Stability of Nonlinear Fractional Neutral Differential Difference Systems

Abstract

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.

1. Introduction

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 [1–5] and the references therein.

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 [6]; the author studied the case of linear Caputo fractional differential equations. Since then, many researchers have done further studies on the stability of fractional differential systems [7–18]. For more details about the stability results and the methods available to analyze the stability of fractional differential systems, the reader may refer to the recent survey papers [19, 20] and the references therein.

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 [13, 14], the fractional Lyapunov’s second method was proposed, and the authors extended the exponential stability of integer order differential system to the Mittag-Leffler stability of fractional differential system. In [15], by using Bihari’s and Bellman-Gronwall’s inequality, an extension of Lyapunov’s second method for fractional-order systems was proposed. In [16–18], Baleanu et al. extended Lyapunov’s method to fractional functional differential systems and developed the Lyapunov-Krasovskii stability theorem, Lyapunov-Razumikhin stability theorem, and Mittag-Leffler stability theorem for fractional functional differential systems.

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. [13, 14], Baleanu et al. [16], and Cruz and Hale [21], we address the stability of fractional neutral systems. Specifically, we extend the Lyapunov-Krasovskii method for the nonlinear fractional neutral differential difference systems.

The rest of the paper is organized as follows. In Section 2, we give some notations and recall some concepts and preparation results. In Section 3, we develop the Lyapunov-Krasovskii theorem for the nonlinear fractional neutral differential difference systems; results of stability, asymptotically stability, uniform stability, uniform asymptotically stability, and instability for the nonlinear fractional neutral systems are presented. Section 4 brings an example to illustrate the results presented and finally Section 5 concludes the work.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts needed here. Throughout this paper, let ℝ^𝑛 be a real 𝑛-dimensional linear vector space with the norm |⋅|; let 𝒞=𝐶([−𝑟,0],ℝ^𝑛) be the space of continuous functions taking [−𝑟,0] into ℝ^𝑛 with ‖𝜙‖, 𝜙∈𝒞 defined by ‖𝜙‖=sup_{𝑠∈[−𝑟,0]}|𝜙(𝑠)|, 𝒞(𝑀)={𝜙∈𝒞∶‖𝜙‖≤𝑀}, 𝑟>0, 𝑀>0 be constants. Let 𝐴𝐶([𝑎,𝑏],ℝ) be the space of functions 𝑓(𝑡) which are absolutely continuous on [𝑎,𝑏] and let 𝐴𝐶^𝑛([𝑎,𝑏],ℝ) be the space of functions 𝑓(𝑡) which have continuous derivatives up to order 𝑛−1 on [𝑎,𝑏] such that 𝑓^(𝑛−1)(𝑡)∈𝐴𝐶([𝑎,𝑏],ℝ). If 𝜎∈ℝ, 𝐴>0 and 𝑥∈𝐶([𝜎−𝑟,𝜎+𝐴],ℝ^𝑛), then, for any 𝑡∈[𝜎,𝜎+𝐴], we let 𝑥_𝑡∈𝒞 be defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), 𝜃∈[−𝑟,0].

Let us recall the following definitions. For more details, we refer the reader to [1–5].

Definition 1. The fractional order integral of a function 𝑓∶[𝑡₀,∞)→ℝ of order 𝛼∈ℝ⁺=[0,+∞) is defined by 𝐼^𝛼_𝑡0𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠, (1) where Γ(⋅) is the gamma function.

Definition 2. For a function 𝑓 given on the interval [𝑡₀,∞), the 𝛼 order Riemann-Liouville fractional derivative of 𝑓 is defined by 𝐷^𝛼_𝑡0𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡_𝑡0(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)𝑑𝑠,𝑛−1<𝛼<𝑛,𝑛∈ℕ, (2) where ℕ={1,2,3,…}.

Definition 3. For a function 𝑓∈𝐴𝐶^𝑛([𝑡₀,𝑏],ℝ), the 𝛼 order Caputo fractional derivative of 𝑓 is defined by ^𝑐𝐷^𝛼_𝑡0𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑡0(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝛼<𝑛,𝑛∈ℕ. (3)

Some properties of the aforementioned operators are recalled as follows [1, 5].

Property 4. The following results are especially interesting.(i)For 𝜈>−1, we have 𝐷^𝛼_𝑡0(𝑡−𝑡₀)^𝜈=(Γ(1+𝜈)/Γ(1+𝜈−𝛼))(𝑡−𝑡₀)^𝜈−𝛼.(ii)When 0<𝛼<1, we have^𝑐𝐷^𝛼_𝑡0𝑓(𝑡)=𝐷^𝛼_𝑡0𝑓(𝑡)−𝑓𝑡₀𝑡−𝑡₀^−𝛼Γ().1−𝛼 (4)(iii)For 𝛼∈(0,1), 𝑇≥𝑡₀, and 𝑓∈𝐶([𝑡₀,𝑇],ℝ^𝑛), we have ^𝑐𝐷^𝛼_𝑡0𝐼^𝛼_𝑡0𝑓(𝑡)=𝑓(𝑡), 𝐼^𝛼_𝑡0^𝑐𝐷^𝛼_𝑡0𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑡₀).

Remark 5. From Property 4, if ^𝑐𝐷^𝛼_𝑡0𝑓(𝑡)≥0, 𝛼∈(0,1), then, for 𝑡≥𝑡₀, we have the following.(i)𝑓(𝑡)≥𝑓(𝑡₀).(ii)In general, it is not true that 𝑓(𝑡) is nondecreasing in 𝑡.

