Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

Abstract

The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.

1. Introduction

We have investigated that the stability of perturbed solution with respect to unperturbed solution with initial time difference of the nonlinear differential equations of fractional-order. The differential operators are taken in the Riemann-Liouville and Caputo’s sense and the initial conditions are specified according to Caputo’s suggestion [1], thus allowing for interpretation in a physically meaningful way [2].

Lyapunov’s second method is a standard technique used in the study of the qualitative behavior of differential systems [3–6] along with a comparison result [2, 7] that allows the prediction of behavior of a differential system when the behavior of the null solution of a comparison system is known. However, there has been difficulty with this approach when trying to apply it to unperturbed fractional differential systems [2, 6, 8] and associated perturbed fractional differential systems with an initial time difference. The difficulty arises because there is a significant difference between initial time difference (ITD) stability [7, 9–16] and the classical notion of stability for fractional differential systems [2, 6]. The classical notions of stability [2–6, 8, 17] are with respect to the null solution, but ITD stability [7, 9–16] is with respect to the unperturbed fractional-order differential system where the perturbed fractional-order differential system and the unperturbed fractional-order differential system differ both in initial position and initial time [7, 9–16].

In this paper, we have dissipated this complexity and have a new comparison result which again gives the null solution a central role in the comparison fractional-order differential system. This result creates many paths for continuing research by direct application and generalization [13, 15, 16].

In Section 2, we present basic definitions and necessary rudimentary material. In Section 3, we discuss and compare the differences between the classical notion of stability and the recent notion of initial time difference (ITD) stability by means of fractional-order differential systems. In Section 4, we have a comparison result in which the stability properties of the null solution of the comparison system imply the corresponding stability properties of the unperturbed fractional-order differential system. In Section 5, we have an other comparison result in which the stability properties of the null solution of the comparison system imply the corresponding (ITD) stability properties of the perturbed fractional-order differential system with respect to the unperturbed fractional-order differential system.

2. Preliminaries

In this section we give relation among the fractional-order derivatives: Caputo, Reimann-Liouville and Grünwald-Letnikov fractional-order derivatives and necessary definition of initial value problems of fractional-order differential equations with these sense.

2.1. Fractional-Order Derivatives: Caputo, Reimann-Liouville and Grünwald-Letnikov

Caputo’s and Reimann-Liouville’s definitions of fractional derivatives, are namely,

^𝑐𝐷^𝑞1𝑥=Γ(1−𝑞)^𝑡_𝜏0(𝑡−𝑠)^−𝑞𝑥^′(𝑠)𝑑𝑠,𝜏₀≤𝑡≤𝑇,𝐷^𝑞1𝑥=(𝑑Γ(𝑝)𝑑𝑡^𝑡_𝜏0(𝑡−𝑠)^𝑝−1𝑥(𝑠)𝑑𝑠),𝜏₀≤𝑡≤𝑇, (2.1)(2.2) respectively, order of 0<𝑞<1 and 𝑝+𝑞=1, where Γ denotes the Gamma function.

The most important advantage for fractional-order differential equations with Caputo’s derivative is the initial conditions that are the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant 𝐶 is zero, while the Riemann-Liouville fractional derivative for a constant 𝐶 is not zero but equals to 𝐷^𝑞𝐶=𝐶(𝑡−𝜏₀)^−𝑞/Γ(1−𝑞). By using (2.1), therefore,

^𝑐𝐷^𝑞𝑥(𝑡)=𝐷^𝑞[𝑥(𝑡)−𝑥(𝜏₀)],_𝑐𝐷^𝑞𝑥(𝑡)=𝐷^𝑞𝑥(𝑡)−𝑥(𝜏₀)Γ(1−𝑞)(𝑡−𝜏₀)^−𝑞. (2.3) In particular, if 𝑥(𝜏₀)=0, we obtain

^𝑐𝐷^𝑞𝑥(𝑡)=𝐷^𝑞𝑥(𝑡). (2.4) Hence, we can see that Caputo’s derivative is defined for functions for which Riemann-Liouville fractional-order derivative exists.

