We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem:

Differential equations of fractional order occur more frequently in different research and engineering areas such as physics, chemistry, economics, and control of dynamical. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, and electromagnetism. (see, e.g., [

For an extensive collection of results about this type of equations we refer the reader to the monograph by Kilbas and Trujillo [

On the other hand, some basic theory for the initial value problems of fractional differential equations involving the Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala [

In [

In [

Motivated by these works, in this paper we discuss the existence and uniqueness of positive solutions for the following nonlinear boundary value problem of fractional order:

This problem was studied in [

Our main interest in this paper is to give an alternative answer to the main results of the paper [

The main tool used in our study is a fixed point theorem in partially ordered sets which gives us uniqueness of the solution.

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

The Riemann-Liouville fractional integral of order

The Riemann-Liouville fractional derivative of order

The following two lemmas can be found in [

Let

Assume that

Using Lemma

Given

In the sequel, we present the fixed-point theorems which we will use later. These results appear in [

Let

Moreover, if

Adding condition (

In our considerations we will work in the Banach space

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

Finally, by

By

The main result of the paper is the following.

Problem (

There exists

There exists

Before the proof of Theorem

The continuity of

One has

Since

In the sequel, we give the proof of Theorem

Consider the cone

Now, we consider the operator

In the sequel we check that

Firstly, we prove that

In fact, for

Finally, as

Now, Theorem

In what follows, we will prove that this solution is positive (this means that

Finally, we will prove that the zero function is not the solution for problem (

This proves that the zero function is not the solution for problem (

In the contrary case, we find

Using a similar reasoning to the one above used we obtain a contradiction.

Therefore,

This finishes the proof.

In Theorem

For the other implication, suppose that

Notice that the assumptions in Theorem

In the sequel we present an example where the results can be applied.

Consider the fractional boundary value problem

Besides, for

Moreover, in this case

In connection with problem (

Problem (

In the sequel, we present an example which can be treated by Theorem

Consider the fractional boundary value problem

In this case,

Besides, if

Moreover, in this case,

On the other hand, we will show that

Our main contribution in this paper is to prove under certain assumptions the existence and the uniqueness of positive solution for problem (

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