Ty C1-β[a, x1]. Also, for any y1, y2 in C1-β[a, x1], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M115">View MathML</a>

Hence by Theorem 15 there exists a unique solution y* ∈ C1-β[a, x1] to the Equation (36) on the interval (a, x1].

If x1 b then we consider the interval [x1, b]. On this interval we consider solutions y C[x1, b] for the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M116">View MathML</a>

(41)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M117">View MathML</a>

Now we select x2 ∈ (x1, b] such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M118">View MathML</a>

where L is given by (25). Since the solution is uniquely defined on the interval (a, x1], we can consider v01(x) to be a known function. For y1, y2 C[x1, x2], it follows from the Lipschitz condition and Lemma 22 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M119">View MathML</a>

Since 0 < w2 < 1, T is a contraction. Since f(x, y(x)) ∈ C[x1, x2] for any y C[x1, x2], then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M120">View MathML</a>. Moreover, clearly v01(x) is in C[x1, x2]. Thus the right-hand side of (41) is in C[x1, x2]. Therefore T maps C[x1, x2] into itself. By Theorem 15, there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M121">View MathML</a> to the equation on the interval [x1, x2]. Moreover, it follows from Lemma 23 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M122">View MathML</a>. Therefore if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M123">View MathML</a>

then by Lemma 14, y* ∈ C1-β[a, x2]. So y* is the unique solution of (36) in C1-β[a, x2] on the interval (a, x2].

If x2 b, we repeat the process as necessary, say M - 2 times, to obtain the unique solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M124">View MathML</a>, where a = x0 < x1 < ··· < xM = b, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M125">View MathML</a>

As a result we have the unique solution y* ∈ C1-β[a, b] of (36) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M126">View MathML</a>

(42)

This solution is also a solution for (28-30).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/58/mathml/M80">View MathML</a> then the uniqueness follows from part (b) of Theorem 27. This completes the proof.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author was grateful for the support provided by the King Fahd University of Petroleum & Minerals and the financial support by the BAE Systems through the PDSR program by the British Council in Saudi Arabia.

References

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

  2. Lakshmikantham, V, Leela, S, Devi, JV: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

  3. Diethelm, K: The Analysis of Fractional Differential Equations. Springer, Heidelberg (2010)

  4. Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput Math Appl. 59, 1095–1100 (2010). Publisher Full Text OpenURL

  5. Avad, HK, Glushak, AV: On perturbations of abstract fractional differential equations by nonlinear operators. J Math Sci. 170(3), 306–323 (2010). Publisher Full Text OpenURL

  6. Kirane, M, Malik, SA: The profile of blowing-up solutions to a nonlinear system of fractional differential equations. Nonlinear Anal. 73, 3723–3736 (2010). Publisher Full Text OpenURL

  7. Chai, G: Existence results for boundary value problems of nonlinear fractional differential equations. Comput Math Appl. 62, 2374–2382 (2011). Publisher Full Text OpenURL

  8. Zhang, S, Su, X: The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order. Comput Math Appl. 62, 1269–1274 (2011). Publisher Full Text OpenURL

  9. Babakhani, A, Baleanu, D: Employing of some basic theory for class of fractional differential equations. Adv Diff Equ. 2011, 1–13 (2011)

  10. Furati, KM, Kassim, MD, Tatar, Ne: Existence and uniqueness for a problem involving Hilfer fractional derivative. In: Comput Math Appl

  11. Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, San Diego (1999)

  12. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

  13. Caponetto, R, Dongola, G, Fortuna, L, Petráš, I: Fractional Order Systems: Modeling and Control Applications. World Scientific Series on Nonlinear Science. World Scientific (2010)

  14. Monje, CA, Chen, Y, Vinagre, BM, Xue, D, Feliu, V: Fractional-order Systems and Controls. Adv Industr Control, Springer, New York (2010)

  15. Hilfer, R: Experimental evidence for fractional time evolution in glass forming materials. Chem Phys. 284, 399–408 (2002). Publisher Full Text OpenURL