In [21], Cruz and Hale studied a class of functional difference operators which are very useful in stability theory and the asymptotic behavior of solutions of functional differential equations of neutral type. Suppose 𝜏∈ℝ, 𝑔∶[𝜏,+∞)×𝒞→ℝ^𝑛 is continuous, 𝑔(𝑡,𝜙) is linear in 𝜙, and there are an 𝑛×𝑛 matrix 𝜇(𝑡,𝜃), 𝑡∈[𝜏,+∞), 𝜃∈[−𝑟,0], of bounded variation in 𝜃 and a scalar function 𝑙(𝑠) continuously nondecreasing for 𝑠∈[0,𝑟], 𝑙(0)=0, such that 𝑔𝑡,𝜙=⁰_−𝑟𝑑_𝜃𝜇(𝑡,𝜃)_0−𝑠𝜙(𝜃),𝑑_𝜃𝜇(𝑡,𝜃)𝜙(𝜃)≤𝑙(𝑠)sup_{[]𝜃∈−𝑠,0}𝜙(𝜃), (5) for all 𝑡∈[𝜏,+∞), 𝜙∈𝒞. Define the linear functional difference operator 𝒟∶ℝ×𝒞⟶ℝ^𝑛 (6) by 𝒟𝑡,𝜙=𝜙(0)−𝑔𝑡,𝜙. (7)

For any 𝐻∈𝐶([𝜏,+∞),ℝ^𝑛), the space of continuous functions taking [𝜏,+∞) into ℝ^𝑛, 𝜎∈[𝜏,+∞) consider the equation 𝒟𝑡,𝑥_𝑡=𝒟𝜎,𝜑+𝐻(𝑡)−𝐻(𝜎),𝑡≥𝜎,𝑥_𝜎=𝜑,𝜑∈𝒞. (8)

Definition 6 (see [21]). Suppose ℋ is a subset of 𝐶([𝜏,+∞),ℝ^𝑛). Ones says the operator 𝒟 is uniformly stable with respect to ℋ if there are constants 𝐾,𝑀 such that, for any 𝜑∈𝒞, 𝜎∈[𝜏,+∞) and ℋ, the solution 𝑥(𝑡,𝜎,𝜑,𝐻) of (8) satisfies 𝑥_𝑡𝜎,𝜑,𝐻≤𝐾𝜑+𝑀sup_{[]𝑢∈𝜎,𝑡}|𝐻(𝑢)−𝐻(𝜎)|,𝑡≥𝜎. (9)

Lemma 7 (see [21]). If 𝒟 is uniformly stable with respect to 𝐶([𝜏,+∞),ℝ^𝑛), then there are positive constants 𝑎,𝑏,𝑐,𝑑 such that, for any ℎ∈𝐶([𝜏,+∞),ℝ^𝑛), 𝜎∈[𝜏,+∞), the solution 𝑥(𝑡,𝜎,𝜑,ℎ) of the equation 𝒟𝑡,𝑥_𝑡=ℎ(𝑡),𝑡≥𝜎,𝑥_𝜎=𝜑, (10) satisfies 𝑥_𝑡𝜎,𝜑,ℎ≤𝑒^{−𝑎(𝑡−𝜎)}𝑏𝜑+𝑐sup_{[]𝑢∈𝜎,𝑡}|ℎ(𝑢)|+𝑑sup_{[]𝑢∈𝜎,𝑡}|ℎ(𝑢)|, (11) for all 𝑡≥𝜎. Furthermore, the constants 𝑎,𝑏,𝑐,𝑑 can be chosen so that for any 𝑠∈[𝜎,+∞)𝑥_𝑡𝜎,𝜑,ℎ≤𝑒^{−𝑎(𝑡−𝑠)}𝑏𝜑+𝑐sup_{[]𝑢∈𝜎,𝑡}|ℎ(𝑢)|+𝑑sup_{[]𝑢∈𝑠,𝑡}|ℎ(𝑢)|, (12) for all 𝑡≥𝑠+𝑟.

Definition 8 (see [22]). Suppose 𝒟∶𝒞→ℝ^𝑛 is linear, continuous, and atomic at 0 and let 𝒞_𝒟={𝜙∈𝒞∶𝒟(𝜙)=0}. The operator 𝒟 is said to be stable if the zero solution of the homogeneous difference equation, 𝒟𝑦_𝑡=0,𝑡≥0,𝑦₀=𝜑∈𝒞_𝒟, (13) is uniformly asymptotically stable.

Definition 9 (see [23]). The matrix 𝐵∈ℝ^𝑛×𝑛 is Schur stable if the spectrum of the matrix lies in the open unit disc of the complex plane.

Consider a simple operator 𝒟∶𝒟(𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), where the matrix 𝐵 is Schur stable. Let 𝑥(𝑡) be a solution of the homogeneous difference equation 𝒟𝑥_𝑡=0,𝑡≥0,𝑥₀=𝜑∈𝒞_𝒟. (14) Then, for a given 𝑡≥0, there must exist a positive integer number 𝑘 such that 𝑡∈[(𝑘−1)𝑟,𝑘𝑟]. It follows from (14) that 𝑥(𝑡)=𝐵^𝑘𝑥(𝑡−𝑘𝑟). (15) Since the matrix 𝐵 is Schur stable, there exist 𝐿≥1 and 0<𝜁<1 such that the inequality ‖𝐵^𝑘‖₀≤𝐿𝜁^𝑘 holds. Therefore we have |𝑥(𝑡)|=𝐵^𝑘𝑥(𝑡−𝑘𝑟)≤𝐵^𝑘₀|𝑥(𝑡−𝑘𝑟)|≤𝐿𝜁^𝑘𝜑. (16) Since lim_𝑘→∞𝜁^𝑘=0, it follows from (16) that the zero solution of the homogeneous difference equation (14) is uniformly asymptotically stable. Therefore we have the following remark.

Remark 10. Let 𝒟(𝑡,𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), where 𝐵 is Schur stable. Then 𝒟 is stable.

