^{1}

^{2, 3}

^{3, 4}

^{1}

^{2}

^{3}

^{4}

We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order

For the last decades, fractional calculus has received a great attention because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes of science and engineering. Indeed, we can find numerous applications in viscoelasticity [

On the other hand, the study of dynamical systems with impulsive effects has been an object of intensive investigations in physics, biology, engineering, and so forth. The interest in the study of them is that the impulsive differential systems can be used to model processes which are subject to abrupt changes and which cannot be described by the classical differential problems (e.g., see [

Fečkan et al. [

In [

Ashyralyev and Sharifov [

Motivated by the papers above, in this paper, we study impulsive fractional differential equations with the two-point and integral boundary conditions in the following form:

The rest of the paper is organized as follows. In Section

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. By

Let us recall the following known definitions and results. For more details see [

If

The Caputo fractional derivative of order

Under natural conditions on

Let

For

Assume that

Let

If

We define a solution problem (

A function

We have the following result which is useful in what follows.

Let

Assume that

Integrating the expression (

Thus if

Conversely, assume that

Let

Our first result is based on Banach fixed point theorem. Before stating and proving the main results, we introduce the following hypotheses.

There are constants

for each

There exist constants

for all

Assume that (H1)–(H3) hold. If

The proof is based on the classical Banach fixed theorem for contractions. Let us set

Let

Next we will show that

The second result is based on the Schaefer fixed point theorem. We introduce the following assumptions.

There exist constants

Assume that (H1), (H4), and (H5) hold. Then the boundary value problem (

We will use Schaefer’s fixed point theorem to prove that

Let

Indeed, it is enough to show that, for any

Let

As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem (Theorem

Now it remains to show that the set

Let then

In this section, we give some examples to illustrate our main results.

Consider

Evidently,

Then, by Theorem

Consider

Here

The authors declare that there is no conflict of interests regarding the publication of this paper.