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A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel'skii fixed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.

Fractional differential equations are a natural generalization of ordinary differential equations. In the last few decades many authors pointed out that differential equations of fractional order are suitable for the metallization of various physical phenomena and that they have numerous applications in viscoelasticity, electrochemistry, control and electromagnetic, and so forth, see [

This work is devoted to the study of the following fractional boundary value problem (P1):

Our mean objective is to investigate the existence, uniqueness, and existence of positive solutions for the fractional boundary value problem (P1), by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cone.

The research in this area has grown significantly and many papers appeared on this subject, using techniques of nonlinear analysis, see [

In [

Liang and Zhang in [

In [

This paper is organized as follows, in the Section

In this section, we present some lemmas and definitions from fractional calculus theory which will be needed later.

If

Let

Let

For

Denote by

The following Lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.

Let

Let

Now we start by solving an auxiliary problem.

Let

Applying Lemmas

In this section we prove the existence and uniqueness of solutions in the Banach space

The function

Let

Assume that there exist nonnegative functions

To prove Theorem

Let

Let

Now we prove Theorem

We transform the fractional boundary value problem to a fixed point problem. By Lemma

Therefore

With the help of hypothesis (

Taking into account (

Now we give an existence result for the fractional boundary value problem (P1).

Assume that

To prove this Theorem, we apply Leray-Schauder nonlinear alternative.

Let

First let us prove that

(i) For

Moreover, we have

(ii)

Let us consider the function

Now we apply Leray Schauder nonlinear alternative to prove that

Letting

In this section we investigate the positivity of solution for the fractional boundary value problem (P1), for this we make the following hypotheses.

Now we give the properties of the Green function.

Let

If

(i) It is obvious that

(ii) Let

We recall the definition of positive of solution.

A function

If

First let us remark that under the assumptions on

From (

with the help of (

The proof is complete.

Define the quantities

The main result of this section is as follows.

Under the assumption of Lemma

To prove Theorem

Let

To prove Theorem

Let us prove the superlinear case. First, since

In view of hypothesis (H2), one can choose

In order to illustrate our results, we give the following examples.

The fractional boundary value problem

In this case we have

The fractional boundary value problem

We apply Theorem