3. Main Results

In this section, we consider the stability of the following nonlinear fractional neutral differential difference system: ^𝑐𝐷^𝛼_𝑡0𝒟𝑡,𝑥_𝑡=𝑓𝑡,𝑥_𝑡,𝑡>𝑡₀, (17) with the initial condition 𝑥_𝑡0=𝜑, (18) where 0<𝛼<1, 𝑓∶ℝ×𝒞→ℝ^𝑛 is continuous, Lipschitz in 𝑥_𝑡, and takes closed bounded sets into bounded sets and 𝑔(𝑡,0)=𝑓(𝑡,0)≡0; the linear difference operator 𝒟 is defined in (7) with 𝑔 satisfying (5); that is, 𝑔(𝑡,𝜙) is uniformly nonatomic at zero [22]. Here, we always assume that fractional order neutral system (17) with initial condition (18) has a unique continuous solution 𝑥(𝑡) which depends continuously upon 𝑡₀,𝜑.

If 𝑉∶ℝ×𝒞→ℝ is continuously differentiable, we define the Caputo fractional derivative ^𝑐𝐷^𝛾_𝑡0𝑉(𝑡,𝜙) along the solution 𝑥_𝑡=𝑥_𝑡(𝑡₀,𝜑) of (17)-(18) as ^𝑐𝐷^𝛾_𝑡0𝑉𝑡,𝑥_𝑡=1Γ1−𝛾^𝑡_𝑡0(𝑡−𝑠)^−𝛾𝑉^′𝑠,𝑥_𝑠𝑑𝑠,0<𝛾<1. (19)

Definition 11. We say that the zero solution 𝑥=0 of (17) is stable if, for any 𝑡₀∈ℝ and any 𝜀>0, there exists a 𝛿=𝛿(𝑡₀,𝜀) such that any solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜑) of (17) with initial value 𝜑 at 𝑡₀, ‖𝜑‖<𝛿, satisfies |𝑥(𝑡)|<𝜀 for 𝑡≥𝑡₀. It is asymptotically stable if it is stable and, for any 𝑡₀∈ℝ and any 𝜀>0, there exists a 𝛿₀=𝛿(𝑡₀,𝜀), 𝑇(𝑡₀,𝜀)>0 such that ‖𝜑‖<𝛿₀ implies |𝑥(𝑡)|≤𝜀, for 𝑡≥𝑡₀+𝑇(𝑡₀,𝜀); that is, lim_𝑡→+∞𝑥(𝑡)=0. It is uniformly stable if it is stable and 𝛿=𝛿(𝜀)>0 can be chosen independently of 𝑡₀. It is uniformly asymptotically stable if it is uniformly stable and there exists a 𝛿₀>0; for any 𝜂>0, there exists a 𝑇=𝑇(𝜂)>0 such that ‖𝜑‖<𝛿₀ implies |𝑥(𝑡)|<𝜂 for 𝑡>𝑡₀+𝑇.

Next, we will address the main core of the paper, the stability of the nonlinear fractional neutral systems (17). Firstly, we consider the case 𝒟(𝑡,𝜙)=𝒟(𝜙); that is, 𝒟 is independent of 𝑡. Now, we give the following Lyapunov-Krasovskii theorems for nonlinear fractional neutral systems (17).

Theorem 12. Let 𝒟(𝑡,𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), let 𝐵 be Schur stable, and let 𝑥=0 be an equilibrium point of system (17). Then the zero solution of system (17) is stable if and only if there exist a functional 𝑉∶ℝ×𝒞(𝑀)→ℝ and a continuous function 𝑢(𝑠) with 𝑢(𝑠)>0 for 𝑠>0 and 𝑢(0)=0 such that the following conditions are satisfied.(1)𝑉(𝑡,0)=0.(2)𝑉(𝑡,𝜙)≥𝑢(|𝒟(𝜙)|).(3)For any given 𝑡₀ the functional 𝑉(𝑡₀,𝜙) is continuous in 𝜙 at the point 0; that is, for any 𝜀>0 there exists 𝛿>0 such that the inequality ‖𝜙‖<𝛿 implies |𝑉(𝑡₀,𝜙)−𝑉(𝑡₀,0)|=𝑉(𝑡₀,𝜙)<𝜀.(4)Along the solutions of system (17) the functional 𝑉(𝑡,𝜙) satisfies 𝑉(𝑡,𝑥_𝑡(𝑡₀,𝜑))≤𝑉(𝑡₀,𝜑) for 𝑡≥𝑡₀.

