This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

In the literature, there exist manifold types of fractional derivatives which are used to solve several mathematical and engineering problems as well as to model the dynamics of many systems existing in various fields of science such as mechanics, chemistry, finance, ecology, biology, and control theory. For instance, Caputo and Fabrizio [

The objective of this study is to establish some properties and formulas of the new generalized fractional derivative introduced in [

In this section, we first recall the definition and special cases of the new generalized fractional derivative with non-singular kernel. After, we present new properties for this fractional derivative that can be used to prove the global dynamics of several fractional-order systems.

Let

The importance of the fractional derivative given in the above definition is the generalization of most cases existing in the literature. These particular cases can be summarized in the following remark.

If

If

If

Now, we establish new properties for the generalized fractional derivative. We begin with the following result.

Let

Moreover, we have the following inequalities:

If

If

By using Definition

Let

Then,

Obviously,

Since

Now, we consider the following function:

It is clear that

Since

Let

Applying (i) of Theorem

This completes the proof.

Corollary

Now, we extend the recent result presented in Lemma 3.2 of [

Let

The application of Theorem

The proof of Corollary

For simplicity, we denote

The generalized fractional integral operator corresponding to

The Atangana–Baleanu fractional integral operator is a special case of (

The fractional derivative

This series converges locally and uniformly in

The Mittag-Leffler function

This completes the proof.

Let

This means that the generalized fractional integrals and derivatives are commutative operators.

First, we prove (i). By applying Lemma

Since the last expression is symmetric in

For (ii), we have

This shows (ii).

Let

According Definition

Hence,

Now, we demonstrate (ii). From Lemma

Thus,

This completes the proof.

By applying Theorem

The generalized fractional integral and the generalized fractional derivative satisfy the following Newton–Leibniz formula:

In many fields of science like mathematical biology, the method of constructing Lyapunov functions is often based on quadratic and Volterra-type functions. So, the first aim of this work was to estimate the generalized fractional derivatives of these specific functions. The obtained results extend many elementary lemmas existing in the recent literature for other types of fractional derivatives. Furthermore, the second purpose of this work was to establish a new formula for the generalized fractional derivative in the form of a series of weighted Riemann–Liouville fractional integrals. By means of this formula, various properties of the generalized fractional derivatives and integrals operators have been carefully established. Moreover, the Newton–Leibniz formula has been rigorously extended.

The results presented in this study form a fundamental basis for extending the classical Lyapunov theorems [

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that he has no conflicts of interest.