^{1, 2}

^{1, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

This paper is concerned with the general solution of linear fractional neutral differential difference equations. The exponential estimates of the solution and the variation of constant formula for linear fractional neutral differential difference equations are derived by using the Gronwall integral inequality and the Laplace transform method, respectively. The obtained results extend the corresponding ones of integer order linear ordinary differential equations and delay differential equations.

Fractional differential equations have been proved to be an excellent tool in the modelling of many phenomena in various fields of engineering, physics, and economics. Many practical systems can be represented more accurately through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [

On the other hand, time delays are present inherently in many interconnected real systems due to transportation of energy and materials. For example, feedback control systems containing time delays and fractional processes and controllers lead to fractional delay systems. While delay differential systems with integer order have been thoroughly investigated during the past decades (see [

Motivated and inspired by the mentioned works, in this paper, we investigate the representation of the general solution to linear fractional neutral differential difference system with the form

As we all know, the Laplace transform method is an effective and convenient method for solving linear fractional differential equations. The exponential estimate of the solution is an indispensable tache, which guarantees the rationality of solving fractional differential equations by the Laplace transform method. In [

The purpose of this paper is to construct the representation of the general solution for system (

This paper is organized as follows. In the next section, we present some definitions and preliminary facts used in the paper. In Section

In this section, we recall some definitions and preliminary facts which are used throughout this paper. For more details, one can see [

The Riemann-Liouville’s fractional integral of order

The Caputo’s fractional derivative of

The Laplace transform of a function

From the above definitions, we know that if the integral (

If

In this section, we derive the exponential estimation of the solution

Assume system (

For

From the proof of Theorem

From Theorem

Next, we consider the general solution of linear homogeneous equation of the form

Let

Let

In terms of the fundamental solution

If

Applying the Laplace transform to system (

For the particular case

For the particular case

Based on the Laplace transform method, we derive the variation of constant formula for the nonhomogeneous system (

If

Applying the Laplace transform to system (

For the particular case

The authors are very grateful to the Editor, Professor Pedro R. S. Antunes, and the two anonymous reviewers for their helpful and valuable comments and suggestions, which significantly contributed to improving the quality of the paper. This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 61272530, 11072059, and 11071001, the Doctoral Fund of Ministry of Education of China under Grant no. 20093401110001, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Key Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2010ZD02 and KJ2011A197.