Proof
Sufficiency. Since the matrix 𝐵 is Schur stable, there exist 𝐿≥1 and 0<𝜁<1 such that the inequality ‖𝐵^𝑘‖₀≤𝐿𝜁^𝑘 holds for 𝑘≥0.
For a given 𝜀>0(𝜀<𝑀), we set 𝜀₁=𝜀(1−𝜁)/𝐿>0. Since for a given 𝑡₀ functional 𝑉(𝑡₀,𝜙) is continuous in 𝜙 at the point 0, there exists 𝛿₁(𝜀,𝑡₀) such that 𝑉(𝑡₀,𝜙)<𝑢(𝜀₁) for any 𝜙∈𝒞, with ‖𝜙‖<𝛿₁(𝜀,𝑡₀). Here, we claim that 𝛿₁(𝜀,𝑡₀)≤𝜀₁. then, there exists an initial function 𝜑∈𝒞 such that ‖𝜑‖<𝛿₁(𝜀,𝑡₀) and |𝒟(𝜑)|=𝜀₁. On the one hand, for this initial function, we have 𝑢(|𝒟(𝜑)|)=𝑢(𝜀₁). On the other hand, 𝑢(|𝒟(𝜑)|)≤𝑉(𝑡₀,𝜑)<𝑢(𝜀₁). The contradiction proves the desired inequality.
Now we take 𝛿(𝜀,𝑡₀)=𝛿₁(𝜀,𝑡₀)/(1+𝐿𝜁). For 𝜑∈𝒞 with ‖𝜑‖<𝛿(𝜀,𝑡₀), we have 𝑢𝒟𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡,𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡₀𝜀,𝜑<𝑢₁,𝑡≥𝑡₀. (20) Next, we wish to show 𝒟𝑥_𝑡𝑡₀,𝜑<𝜀₁,𝑡≥𝑡₀. (21) Assume by contradiction that there exists a 𝑡₁≥𝑡₀ for which 𝒟𝑥_𝑡1𝑡₀,𝜑≥𝜀₁. (22) Since 𝒟𝑥_𝑡0𝑡₀,𝜑=𝒟𝜑≤(1+𝐿𝜁)𝜑<𝛿₁𝜀,𝑡₀≤𝜀₁, (23) and |𝒟(𝑥_𝑡(𝑡₀,𝜑))| is a continuous function of 𝑡, there exists 𝑡∈[𝑡₀,𝑡₁] such that 𝒟𝑥_𝑡𝑡₀,𝜑=𝜀₁. (24) On the one hand, we have 𝑢𝒟𝑥_𝑡𝑡₀,𝜑=𝑢𝜀₁. (25) On the other hand, relation (20) provides the following inequality: 𝑢𝒟𝑥_𝑡𝑡₀,𝜑<𝑢𝜀₁. (26) The contradiction proves that inequality (22) is wrong and relation (21) is true. Then there exists a function 𝑧(𝑡) with |𝑧(𝑡)|<𝜀₁, 𝑡≥𝑡₀ such that 𝑥𝑡,𝑡₀,𝜑=𝐵𝑥𝑡−𝑟,𝑡₀,𝜑+𝑧(𝑡),𝑡≥𝑡₀. (27)
For a given 𝑡≥𝑡₀, there must exist a positive integer number 𝑘 such that 𝑡∈[𝑡₀+(𝑘−1)𝑟,𝑡₀+𝑘𝑟]. Iterating equality (27) 𝑘−1 times we obtain 𝑥𝑡,𝑡₀,𝜑=𝐵^𝑘𝑥𝑡−𝑘𝑟,𝑡₀,𝜑+^𝑘−1_𝑖=0𝐵^𝑖𝑧(𝑡−𝑖𝑟). (28) Since 𝑡−𝑘𝑟∈[𝑡₀−𝑟,𝑡₀], 𝑥𝑡−𝑘𝑟,𝑡₀,𝜑≤𝜑<𝛿𝜀,𝑡₀≤𝜀₁, (29) and we obtain the following inequality: 𝑥𝑡,𝑡₀,𝜑≤𝐵^𝑘₀𝑥𝑡−𝑘𝑟,𝑡₀+,𝜑_{𝑘−1𝑖=0}𝐵^𝑖₀|𝑧(𝑡−𝑖𝑟)|<𝐿𝜁^𝑘𝛿𝜀,𝑡₀+^𝑘−1_𝑖=0𝐿𝜁^𝑖𝜀₁<𝐿𝜀1−𝜁¹=𝜀,𝑡≥𝑡₀. (30) Therefore, the zero solution of system (17) is stable.
Necessity. Now, the zero solution of system (17) is stable, and we must prove that there exist a function 𝑢(𝑠) and a functional 𝑉(𝑡,𝜙) that satisfy the conditions (1)–(4).
Since the zero solution of system (17) is stable, for 𝜀=𝑀 there exists 𝛿(𝑀,𝑡₀) such that the inequality ‖𝜑‖<𝛿(𝑀,𝑡₀) implies that |𝑥(𝑡,𝑡₀,𝜑))|<𝑀 for 𝑡≥𝑡₀. We define the function 𝑢(𝑠)=𝑠, 𝑠∈ℝ and the functional 𝑉(𝑡,𝜙) as follows: 𝑉𝑡,𝑥_𝑡𝑡₀=,𝜑sup_𝑠≥𝑡𝒟𝑥_𝑠𝑡₀,𝑥,𝜑if𝑠,𝑡₀,𝜑<𝑀,for𝑠≥𝑡₀,(1+𝐿𝜁)𝑀,if∃𝑇≥𝑡₀suchthat𝑥𝑇,𝑡₀,𝜑≥𝑀. (31) Since for 𝜑=0 the corresponding solution is trivial, 𝑥(𝑡,𝑡₀,0)=0, 𝑡≥𝑡₀, we obtain 𝑉(𝑡,0)=0. In the case where |𝑥(𝑠,𝑡₀,𝜑)|<𝑀, 𝑠≥𝑡₀, we have 𝑢𝒟𝑥_𝑡𝑡₀,𝜑=𝒟𝑥_𝑡𝑡₀,𝜑≤sup_𝑠≥𝑡𝒟𝑥_𝑠𝑡₀,𝜑=𝑉𝑡,𝑥_𝑡𝑡₀,𝜑. (32) In the other case where there exists 𝑇≥𝑡₀ such that |𝑥(𝑇,𝑡₀,𝜑)|≥𝑀, the following inequality holds: 𝑢𝒟𝑥_𝑡𝑡₀,𝜑=𝒟𝑥_𝑡𝑡₀≤(),𝜑1+𝐿𝜁𝑀=𝑉𝑡,𝑥_𝑡𝑡₀,𝜑. (33) Further, for a given 𝑡₀, the stability of the zero solution means that for any 𝜀>0 there exists 𝛿₁=𝛿(𝜀/(1+𝐿𝜁),𝑡₀) such that ‖𝜑‖≤𝛿₁ implies 𝑥𝑡,𝑡₀<𝜀,𝜑1+𝐿𝜁,𝑡≥𝑡₀. (34) Then 𝒟𝑥_𝑡𝑡₀,𝜑≤𝑥𝑡,𝑡₀,𝜑+𝐿𝜁𝑥𝑡−𝑟,𝑡₀,𝜑<𝜀,𝑡≥𝑡₀. (35) Therefore, 𝑉𝑡₀,𝜑−𝑉𝑡₀,0=𝑉𝑡₀,0≤𝜀. (36) That is, for a fixed 𝑡₀ the functional 𝑉(𝑡₀,𝜑) is continuous in 𝜑 at the point 0.
Finally, we need to show 𝑉(𝑡,𝑥_𝑡(𝑡₀,𝜑))≤𝑉(𝑡₀,𝜑), 𝑡≥𝑡₀. First, if |𝑥(𝑠,𝑡₀,𝜑)|<𝑀 for 𝑠≥𝑡₀, 𝑉𝑡₀,𝜑=sup_𝑠≥𝑡0𝒟𝑥_𝑠𝑡₀,𝑉,𝜑𝑡,𝑥_𝑡𝑡₀,𝜑=sup_𝑠≥𝑡𝒟𝑥_𝑠𝑡₀,𝜑. (37) Note that {𝑠∶𝑠≥𝑡}⊆{𝑠∶𝑠≥𝑡₀} for 𝑡≥𝑡₀; then we have 𝑉𝑡,𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡₀,𝜑. (38) In the second case, there exists 𝑇≥𝑡₀ such that |𝑥(𝑇,𝑡₀,𝜑)|≥𝑀; we have 𝑉𝑡,𝑥_𝑡𝑡₀,𝜑=𝑉𝑡₀,𝜑=(1+𝐿𝜁)𝑀. (39) Therefore, we have 𝑉𝑡,𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡₀,𝜑,𝑡≥𝑡₀. (40) Then, the proof is complete.

