On the Stability of Some Discrete Fractional Nonautonomous Systems

Abstract

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

1. Introduction

The fractional calculus which is as old as the usual calculus deals with the generalization of integration and differentiation of integer order to any order. Recently, there has been great interest in this calculus as it turned out that it has many applications in many fields of science and engineering [1–6].

The analogous theory for discrete fractional calculus was initiated by Miller and Ross in [7]. Building this theory was continued in [8–12].

The stability of fractional-order linear and nonlinear dynamic systems was a subject of many reports [13–19], while the stability of discrete dynamic systems was discussed in many books and articles (see, e.g., [20] and the references therein). But the stability of discrete fractional dynamic systems, to our knowledge, has not been reported.

The analysis of the dynamics of complex or hypercomplex systems requires new methods and techniques to be developed. Particularly, the systems which have both continuous evolution and discrete evolution represents a new direction in many fields of science and engineering. For these type of systems the classical or the improved techniques and methods (e.g., with or without nonlocal terms described usually by the fractional derivatives and integrals) are not enough to describe accurately their dynamics. In this line of thought the combination of the fractional and discrete operators leads to a better description of the systems mentioned above. However, these new hybrid methods are at the beginning of their evolutions and new efforts are required to be developed and implemented in practical applications.

In [19], to demonstrate the advantage of using fractional-order derivatives instead of integer-order derivatives, the authors considered 2 systems. The system with integer-order derivative turned to be unstable. But the second system, where the integer-order derivative was replaced by fractional-order derivative, turned out to be stable. The same argument still holds for integer difference and fractional difference systems. To illustrate this, we consider the following two systems:Δ𝑥(𝑡)=𝜇𝑡^(𝜇−1)0<𝜇<1𝑥(0)=𝑥₀,_𝐶Δ^𝛼_𝑎𝑥(𝑡)=𝜇𝑡^(𝜇−1)0<𝛼<1,𝑎=1−𝛼𝑥(𝑎)=𝑥₀,𝑡∈ℕ₀, (1.1)(1.2) where 𝑡^(𝜇)=Γ(𝑡+1)/Γ(𝑡+1−𝜇) and the operator ^𝐶Δ^𝛼_𝑎 is defined by (2.10). The solution of (1.1) is 𝑥(𝑡)=𝑡^(𝜇)+𝑥₀ which clearly tends to ∞ as 𝑡→∞ for 0<𝜇<1 and thus is unstable. The solution of (1.2) which is defined on ℕ_𝛼 has the form𝑥(𝑡)=𝑥(𝑎)+𝜇Δ₀^−𝛼𝑡^(𝜇−1), (1.3) where the operator Δ₀^−𝛼 is defined by (2.6). The value of Δ₀^−𝛼𝑡^(𝜇−1) at 𝑡=𝛼+𝑘∈ℕ_𝛼 is computed as follows:Δ₀^−𝛼𝑡^(𝜇−1)()=1𝛼+𝑘Γ(𝛼)^𝑘_𝑠=0(𝛼+𝑘−𝑠−1)^(𝛼−1)𝑠^(𝜇−1)=1Γ(𝛼)^𝑘_𝑠=1Γ(𝛼+𝑘−𝑠)Γ()Γ()𝑘−𝑠+1𝑠+1Γ=1𝑠−𝜇+2Γ(𝛼)Γ(𝛼+𝑘−1)Γ(𝑘)Γ(2)Γ+Γ()3−𝜇𝛼+𝑘Γ()Γ(3)𝑘+2ΓΓ(𝛼)3−𝜇+⋯+Γ(1)Γ(𝑘+1)Γ.𝑘+2−𝜇 (1.4)

The quantity in (1.4)→0 as 𝑘→∞, and thus the solution (1.3) of (1.2) →𝑥₀ for 0<𝜇<1 as 𝑡→∞ and therefore is stable.

The purpose of this paper is to state the stability theorem for discrete nonlinear dynamic systems in the sense of the Caputo fractional differences.

The paper is organized as follows. In Section 2 basic definitions of fractional calculus and discrete fractional calculus are mentioned. Section 3 presents our main results on the stability of discrete fractional nonautonomous systems.

