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.