Remark 13. The functional (31) has only an academic value. Obviously, we cannot use such functionals in applications. The computation of practically useful Lyapunov functionals is a very difficult task.

Theorem 14. Let 𝒟(𝑡,𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), 𝐵 be Schur stable and let 𝑥=0 be an equilibrium point of system (17). Suppose 𝑢(𝑠) is a continuous function with 𝑢(𝑠)>0 for 𝑠>0 and 𝑢(0)=0. If there exists a continuous functional 𝑉∶ℝ×𝒞(𝑀)→ℝ such that the following conditions are satisfied: (1)𝑉(𝑡,0)=0,(2)𝑉(𝑡,𝜙)≥𝑢(|𝒟(𝜙)|),(3)along the solutions of the system (5) the functional 𝑉(𝑡,𝜙) is continuously differentiable and satisfies ^𝑐𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡𝑡₀,𝜑≤0,𝑡≥𝑡₀, (41) where 𝛽∈(0,1], then the zero solution of system (17) is stable.

Proof. Note that the theorem conditions imply that of Theorem 12; therefore, the zero solution of system (17) is stable.

Theorem 15. Suppose that the assumptions in Theorem 14 are satisfied except replacing ^𝑐𝐷^𝛽_𝑡0 by 𝐷^𝛽_𝑡0; then one has the same result for stability.

Proof. By using Property 4 we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡=𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡−𝑉𝑡₀,𝜑𝑡−𝑡₀^−𝛽Γ.1−𝛽 (42) Since 𝑉(𝑡₀,𝜑)≥0, then ^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡)≤𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡). Then we can obtain the same result for stability.

Next, we present the following sufficient conditions for the asymptotic stability of the zero solution of system (17).

Theorem 16. Let 𝒟(𝑡,𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), let 𝐵 be Schur stable, and let 𝑥=0 be an equilibrium point of system (17). Suppose 𝑢(𝑠),𝑤(𝑠) are continuous functions, 𝑢(𝑠),𝑤(𝑠) are positive for 𝑠>0, and 𝑢(0)=𝑤(0)=0. If there exists a continuously differentiable functional 𝑉∶ℝ×𝒞(𝑀)→ℝ such that the following conditions are satisfied: (1)𝑉(𝑡,0)=0,(2)𝑉(𝑡,𝜙)≥𝑢(|𝒟(𝜙)|),(3)^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡(𝑡₀,𝜑))≤−𝑤(|𝒟(𝑥_𝑡(𝑡₀,𝜑))|), where 𝛽∈(0,1],then the zero solution of system (17) is asymptotically stable.

