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We are concerned with the existence and uniqueness of a positive and nondecreasing solution for the following nonlinear fractional

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., [

Recently, El-Shahed [

In [

The question of uniqueness of the solution is not treated in [

Recently, in [

In this paper we discuss the existence and uniqueness of a positive and nondecreasing solution for the following

Recently, this problem has been studied in [

Our study is based on a different fixed point theorem in partially ordered sets than the one used in [

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

Existence of fixed points in partially ordered sets has been considered recently in [

For existence theorems for fractional differential equations and applications, we refer to the survey [

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

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

Notice that Lemma 3 appears in [

The following result is proved in [

Under the assumption

It is easily checked that

The following lemmas appear in [

The function

The function

For convenience, we will denote by

Firstly, we need to introduce the following class of functions. By

Let

Suppose that there exists

Assume that either

In our considerations, we will work in the space

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

In [

Our starting point in this section is to present the class of functions

For any

Examples of functions in

In what follows, we formulate our main result.

Suppose that the following assumptions are satisfied:

there exists

Then problem (

Consider the cone

Notice that, as

Now, for

By Lemma

Now, we will check that assumptions in Theorem

Firstly, the operator

In fact, by assumption (b), for

Obviously, the last inequality is satisfied for

Thus condition (

In what follows, we will prove that the unique nonnegative solution

In fact, by Lemmas

Now, we present a sufficient condition for the existence and uniqueness of a positive and strictly increasing solution for problem (

Under assumptions of Theorem

Consider the nonnegative solution

Notice that

In fact, in contrary case we can find

This fact and the continuity of

This contradicts (

Therefore,

In the sequel, we will show that

In fact, since

We can consider two cases.

Suppose that

Using a similar argument similar o the one used in the proof of the positive character of

Suppose that

In this case, we have

Since

Again, the same reasoning that we use earlier gives us a contradiction.

Therefore,

In Theorem

The reverse implication is obvious.

Notice that assumptions in Theorem

In the sequel, we present an example which illustrates our results.

Consider the following boundary value problem:

It is easily seen that

Since

Moreover, for

In [

Problem (

there exists

The main tool used by the authors in [

By

The same proof used by the authors in [

In what follows we will prove that the classes of functions

This example appears in [

Consider the function

On the other hand, it is easily seen that

Consider the function

It is easily proved that

On the other hand, since

Examples

In [

Problem (

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

Consider the fractional boundary value problem

It is easily seen that

Moreover, for

Since

On the other hand, we will show that condition

In fact, suppose that there exists

Since

Therefore, Problem (

In [

Assume that

In what follows we present an example which can be treated by our results and it cannot be studied by Theorem

Consider Problem (

We prove that this example can be treated by Theorem

On the other hand, in this case, since

Consider the following boundary value problem

It is easily seen that

Moreover, in [

In [

Since

On the other hand, since

Our main contribution is that for

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