  16. Wenchang, T, Wenxiao, P, Mingyu, X: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int J Non-Linear Mech. 38, 645–650 (2003). Publisher Full Text OpenURL

  17. Mainardi, F, Gorenflo, R: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract Calc Appl Anal. 10(3), 269–308 (2007)

  18. Mainardi, F: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

  19. Baleanu, D, Mustafa, OG, Agarwal, RP: Asymptotically linear solutions for some linear fractional differential equations. Abstr Appl Anal. 2010, 8 (2010)

  20. Sandev, T, Tomovski, Z: General time fractional wave equation for a vibrating string. J Phys A Math Theor. 43, 055204 (2010). Publisher Full Text OpenURL

  21. Klages, R, Radons, G, Sokolov, I: Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008)

  22. Gerolymatou, E, Vardoulakis, I, Hilfer, R: Modelling infiltration by means of a nonlinear fractional diffusion model. J Phys D Appl Phys. 39, 4104–4110 (2006). Publisher Full Text OpenURL

  23. Rao, BLSP: Statistical Inference for Fractional Diffusion Processes. Wiley, Chichester (2010)

  24. Meerschaertm, MM, Sikorskii, A: Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics. De Gruyter, Berlin (2012)

  25. Hilfer, R, Anton, L: Fractional master equations and fractal time random walks. Phys Rev E. 51, R848–R851 (1995). Publisher Full Text OpenURL

  26. Zhang, Y, Benson, DA, Meerschaert, MM, LaBolle, EM, Scheffler, HP: Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys Rev E. 74, 026706–026715 (2006)

  27. Baeumer, B, Meerschaert, MM, Nane, E: Brownian subordinators and fractional cauchy problems. Trans Am Math Soc. 361, 3915–3930 (2009). Publisher Full Text OpenURL

  28. Scalas, E, Gorenflo, R, Mainardi, F: Fractional calculus and continuous-time finance. Phys A. 284, 376–384 (2000). Publisher Full Text OpenURL

  29. Scalas, E, Gorenflo, R, Mainardi, F, Meerschaert, M: Speculative option valuation and the fractional diffusion equation. In: Sabatier, J, Machado, JT (eds.) Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, (FDA 04), Bordeaux (2004)

  30. Fulger, D, Scalas, E, Germano, G: Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys Rev E Stat,p. 021122. Nonlinear Soft Matter Phys (2008)

  31. Ortigueira, MD: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering. Springer, Netherlands (2011)

  32. Petráš, I: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, New York (2011)

  33. Glushak, AV: Cauchy-type problem for an abstract differential equation with fractional derivative. Math Notes. 77(1), 26–38 (2005) Translated from Matematicheskie Zametki 77(1) 28-41 (2005)

    Translated from Matematicheskie Zametki 77(1) 28-41 (2005)

    Publisher Full Text OpenURL

  34. Glushak, AV: On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives. Math Notes. 82(5), 596–607 (2007) Translated from Matematicheskie Zametki 82(5), 665-677 (2007)

    Translated from Matematicheskie Zametki 82(5), 665-677 (2007)

    Publisher Full Text OpenURL

  35. Glushak, AV: Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives. Russ Math. 53(9), 1–19 (2009) Translated from Izvestiya Vysshikh Uchebnykh sZavedenii. Matematika 2009(9), 13-24 (2009)

    Translated from Izvestiya Vysshikh Uchebnykh sZavedenii. Matematika 2009(9), 13-24 (2009)

    Publisher Full Text OpenURL

  36. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Amsterdam (1993). Engl. Trans. from the Russian (1987)

  37. Kou, C, Liu, J, Ye, Y: Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations. Discr Dyn Nature Soc. 2010, 1–15 (2010)

  38. Al-Jaser, A, Furati, KM: Singular fractional integro-differential inequalities and applications. J Inequal Appl. 2011, 110 (2011)

  39. Kilbas, AA, Bonilla, B, Trujillo, JJ: Fractional integrals and derivatives, and differential equations of fractional order in weighted spaces of continuous functions (russian). Dokl Nats Akad Nauk Belarusi. 44(6), 18–22 (2000)