Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.

1. Introduction

Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, e.g., [1, 2]) and to problems of analyticity in the complex domain [3]. Moreover, Delbosco and Rodino [4] considered the existence of a solution for the nonlinear fractional differential equation

In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation

where

Note that this problem was considered in [6] where the authors proved the existence of one positive solution for (1.1) by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function

In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem (1.1) by using a fixed point theorem in partially ordered sets.

Existence of fixed point in partially ordered sets has been considered recently in [7–12]. This work is inspired in the papers [6, 8].

For existence theorems for fractional differential equation and applications, we refer to the survey [13]. Concerning the definitions and basic properties we refer the reader to [14].

Recently, some existence results for fractional boundary value problem have appeared in the literature (see, e.g., [15–17]).

2. Preliminaries and Previous Results

For the convenience of the reader, we present here some notations and lemmas that will be used in the proofs of our main results.

Definition 2.1.

The Riemman-Liouville fractional integral of order

provided that the right-hand side is pointwise defined on

Definition 2.2.

The Caputo fractional derivative of order

where

The following lemmas appear in [14].

Lemma 2.3.

Let

where

Lemma 2.4.

The relation

is valid when

The following lemmas appear in [6].

Lemma 2.5.

Given

is given by

where

Remark 2.6.

Note that

Lemma 2.7.

Let

is continuous on [0,1], where

Now, we present some results about the fixed point theorems which we will use later. These results appear in [8].

Theorem 2.8.

Let

where

If we consider that

then we have the following theorem [8].

Theorem 2.9.

Adding condition (2.10) to the hypotheses of Theorem 2.8 one obtains uniqueness of the fixed point of

In our considerations, we will work in the Banach space

Note that this space can be equipped with a partial order given by

In [10] it is proved that

satisfies condition (2) of Theorem 2.8. Moreover, for

3. Main Result

Theorem 3.1.

Let

Then one's problem (1.1) has an unique nonnegative solution.

Proof.

Consider the cone

Note that, as

Now, for

By Lemma 2.7,

Hence,

In what follows we check that hypotheses in Theorems 2.8 and 2.9 are satisfied.

Firstly, the operator

Besides, for

As the function

and from last inequality we get

Put

Thus, for

Finally, take into account that for the zero function,

Remark 3.2.

In [6, lemma 3.2] it is proved that

In the sequel we present an example which illustrates Theorem 3.1.

Example 3.3.

Consider the fractional differential equation (this example is inspired in [6])

In this case,

because

Note that

Theorem 3.1 give us that our fractional differential (3.10) has an unique nonnegative solution.

This example give us uniqueness of the solution for the fractional differential equation appearing in [6] in the particular case

Remark 3.4.

Note that our Theorem 3.1 works if the condition (3.1) is changed by, for

where

(a)

(b)

(c)

(d)

Examples of such functions are

Remark 3.5.

Note that the Green function

Case 1.

For

It is trivial that

Case 2.

For

Now,

Hence, taking into account the last inequality and (3.16), we obtain

Case 3.

For

and, as

This completes the proof.

Remark 3.5 gives us the following theorem which is a better result than that [6, Theorem 3.3] because the solution of our problem (1.1) is positive in

Theorem 3.6.

Under assumptions of Theorem 3.1, our problem (1.1) has a unique nonnegative and strictly increasing solution.

Proof.

By Theorem 3.1 we obtain that the problem (1.1) has an unique solution

Taking into account Remark 3.4 and the fact that

Now, if we suppose that

On the other hand, if

Now, as

and this contradicts that

Thus,

Acknowledgment

This research was partially supported by "Ministerio de Educación y Ciencia" Project MTM 2007/65706.