Proof. Since the matrix 𝐵 is Schur stable, there exist 𝐿≥1 and 0<𝜁<1 such that the inequality ‖𝐵^𝑘‖₀≤𝐿𝜁^𝑘 holds for 𝑘≥0.
Note that the conditions (1)–(3) of the theorem imply that of Theorem 14; then the zero solution of system (17) is stable; that is, for any 𝜀>0 and 𝑡₀, there exists 𝛿(𝜀,𝑡₀)>0 such that for every initial function 𝜑∈𝒞, with ‖𝜑‖<𝛿(𝜀,𝑡₀), the following inequality holds: 𝑥𝑡,𝑡₀,𝜑<𝜀,𝑡≥𝑡₀. (43)
Now, let 𝛿(𝜀,𝑡₀)=𝛿(𝜀/(1+𝐿𝜁),𝑡₀)>0. Given 𝑡₀ and an initial function 𝜑∈𝒞 with ‖𝜑‖<𝛿(𝜀,𝑡₀), we have 𝑥𝑡,𝑡₀<𝜀,𝜑1+𝐿𝜁,𝑡≥𝑡₀,𝒟𝑥_𝑡𝑡₀,𝜑<𝜀,𝑡≥𝑡₀+𝑟. (44) Next, we wish to show lim_𝑡→∞𝒟𝑥_𝑡𝑡₀,𝜑=lim_𝑡→∞𝑥𝑡,𝑡₀,𝜑−𝐵𝑥𝑡−𝑟,𝑡₀,𝜑=0. (45) Suppose not, then there exist 𝜀₀>0 and a sequence {𝑡_𝑘}^∞_𝑘=1, 𝑡_𝑘→∞, as 𝑘→∞ such that 𝒟𝑥_𝑡𝑘𝑡₀,𝜑=𝑥𝑡_𝑘,𝑡₀,𝜑−𝐵𝑥𝑡_𝑘−𝑟,𝑡₀,𝜑≥𝜀₀,𝑘≥1. (46) Without loss of generality we may assume that 𝑡_𝑘+1−𝑡_𝑘≥𝑟 for 𝑘≥0. It follows from system (17) that 𝒟𝑥_𝑡𝑡₀,𝜑=𝒟𝜑+𝐼^𝛼_𝑡0𝑓𝑡,𝑥_𝑡𝑡₀,𝜑,𝑡≥𝑡₀,𝒟𝑥_𝑡𝑘𝑡₀,𝜑=𝒟𝜑+𝐼^𝛼_𝑡0𝑓𝑡_𝑘,𝑥_𝑡𝑘𝑡₀,𝜑. (47) Then, we have 𝒟𝑥_𝑡𝑡₀,𝜑=𝒟𝑥_𝑡𝑘𝑡₀,𝜑+𝐼^𝛼_𝑡0𝑓𝑡,𝑥_𝑡𝑡₀,𝜑−𝐼^𝛼_𝑡0𝑓𝑡_𝑘,𝑥_𝑡𝑘𝑡₀,𝜑. (48) Since 𝑓 takes bounded sets into bounded sets, there is a constant 𝐿₀ such that |𝑓(𝑡,𝑥_𝑡(𝑡₀,𝜑))|≤𝐿₀ for 𝑡≥𝑡₀, ‖𝜑‖≤𝛿(𝜀,𝑡₀). Then 𝒟𝑥_𝑡𝑡₀,𝜑≥𝜀₀−2𝐿₀𝑡−𝑡_𝑘^𝛼Γ(1+𝛼),𝑡≥𝑡_𝑘. (49) Hence, for any 𝑘≥1, we have 𝒟𝑥_𝑡𝑡₀≥𝜀,𝜑⁰2,𝑡∈𝑡_𝑘,𝑡_𝑘+𝜃, (50) where 𝜃=min{𝑟,[Γ(1+𝛼)𝜀₀/4𝐿₀]^1/𝛼}. Let 𝜉=min_{𝜀0/2≤𝑠≤𝜀}𝑤(𝑠)>0. Then there exists some 𝑡_𝑛∈{𝑡_𝑘}^∞_𝑘=1 such that 𝑉𝑡₀<𝜉,𝜑Γ𝑡1+𝛽^𝑛−𝑡₀^𝛽. (51) From the third condition of the theorem, we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡_𝑛,𝑥_𝑡𝑛𝑡₀,𝜑≤−𝑤𝒟𝑥_𝑡𝑛𝑡₀,𝜑≤−𝜉. (52) That is, 𝑉𝑡_𝑛,𝑥_𝑡𝑛𝑡₀,𝜑≤𝑉𝑡₀−𝜉,𝜑Γ𝑡1+𝛽^𝑛−𝑡₀^𝛽<0, (53) which contradicts the condition (2) of the theorem. Therefore, relation (45) is true. This means that there exists a function 𝑧(𝑡) with lim_𝑡→∞|𝑧(𝑡)|=0 such that 𝑥𝑡,𝑡₀,𝜑=𝐵𝑥𝑡−𝑟,𝑡₀,𝜑+𝑧(𝑡),𝑡≥𝑡₀. (54) And given a positive value 𝜀₁<𝜀, there exists 𝑡₁>𝑡₀ such that |𝑧(𝑡)|≤1−𝜁𝜀2𝐿¹,𝑡≥𝑡₁. (55) Then there exists 𝑘₀ such that 𝐿𝜁^𝑘𝜀<(1/2)𝜀₁ for 𝑘≥𝑘₀. Thus, we have 𝑥𝑡,𝑡₀,𝜑=𝐵^𝑘0𝑥𝑡−𝑘₀𝑟,𝑡₀+,𝜑_{𝑘0−1𝑖=0}𝐵^𝑖𝑧(𝑡−𝑖𝑟),𝑡≥𝑡₁+𝑘₀𝑥𝑟,𝑡,𝑡₀,𝜑≤𝐵^𝑘0₀𝑥𝑡−𝑘₀𝑟,𝑡₀+,𝜑_{𝑘0−1𝑖=0}𝐵^𝑖₀|𝑧(𝑡−𝑖𝑟)|<𝐿𝜁^𝑘0𝜀𝜀+¹2<𝜀₁,𝑡≥𝑡₁+𝑘₀𝑟. (56) Therefore, we have lim_𝑡→∞𝑥(𝑡,𝑡₀,𝜑)=0; that is, the zero solution of system (17) is asymptotically stable.

Theorem 17. Suppose that the assumptions in Theorem 16 are satisfied except replacing ^𝑐𝐷^𝛽_𝑡0 by 𝐷^𝛽_𝑡0; then one has the same result for asymptotical stability.

Proof. By using Property 4 we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡=𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡−𝑉𝑡₀,𝜑𝑡−𝑡₀^−𝛽Γ.1−𝛽 (57) Since 𝑉(𝑡₀,𝜑)≥0, then ^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡)≤𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡). Then we can obtain the same result for asymptotical stability.

Here, we present the following sufficient conditions for the instability of the zero solution of system (17).

Theorem 18. Suppose 𝒟 in (17) is independent of 𝑡, 𝑉∶𝒞→ℝ is continuous and maps bounded sets into bounded sets, and there exist continuous, nondecreasing nonnegative functions 𝑎(𝑠),𝑏(𝑠), 𝑠≥0, positive for 𝑠>0, an open set 𝑆 in 𝒞, and a bounded open neighborhood 𝑈 of zero in 𝒞 such that 𝑉(𝜙) satisfies the following.(i)𝑉(𝜙)>0 on 𝑆, 𝑉(𝜙)=0 on the boundary 𝜕𝑆 of 𝑆.(ii)0 belongs to the closure of 𝑆∩𝑈.(iii)𝑉(𝜙)≤𝑎(|𝒟(𝜙)|) on ℝ×(𝑆∩𝑈).(iv)^𝑐𝐷^𝛽_𝑡0𝑉(𝜙)≥𝑏(|𝒟(𝜙)|) on ℝ×(𝑆∩𝑈), 𝛽∈(0,1].Under these conditions, the zero solution of (17) is unstable. More specifically, each solution 𝑥=𝑥(𝑡,𝑡₀,𝜑) of (17) with initial value 𝜑 in 𝑆∩𝑈 at 𝑡₀∈ℝ must reach the boundary of 𝑈 in finite time.