2. Preliminaries

For a function 𝑓 defined on the interval [𝑎,𝑏], the left and right Riemann-Liouville integrals of order 𝛼∈ℂ,Re(𝛼)>0, are defined, respectively, by [1–3]_𝑎𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝐼𝑑𝑠,_𝛼𝑏𝑓𝑡)=1(Γ(𝛼)^𝑏_𝑡𝑓(𝑠)(𝑠−𝑡)^1−𝛼𝑑𝑠, (2.1) while the left and right Riemann-Liouville derivatives of order 𝛼 are defined, respectively, by [1–3]_𝑎𝐷^𝛼𝑓(𝑡)=𝑑𝑑𝑡^𝑛𝑎𝐼^𝑛−𝛼𝑓(𝑡)=𝑑𝑑𝑡^𝑛1Γ(𝑛−𝛼)^𝑡_𝑎𝑓(𝑠)(𝑡−𝑠)^{1−𝑛+𝛼}𝐷𝑑𝑠,_𝛼𝑏𝑓𝑡)=−𝑑(𝑑𝑡^𝑛𝐼_𝑏^𝑛−𝛼𝑓𝑡)=−𝑑(𝑑𝑡^𝑛1Γ(𝑛−𝛼)^𝑏_𝑡𝑓(𝑠)(𝑠−𝑡)^{1−𝑛+𝛼}𝑑𝑠, (2.2) where 𝑛=[Re(𝛼)]+1 and Γ(⋅) is the Gamma function.

The left and right Caputo fractional derivatives of order 𝛼 on the interval [𝑎,𝑏] are defined, respectively, by^𝐶_𝑎𝐷^𝛼𝑓(𝑡)=_𝑎𝐼^𝑛−𝛼𝑓^(𝑛)(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,_𝐶𝐷^𝛼_𝑏𝑓(𝑡)=𝐼_𝑏^𝑛−𝛼𝑓^(𝑛)(𝑡)=1Γ(𝑛−𝛼)^𝑏_𝑡𝑓^(𝑛)(𝑠)(𝑠−𝑡)^1−𝛼𝑑𝑠. (2.3)

Some properties of the Riemann-Liouville and Caputo fractional derivatives are stated below [1–3].

Property 1. We have^𝐶_𝑎𝐷^𝛼𝑓(𝑡)=_𝑎𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)(𝑎)Γ()()𝑘−𝛼+1𝑡−𝑎^𝑘−𝛼,_𝐶𝐷^𝛼_𝑏𝑓(𝑡)=𝐷^𝛼_𝑏𝑓(𝑡)−^𝑛−1_𝑘=0(−1)^𝑘𝑓^(𝑘)(𝑏)Γ()()𝑘−𝛼+1𝑏−𝑡^𝑘−𝛼. (2.4)

