^{1, 2}

^{2}

^{1}

^{2}

We study the stability of a class of nonlinear fractional neutral differential difference systems equipped with the Caputo derivative. We extend Lyapunov-Krasovskii theorem for the nonlinear fractional neutral systems. Conditions of stability and instability are obtained for the nonlinear fractional neutral systems.

Recently, fractional differential equations have attracted great attention. It has been proved that fractional differential equations are valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. For details and examples, see [

Stability analysis is always one of the most important issues in the theory of differential equations and their applications for both deterministic and stochastic cases. Recently, stability of fractional differential equations has attracted increasing interest. Since fractional derivatives are nonlocal and have weakly singular kernels, the analysis on stability of fractional differential equations is more complex than that of classical differential equations. The earliest study on stability of fractional differential equations started in [

As we all know, Lyapunov’s second method provides a way to analyze the stability of a system without explicitly solving the differential equations. It is necessary to extend Lyapunov’s second method to fractional systems. In [

As far as we know, there are few papers with respect to the stability of fractional neutral systems. In this paper, we consider the stability of a class of nonlinear fractional neutral differential difference equations with the Caputo derivative. Motivated by Li et al. [

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

In this section, we introduce notations, definitions, and preliminary facts needed here. Throughout this paper, let

Let us recall the following definitions. For more details, we refer the reader to [

The fractional order integral of a function

For a function

For a function

Some properties of the aforementioned operators are recalled as follows [

The following results are especially interesting.

From Property

In general, it is not true that

In [

For any

Suppose

If

Suppose

The matrix

Consider a simple operator

Let

In this section, we consider the stability of the following nonlinear fractional neutral differential difference system:

If

We say that the zero solution

Next, we will address the main core of the paper, the stability of the nonlinear fractional neutral systems (

Let

For any given

Along the solutions of system (

For a given

Now we take

For a given

Since the zero solution of system (

Finally, we need to show

The functional (

Let

along the solutions of the system (

Note that the theorem conditions imply that of Theorem

Suppose that the assumptions in Theorem

By using Property

Next, we present the following sufficient conditions for the asymptotic stability of the zero solution of system (

Let

Since the matrix

Note that the conditions (1)–(3) of the theorem imply that of Theorem

Now, let

Suppose that the assumptions in Theorem

By using Property

Here, we present the following sufficient conditions for the instability of the zero solution of system (

Suppose

Suppose

Finally, we consider that the case

Suppose

Suppose the constants

To prove uniform asymptotic stability, let

Next, for any

For

This contradiction proves that there exists a

Suppose that the assumptions in Theorem

By using Property

A fractional neutral differential difference system is considered in the following state-space description:

Let the Lyapunov candidate be

In this paper, we have studied the stability of a class of nonlinear fractional neutral differential difference systems. We introduce the Lyapunov-Krasovskii approach for fractional neutral systems, which enrich the knowledge of both the system theory and the fractional calculus. By using Lyapunov-Krasovskii technique, stability and instability criteria are obtained for the nonlinear fractional neutral differential difference systems. Finally, we point out that since the computation of practically useful Lyapunov functionals is a very difficult task, fractional Lyapunov method has its own limitations. In other words, the present paper is only an introduction to the topic, and there remains a lot of work to do.

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

This work is supported by the National Natural Science Foundation of China (11371027), the Fundamental Research Funds for the Central Universities (2013HGXJ0226), and the Fund of Anhui University Graduate Academic Innovation Research (10117700004).