Proof. Suppose 𝜑₀∈𝑆∩𝑈, 𝑡₀∈ℝ. Then 𝑉(𝜑₀)>0. From (iv), the solution 𝑥(𝑡,𝑡₀,𝜑₀) satisfies 𝑉(𝑥_𝑡)≥𝑉(𝜑₀) as long as 𝑥_𝑡∈𝑆∩𝑈. From (iii) and (iv), this implies ^𝑐𝐷^𝛽_𝑡0𝑉𝑥_𝑡≥𝑏𝒟𝑥_𝑡≥𝑏𝑎⁻¹𝑉𝑥_𝑡≥𝑏𝑎⁻¹𝑉𝜑₀>0, (58) as long as 𝑥_𝑡∈𝑆∩𝑈. Then this relation implies 𝑉𝑥_𝑡≥𝑉𝜑₀+𝑏𝑎⁻¹𝑉𝜑₀𝑡−𝑡₀^𝛽Γ,1+𝛽 (59) as long as 𝑥_𝑡∈𝑆∩𝑈. Since 𝑈 is bounded and 𝑉 is bounded on 𝑈, there must be a 𝑡₁ such that 𝑥_𝑡1∈𝜕(𝑆∩𝑈). But hypothesis (i) implies that 𝑥_𝑡1∈𝜕(𝑈). This proves the last assertion of the theorem. Hypothesis (ii) implies that each neighborhood of zero contains a 𝜑₀ in 𝑆∩𝑈. Thus, 𝑥=0 is unstable and the theorem is proved.

Finally, we consider that the case 𝒟 is dependent of 𝑡 and give the following Lyapunov-Krasovskii theorems for nonlinear fractional neutral systems (17).

Theorem 19. Suppose 𝒟 is uniformly stable with respect to 𝐶(ℝ,ℝ^𝑛), 𝑢(𝑠), 𝑣(𝑠), 𝑤(𝑠) are continuous nondecreasing functions, 𝑢(𝑠),𝑣(𝑠) are positive for 𝑠>0, 𝑢(0)=𝑣(0)=0, and 𝑤(𝑠) is nonnegative. If there exists a continuously differentiable functional 𝑉∶ℝ×𝒞→ℝ such that (i)𝑢(|𝒟(𝑡,𝜙)|)≤𝑉(𝑡,𝜙)≤𝑣(‖𝜙‖),(ii)^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡)≤−𝑤(|𝒟(𝑡,𝑥_𝑡)|), 𝛽∈(0,1],then the zero solution of (17) is uniformly stable. If, in addition, 𝑤(𝑠)>0 for 𝑠>0, then it is uniformly asymptotically stable.