Property 2. If 𝑓∈𝐴𝐶^𝑛[𝑎,𝑏]or𝐶^𝑛[𝑎,𝑏], then _𝑎𝐼_𝑎^𝛼𝐶𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)(𝑎)Γ()()𝑘−𝛼+1𝑡−𝑎^𝑘−𝛼,𝐼^𝛼_𝑏^𝐶𝐷^𝛼_𝑏𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(−1)^𝑘𝑓^(𝑘)(𝑏)Γ()()𝑘−𝛼+1𝑏−𝑡^𝑘−𝛼. (2.5)
Below we present basic definitions and properties of fractional sums and differences.
First we define the set ℕ_𝑎={𝑎,𝑎+1,𝑎+2,…} and _𝑏ℕ=𝑏,𝑏−1,𝑏−2,…. The (left) fractional sum of 𝑓, defined on ℕ_𝑎, of order 𝛼 is defined by [7–12]Δ_𝑎^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡−𝛼_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝛼−1)𝑓(𝑠), (2.6) where 𝛼>0, 𝜎(𝑠)=𝑠+1, and 𝑡^(𝛼)=Γ(𝑡+1)/Γ(𝑡+1−𝛼). The (right) fractional sum of 𝑓, defined on _𝑏ℕ, of order 𝛼 is defined by ∇_𝑏^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑏_{𝑠=𝑡+𝛼}𝜌(𝑠)−𝑡^(𝛼−1)𝑓(𝑠), (2.7) where 𝜌(𝑠)=𝑠−1. Analogous of the case of fractional derivatives, the Riemann left and right fractional differences of order 𝛼 are defined, respectively, by Δ^𝛼_𝑎𝑓(𝑡)=Δ^𝑛Δ_𝑎^{−(𝑛−𝛼)}𝑓(𝑡)=Δ^𝑛1Γ(𝑛−𝛼)^{𝑡−𝑛+𝛼}_𝑠=𝑎(𝑡−𝜎(𝑠))^{(𝑛−𝛼−1)}𝑓(𝑠,∇_𝛼𝑏)𝑓(𝑡)=(−1)^𝑛∇^𝑛∇_𝑏^{−(𝑛−𝛼)}𝑓(𝑡)=(−1)^𝑛∇^𝑛1Γ(𝑛−𝛼)^𝑏_{𝑠=𝑡+𝑛−𝛼}𝜌(𝑠)−𝑡^{(𝑛−𝛼−1)}𝑓(𝑠), (2.8) where 𝑛=[𝛼]+1. It can be obviously noticed that Δ_𝑎^−𝛼 maps functions defined on ℕ_𝑎 to functions defined on ℕ_𝑎+𝛼 and Δ^𝛼_𝑎 maps functions defined on ℕ_𝑎 to functions defined on ℕ_{𝑎+𝑛−𝛼}, while ∇_𝑏^−𝛼 maps functions defined on _𝑏ℕ to functions defined on _𝑏−𝛼ℕ_𝑎+𝛼 and ∇^𝛼_𝑏 maps functions defined on _𝑏ℕ to functions defined on _{𝑏−𝑛+𝛼}ℕ.
Similar to the case of fractional derivatives, the left and right Caputo fractional differences of order 𝛼 of a function defined on ℕ_𝑎 and _𝑏ℕ are defined, respectively, by [12]^𝐶Δ^𝛼_𝑎𝑓(𝑡)=Δ_𝑎^{−(𝑛−𝛼)}Δ^𝑛𝑓(𝑡)=1Γ(𝑛−𝛼)^{𝑡−𝑛+𝛼}_𝑠=𝑎(𝑡−𝜎(𝑠))^{(𝑛−𝛼−1)}Δ^𝑛𝑓(𝑠),_𝐶∇^𝛼_𝑏𝑓(𝑡)=(−1)^𝑛∇_𝑏^{−(𝑛−𝛼)}∇^𝑛𝑓(𝑡)=(−1)^𝑛Γ(𝑛−𝛼)^𝑏_{𝑠=𝑡+𝑛−𝛼}𝜌(𝑠)−𝑡^{(𝑛−𝛼−1)}∇^𝑛𝑓(𝑠), (2.9) where 𝑛=[𝛼]+1. The following property gives the relation between the Riemann and Caputo fractional differences [12].

Property 3. For any 𝛼>0, we have the following: ^𝐶Δ^𝛼_𝑎𝑓(𝑡)=Δ^𝛼_𝑎𝑓(𝑡)−^𝑛−1_𝑘=0(𝑡−𝑎)^{(𝑘−𝛼)}Γ()Δ𝑘−𝛼+1^𝑘𝑓(𝑎),_𝐶∇^𝛼_𝑏𝑓(𝑡)=∇^𝛼_𝑏𝑓(𝑡)−^𝑛−1_𝑘=0(𝑏−𝑡)^{(𝑘−𝛼)}Γ()()𝑘−𝛼+1−1^𝑘∇^𝑘𝑓(𝑏). (2.10)(2.11) Particularly, when 0<𝛼<1 we have ^𝐶Δ^𝛼_𝑎𝑓(𝑡)=Δ^𝛼_𝑎𝑓(𝑡)−(𝑡−𝑎)^(−𝛼)Γ()𝑓(𝑎),1−𝛼_𝐶∇^𝛼_𝑏𝑓(𝑡)=∇^𝛼_𝑏𝑓(𝑡)−(𝑏−𝑡)^(−𝛼)Γ()𝑓(𝑏).1−𝛼 (2.12)

Property 4. Let 𝛼>0, and let 𝑓 be a function defined on suitable subsets of ℕ_𝑎 and _𝑏ℕ. Then, Δ^𝐶−𝛼_{𝑎+𝑛−𝛼}Δ^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(𝑡−𝑎)^(𝑘)Δ𝑘!^𝑘𝑓(𝑎),∇^𝐶−𝛼_{𝑏−𝑛+𝛼}∇^𝛼_𝑏𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(𝑏−𝑡)^(𝑘)()𝑘!−1^𝑘∇^𝑘𝑓(𝑏). (2.13) Particularly, when 0<𝛼≤1 we have Δ^𝐶−𝛼_{𝑎+1−𝛼}Δ^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑎),∇^𝐶−𝛼_{𝑏−1+𝛼}∇^𝛼_𝑏𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑏). (2.14)

3. Lyapunov Stability Theorems for Nonautonomous Systems of Some Fractional Difference Equations

The stability theory of dynamic systems has a crucial role within the system theory and engineering. The stability of equilibrium points is usually done by utilizing the Lyapunov stability. Hypercomplex systems, namely, systems with both discrete and continuous behavior, need new methods and techniques to define their Lyapunov stability.

