We prove the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions. The first existence and uniqueness result is based on Banach’s contraction principle. Moreover, other existence results are also obtained by using the Krasnoselskii fixed point theorem. An example is given to illustrate the main results.

Very recently, fractional differential equations have gained much attention due to extensive applications of these equations in the mathematical modelling of physical, engineering, biological phenomena and viscoelasticity (see, e.g., [

Moreover, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, population dynamics [

In this paper, we study the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions of the following form:

Moreover,

We now gather some definitions and preliminary facts which will be used throughout this paper. Denote by

We need some basic definitions and properties [

The Riemann-Liouville fractional integral of order

For a function

From the definition of the Caputo derivative, the following auxiliary results have been established in [

Let

Let

To study the boundary value problem (

For a given

By Lemma

Applying the boundary conditions in (

If a sequence

Let

In view of Lemma

The function

Let

where

Our first result is based on Banach’s contraction principle.

Assume that conditions (H1) and (H2) are satisfied. Then the boundary value problem (

We need to prove that the operator

Now, we will prove that

The following result is based on the Krasnoselskii fixed point theorem. To apply this theorem, we need the following hypothesis:

There exist

Assume that conditions (H1) and (H3) are satisfied. Then the boundary value problem (

Letting

On the other hand, from

Moreover, continuity of

Now we prove the compactness of the operator

In view of

Consider the following fractional boundary value problem:

Here,

Moreover,

Also, we get

Finally, simple calculations give

In this paper, we study the existence and uniqueness for fractional order differential equations with Riemann-Liouville integral boundary conditions (

The author declares that there is no conflict of interests regarding the publication of this paper.