Proof. Suppose the constants 𝑏,𝑐,𝑑 are defined as in Lemma 7, and 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜑) is solution of (17)-(18). For any 𝜀>0, we can find a sufficiently small 𝛿 such that 𝑏𝛿<𝜀/2, 𝑣(𝛿)<𝑢(𝜀/2(𝑐+𝑑)). Hence, for any initial time 𝑡₀ and any initial condition 𝑥_𝑡0=𝜑 with ‖𝜑‖<𝛿, we have ^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡(𝑡₀,𝜑))≤0, and therefore 𝑉(𝑡,𝑥_𝑡(𝑡₀,𝜑))≤𝑉(𝑡₀,𝜑) for any 𝑡≥𝑡₀. This implies that 𝑢𝒟𝑡,𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡,𝑥_𝑡𝑡₀,𝜑≤𝑉𝑡₀(𝛿)𝜀,𝜑≤𝑣<𝑢2(),𝑐+𝑑 (60) which implies that |𝒟(𝑡,𝑥_𝑡(𝑡₀,𝜑))|<𝜀/2(𝑐+𝑑) for 𝑡≥𝑡₀. Since 𝒟 is uniformly stable, Lemma 7 implies 𝑥_𝑡𝑡₀𝜑+()𝜀,𝜑≤𝑏𝑐+𝑑2()𝜀𝑐+𝑑<𝑏𝛿+2<𝜀. (61) Therefore, the zero solution is uniformly stable.
To prove uniform asymptotic stability, let 𝜀₀∶=𝜀=1; choose 𝛿₀∶=𝛿(1)>0 which correspond to uniform stability. Then, for any 𝑡₀∈ℝ, ‖𝜑‖<𝛿₀ implies 𝑥_𝑡𝑡₀,𝜑<𝜀₀,𝒟𝑡,𝑥_𝑡𝑡₀<𝜀,𝜑⁰2(),𝑐+𝑑𝑡≥𝑡₀. (62)
Next, for any 𝜂>0, we wish to show that there is a 𝑇(𝛿₀,𝜂) such that any solution 𝑥(𝑡,𝑡₀,𝜑) of (17) with ‖𝜑‖<𝛿₀ satisfies ‖𝑥_𝑡(𝑡₀,𝜑)‖<𝜂 for 𝑡≥𝑡₀+𝑇(𝛿₀,𝜂). To do this, we show that there is a 𝑇(𝛿₀,𝜂) and 𝑡₁ in [𝑡₀,𝑡₀+𝑇(𝛿₀,𝜂)] such that ‖𝑥_𝑡1(𝑡₀,𝜑)‖<𝛿, where 𝛿=𝛿(𝜂) is the above constant for uniform stability. The uniform stability then implies that ‖𝑥_𝑡(𝑡₀,𝜑)‖<𝜂 for 𝑡≥𝑡₁ and, in particular, for 𝑡≥𝑡₀+𝑇(𝛿₀,𝜂).
For 𝑎,𝑏,𝑐,𝑑 as in Lemma 7, choose 𝑁=𝑁(𝛿₀,𝜂) so that 𝑒^−𝑎𝑁𝑏𝛿₀+𝑐2()≤𝛿𝑐+𝑑2. (63) Let 𝑇(𝛿₀,𝜂)=𝑁+𝑟+[𝑣(𝛿₀)Γ(1+𝛽)/𝑤(𝛿/2𝑑)]^1/𝛽. Suppose there is a solution 𝑥=𝑥(𝑡,𝑡₀,𝜑) of (17) with ‖𝜑‖<𝛿₀ and ‖𝑥_𝑡(𝑡₀,𝜑)‖≥𝛿 for 𝑡≥𝑡₀. Taking 𝑠=𝑡₀+[𝑣(𝛿₀)Γ(1+𝛽)/𝑤(𝛿/2𝑑)]^1/𝛽, 𝑡=𝑠^′∶=𝑠+𝑁+𝑟 in relation (12), we have 𝛿≤𝑥_𝑠′≤𝑒^−𝑎𝑁𝑏𝜑+𝑐sup_{𝑢∈[𝑡0,𝑠′]}𝒟𝑢,𝑥_𝑢+𝑑sup_{𝑢∈[𝑠,𝑠′]}𝒟𝑢,𝑥_𝑢≤𝑒^−𝑎𝑁𝑏𝛿₀+𝑐2(𝑐+𝑑)+𝑑sup_{[]𝑢∈𝑠,𝑠+𝑁+𝑟}𝒟𝑢,𝑥_𝑢≤𝛿2+𝑑sup_{𝑢∈[𝑠,𝑠′]}𝒟𝑢,𝑥_𝑢. (64) Therefore, there exists a 𝑡^′∈[𝑠,𝑠^′] such that |𝒟(𝑡^′,𝑥_𝑡′)|>𝛿/2𝑑. Then, by the condition (ii) of the theorem, we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡^′,𝑥_𝑡′≤−𝑤𝒟𝑡^′,𝑥_𝑡′𝛿≤−𝑤,2𝑑 (65) and hence by Property 4 we conclude ^𝑐𝐷^𝛽_𝑡0𝑉𝑡^′,𝑥_𝑡′𝛿+𝑤𝑡2𝑑^′−𝑡₀^𝛽Γ1+𝛽≤0. (66) Then by using Property 4 and condition (i) of the theorem, we have 𝑉𝑡^′,𝑥_𝑡′𝛿+𝑤𝑡2𝑑^′−𝑡₀^𝛽Γ𝑡1+𝛽≤𝑉₀,𝜑≤𝑣𝜙≤𝑣𝛿₀. (67) As a result we obtain 𝛿0<𝑢𝑡2𝑑≤𝑉^′,𝑥_𝑡′≤𝑣𝛿₀𝛿−𝑤𝑡2𝑑^′−𝑡₀^𝛽Γ𝛿1+𝛽≤𝑣₀𝛿−𝑤2𝑑𝑠−𝑡₀^𝛽Γ1+𝛽=0. (68)
This contradiction proves that there exists a 𝑡₁∈[𝑡₀,𝑡₀+𝑇(𝛿₀,𝜂)] such that ‖𝑥_𝑡1(𝑡₀,𝜑)‖<𝛿. Thus, we have |𝑥(𝑡,𝑡₀,𝜑)|<𝜂, 𝑡≥𝑡₀+𝑇(𝛿₀,𝜂), whenever ‖𝜑‖<𝛿₀. This proves the uniform asymptotic stability of the zero solution of (17).

Theorem 20. Suppose that the assumptions in Theorem 19 are satisfied except replacing ^𝑐𝐷^𝛽_𝑡0 by 𝐷^𝛽_𝑡0; then one has the same result for uniform stability and uniform asymptotic stability.

Proof. By using Property 4 we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡=𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡−𝑉𝑡₀,𝜑𝑡−𝑡₀^−𝛽Γ.1−𝛽 (69) Since 𝑉(𝑡₀,𝜑)≥0, then ^𝑐𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡)≤𝐷^𝛽_𝑡0𝑉(𝑡,𝑥_𝑡). Then we can obtain the same result for uniform stability and uniform asymptotic stability.

4. An Illustrative Example

A fractional neutral differential difference system is considered in the following state-space description: ^𝑐𝐷^𝛼_𝑡0𝒟𝑥_𝑡=𝑓𝒟𝑥_𝑡,𝑥_𝑡0=𝜑, (70) where 0<𝛼<1, 𝒟(𝑥_𝑡)=𝑥(𝑡)−𝐵𝑥(𝑡−𝑟), 𝐵 is Schur stable, 𝑓(0)=0, 𝑓(𝑥)(𝑑𝑓(𝑥)/𝑑𝑥)𝑥̇≤0, and there exist a continuous function 𝑢(𝑠) with 𝑢(𝑠)>0 for 𝑠>0 and 𝑢(0)=0 such that 𝑢(|𝒟(𝑥_𝑡)|)≤|𝑓(𝒟(𝑥_𝑡))|; then the equilibrium point 𝑥=0 of system (70) is stable.

Proof. Let the Lyapunov candidate be 𝑉(𝑡,𝑥_𝑡)=𝑓²(𝒟(𝑥_𝑡)), then, for 0<𝛽<1, we have ^𝑐𝐷^𝛽_𝑡0𝑉𝑡,𝑥_𝑡=𝐼_𝑡^1−𝛽₀𝑑𝑉𝑡,𝑥_𝑡𝑑𝑡=2𝐼_𝑡^1−𝛽₀𝑓𝒟𝑥_𝑡𝑑𝑓(𝑥)𝑑𝑥_{𝑥=𝒟(𝑥𝑡)}𝑑𝒟𝑥_𝑡𝑑𝑡=2𝐼_𝑡^1−𝛽₀𝑓𝒟𝑥_𝑡𝑓^′𝒟𝑥_𝑡𝑑𝒟𝑥_𝑡𝑑𝑡≤0. (71) Then, it follows from Theorem 14 that the equilibrium point 𝑥=0 of system (70) is stable.

5. Conclusions

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.

Conflict of Interests

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

Acknowledgments

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).