In this section we extend the method of the Lyapunov functions to study the stability of solutions of the following system:^𝐶Δ^𝛼_𝑡0𝑥(𝑡)=𝑔(𝑡+𝛼−1,𝑥(𝑡+𝛼−1))𝑥𝑡₀=𝑥₀, (3.1) where 𝑡₀=𝑎+𝑛₀∈ℕ_𝑎(𝑛₀∈ℕ), 𝑡∈ℕ_𝑛0, 𝑎=𝛼−1, 𝑔∶ℕ_𝑎×ℝ^𝑛→ℝ^𝑛 is continuous, and 0<𝛼≤1. We note that a Lyapunov function 𝑉 for the system (3.1) must depend on 𝑡 and 𝑥. Now let 𝑔(𝑡,0)=0, for all 𝑡∈ℕ_𝑎 so that the system (3.1) admits the trivial solution. Next we list some definitions that will be used in studying the stability properties of (3.1).

Definition 3.1. The trivial solution 𝑥(𝑡)=0 of (3.1) is said to be(i)stable if, for each 𝜖>0 and 𝑡₀∈ℕ_𝑎, there exists a 𝛿=𝛿(𝜖,𝑡₀)>0 such that for any solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥₀) with ‖𝑥₀‖<𝛿 one has ‖𝑥(𝑡)‖<𝜖, for all 𝑡∈ℕ_𝑡0⊆ℕ_𝑎,(ii)uniformly stable if it is stable and 𝛿 depends solely on 𝜖,(iii)asymptotically stable if it is stable and for all 𝑡₀∈ℕ_𝑎 there exists 𝛿=𝛿(𝑡₀)>0 if ‖𝑥₀‖<𝛿 implies that lim_𝑡→∞𝑥(𝑡,𝑡₀,𝑥₀)=0. (iv)uniformly asymptotically stable if it is uniformly stable and, for each 𝜖>0, there exists 𝑇=𝑇(𝜖)∈ℕ₀ and 𝛿₀>0 such that ‖𝑥₀‖<𝛿₀ implies ‖𝑥(𝑡)‖<𝜖 for all 𝑡∈ℕ_𝑡0+𝑇 and for all 𝑡₀∈ℕ_𝑎,(v)globally asymptotically stable if it is asymptotically stable for all 𝑥₀∈ℝ^𝑛,(vi)globally uniformly asymptotically stable if it is uniformly asymptotically stable for all 𝑥₀∈ℝ^𝑛.

Definition 3.2. A function 𝜙(𝑟) is said to belong to the class 𝒦 if and only if 𝜙∈𝐶[[0,𝜌),ℝ₊], 𝜙(0)=0, and 𝜙(𝑟) is strictly monotonically increasing in 𝑟. If 𝜙∶ℝ₊→ℝ₊, 𝜙∈𝒦, and lim_𝑟→∞𝜙(𝑟)=∞, then 𝜙 is said to belong to class 𝒦ℛ.

Definition 3.3. A real valued function 𝑉(𝑡,𝑥) defined on ℕ_𝑎×𝑆_𝜌, where 𝑆_𝜌={𝑥∈ℝ^𝑛∶‖𝑥‖≤𝜌}, is said to be positive definite if and only if 𝑉(𝑡,0)=0 for all 𝑡∈ℕ_𝑎 and there exists 𝜙(𝑟)∈𝒦 such that 𝜙(𝑟)≤𝑉(𝑡,𝑥),‖𝑥‖=𝑟,(𝑡,𝑥)∈ℕ_𝑎×𝑆_𝜌.

Definition 3.4. A real valued function 𝑉(𝑡,𝑥) defined on ℕ_𝑎×𝑆_𝜌, where 𝑆_𝜌={𝑥∈ℝ^𝑛∶‖𝑥‖≤𝜌}, is said to be decrescent if and only if 𝑉(𝑡,0)=0 for all 𝑡∈ℕ_𝑎 and there exists 𝜑(𝑟)∈𝒦 such that 𝑉(𝑡,𝑥)≤𝜑(𝑟),‖𝑥‖=𝑟,(𝑡,𝑥)∈ℕ_𝑎×𝑆_𝜌.
Now, we can state the theorems regarding the stability of solutions of the system (3.1).

Theorem 3.5. If there exists a positive definite and decrescent scalar function 𝑉(𝑡,𝑥)∈𝐶[ℕ_𝑎×𝑆_𝜌,ℝ₊] such that ^𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤0forall𝑡₀∈ℕ_𝑎 and (𝑡,𝑥)∈ℕ₀×𝑆_𝜌, then the trivial solution of (3.1) is uniformly stable.

Proof. Let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥₀) be a solution of system (3.1). Since 𝑉(𝑡,𝑥) is positive definite and decrescent, there exist 𝜙,𝜑∈𝒦 such that 𝜙(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝜑(‖𝑥‖) for all (𝑡,𝑥)∈ℕ_𝑎×𝑆_𝜌. For each 𝜖>0, 0<𝜖<𝜌, we choose a 𝛿=𝛿(𝜖) such that 𝜑(𝛿)<𝜙(𝜖). For any solution 𝑥(𝑡) of (3.1) we have 𝜙(‖𝑥(𝑡)‖)≤𝑉(𝑡,𝑥(𝑡)) with ‖𝑥₀‖<𝛿(𝜖). Since Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤0, by using (2.14) in Property 4 we have 𝑉(𝑡,𝑥(𝑡))≤𝑉(𝑡₀,𝑥₀)forall𝑡∈ℕ_𝑡0. Consequently, 𝜙(‖𝑥(𝑡)‖)≤𝑉(𝑡,𝑥(𝑡))≤𝑉𝑡₀,𝑥₀≤𝜑𝑥₀<𝜑(𝛿)<𝜙(𝜖), (3.2) and thus ‖𝑥(𝑡)‖<𝜖forall𝑡∈ℕ_𝑡0.

Theorem 3.6. If there exists a positive definite and decrescent scalar function 𝑉(𝑡,𝑥)∈𝐶[ℕ_𝑎×𝑆_𝜌,ℝ₊] such that ^𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤−𝜓(‖𝑥(𝑡+𝛼−1)‖),∀𝑡₀∈ℕ_𝑎,(𝑡,𝑥)∈ℕ₀×𝑆_𝜌, (3.3) where 𝜓∈𝒦, then the trivial solution of (3.1) is uniformly asymptotically stable.

Proof. Since all the conditions of Theorem 3.5 are satisfied, the trivial solution of the system (3.1) is uniformly stable. Let 0<𝜖<𝜌 and 𝛿=𝛿(𝜖) correspond to uniform stability. Choose a fixed 𝜖₀≤𝜌 and 𝛿₀=𝛿(𝜖₀)>0. Now, choose ‖𝑥₀‖<𝛿₀ and 𝑇(𝜖) large enough such that (𝑇+𝑎)^(𝛼)≥(𝜙(𝛿₀)/𝜓(𝛿(𝜖)))Γ(𝛼+1). Such a large 𝑇 can be chosen since lim_𝑇→∞(Γ(𝑇+𝛼)/Γ(𝑇))=∞. Now, we claim that ‖𝑥(𝑡,𝑡₀,𝑥₀)‖<𝛿(𝜖) for all 𝑡∈[𝑡₀,𝑡₀+𝑇]⋂ℕ_𝑡0. If this is not true, due to (3.3) and Property 4, we get 𝑉𝑡,𝑥𝑡,𝑡₀,𝑥₀≤𝑉𝑡₀,𝑥₀−1Γ(𝛼)^𝑡−𝛼_{𝑠=𝑡0+1−𝛼}(𝑡−𝜎(𝑠))^(𝛼−1)𝜓𝑥(𝑥(‖𝑠+𝛼−1)‖)≤𝜙₀−𝜓(𝛿)Γ(𝛼)^𝑡−𝛼_𝑠=𝑛0(𝑡−𝜎(𝑠))^(𝛼−1)≤𝜙𝛿₀−𝜓(𝛿)Γ(𝛼+1)𝑡−𝑛₀^(𝛼). (3.4) Substituting 𝑡=𝑡₀+𝑇, we get 0<𝜑(𝛿(𝜖))≤𝑉𝑡₀+𝑇,𝑥𝑡₀+𝑇,𝑡₀,𝑥₀≤𝜙𝛿₀−𝜓(𝛿)Γ(𝛼+1)𝑇+𝑡₀−𝑛₀^(𝛼)≤0, (3.5) which is a contradiction. Thus, there exists a 𝑡∈[𝑡₀,𝑡₀+𝑇] such that ‖𝑥(𝑡)‖<𝛿(𝜖). But in this case, since the trivial solution is uniformly stable and 𝑡 is arbitrary, ‖𝑥(𝑡)‖<𝜖 for all 𝑡≥𝑡₀+𝑇 whenever ‖𝑥₀‖<𝛿₀.

Theorem 3.7. If there exists a function 𝑉(𝑡,𝑥)∈𝐶[ℕ_𝑎×ℝ^𝑛,ℝ₊] such that 𝜙(‖𝑥(𝑡)‖)≤𝑉(𝑡,𝑥)≤𝜑(‖𝑥(𝑡)‖)∀(𝑡,𝑥)∈ℕ_𝑎×ℝ^𝑛,_𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤−𝜓(‖𝑥(𝑡+𝛼−1)‖)∀𝑡₀∈ℕ_𝑎,(𝑡,𝑥)∈ℕ₀×ℝ^𝑛, (3.6) where 𝜙,𝜑,and𝜓∈𝒦ℛ hold for all (𝑡,𝑥)∈ℕ_𝑎×ℝ^𝑛, then the trivial solution of (3.1) is globally uniformly asymptotically stable.

Proof. Since the conditions of Theorem 3.6 are satisfied, the trivial solution of (3.1) is uniformly asymptotically stable. It remains to show that the domain of attraction of 𝑥=0 is all of ℝ^𝑛. Since lim_𝑟→∞𝜙(𝑟)=∞, 𝛿₀ in the proof of Theorem 3.6 may be chosen arbitrary large and 𝜖 can be chosen such that it satisfies 𝜑(𝛿₀)<𝜙(𝜖). Thus, the globally uniformly asymptotic stability of 𝑥=0 is concluded.

Lemma 3.8. If 𝑉(𝑡₀,𝑥(𝑡₀))≥0, then, for 0<𝛼≤1, one has

^𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡)). (3.7)

Proof. From Property 3, we have ^𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))=Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))−𝑡−𝑡₀^(−𝛼)Γ()𝑉𝑡1−𝛼⁰,𝑥𝑡₀∀𝑡∈ℕ_𝑡0. (3.8) Since 𝑉(𝑡₀,𝑥(𝑡₀))≥0 and (𝑡−𝑡₀)^(−𝛼)/Γ(1−𝛼)≥0, we obtain ^𝐶Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡))≤Δ^𝛼_𝑡0𝑉(𝑡,𝑥(𝑡)). (3.9)

Theorem 3.9. (i) If the assumptions in Theorem 3.5 are satisfied except replacing ^𝐶Δ^𝛼_𝑡0 by Δ^𝛼_𝑡0, then the trivial solution of (3.1) is uniformly stable.
(ii) If the assumptions in Theorem 3.6 are satisfied except replacing ^𝐶Δ^𝛼_𝑡0 by Δ^𝛼_𝑡0, then the trivial solution of (3.1) is uniformly asymptotically stable.
(iii) If the assumptions in Theorem 3.7 are satisfied except replacing ^𝐶Δ^𝛼_𝑡0 by Δ^𝛼_𝑡0, then the trivial solution of (3.1) is globally uniformly asymptotically stable.

Proof. The proof is done by using Lemma 3.8 and following the same arguments as in the proofs of Theorem 3.5, Theorem 3.6, and Theorem 3.7.

4. Conclusion

In this paper using the Lyapunov direct method, we studied the stability of discrete nonautonomous systems in the sense of the Caputo fractional difference. We listed the conditions for uniform stability, uniformly asymptotic stability, and globally uniformly asymptotic stability for such systems. We will consider the Mittag-Leffler stability for such systems in future works.

Acknowledgments

This work is partially supported by the Scientific and Technical Research Council of Turkey. The authors would like to thank to the anonymous referee for her/his valuable comments.