Let us write that Grünwald-Letnikov's notion of fractional-order derivative in a convenient form

𝐷^𝑞₀𝑥(𝑡)=limlim_{ℎ→00𝑛ℎ=𝑡−𝜏}1ℎ^𝑞[𝑥(𝑡)−𝑆(𝑥,ℎ,𝑟,𝑞)], (2.5) where 𝑆(𝑥,ℎ,𝑟,𝑞)=∑^𝑛_𝑟=1(−1)^𝑟+1(^𝑞_𝑟)𝑥(𝑡−𝑟ℎ). If we know that 𝑥(𝑡) is continuous and 𝑑𝑥(𝑡)/𝑑𝑡 exist and integrable, then Riemann-Liouville and Grünwald-Letnikov fractional-order derivatives are connected by the relation

𝐷^𝑞𝑥(𝑡)=𝐷^𝑞₀𝑥(𝑡)=𝑥(𝜏₀)(𝑡−𝜏₀)^−𝑞+Γ(1−𝑞)^𝑡_𝜏0(𝑡−𝑠)^−𝑞𝑑Γ(1−𝑞)𝑑𝑠𝑥(𝑠)𝑑𝑠. (2.6)

Using (2.3)implies that we have the following relations among the Caputo, Riemann-Liouville and Grünwald-Letnikov fractional derivatives

^𝑐𝐷^𝑞𝑥(𝑡)=𝐷^𝑞[𝑥(𝑡)−𝑥(𝜏₀)]=𝐷^𝑞₀[𝑥(𝑡)−𝑥(𝜏₀1)]=Γ(1−𝑞)^𝑡_𝜏0(𝑡−𝑠)^−𝑞𝑑𝑥(𝑠)𝑑𝑠𝑑𝑠. (2.7) The foregoing equivalent expressions are very useful in the study of qualitative properties of solutions of fractional differential equations.

2.2. Existence of Euler Solution

We consider the initial value problem of the fractional-order differential equation with Reimann-Liouville's derivative

𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥(𝑡)(𝑡−𝑡₀)^1−𝑞∣_𝑡=𝑡0=𝑥⁰for𝑡≥𝑡₀,𝑡₀∈ℝ₊, (2.8) where 𝑓 is any function from [𝑡₀,𝑇]×ℝ^𝑛→ℝ^𝑛. Let

𝜋=[𝑡₀,𝑡₁,…,𝑡_𝑁] (2.9) be a partition of [𝑡₀,𝑇].

Consider the interval [𝑡₀,𝑡₁] and observe that the right hand side of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative

𝐷^𝑞𝑥=𝑓(𝑡₀,𝑥⁰),𝑥(𝑡)(𝑡−𝑡₀)^1−𝑞∣_𝑡=𝑡0=𝑥⁰for𝑡≥𝑡₀ (2.10) on [𝑡₀,𝑡₁] is constant.

Therefore, the initial value problem has a unique solution of (2.10) of the fractional-order differential equation with Reimann-Liouville’s derivative given by

𝑥𝑥(𝑡)=⁰(𝑡−𝑡₀)^𝑞−1Γ(𝑞)+𝑓(𝑡₀,𝑥⁰)(𝑡−𝑡₀)^𝑞Γ(1+𝑞),𝑡∈[𝑡₀,𝑡₁]. (2.11) Define the node 𝑥₁=𝑥(𝑡₁) and iterate next by considering on [𝑡₁,𝑡₂] the initial value problem

𝐷^𝑞𝑥=𝑓(𝑡₁,𝑥₁),𝑥₁(𝑡)(𝑡−𝑡₁)^1−𝑞∣_𝑡=𝑡1=𝑥⁰₁for𝑡≥𝑡₁. (2.12) The next node is 𝑥₂=𝑥(𝑡₂) and we proceed this way till an arc 𝑥_𝜋=𝑥_𝜋(𝑡) has been defined on all [𝑡₀,𝑇]. Let us employ the notation 𝑥_𝜋 to emphasize the role played by the particular partition 𝜋 in determining 𝑥_𝜋 which is the Euler curved arc corresponding to the partition 𝜋. The diameter of the partition 𝜋 is given by

𝜇_𝜋=max[𝑡_𝑖−𝑡_𝑖−1∶1≤𝑖≤𝑁]. (2.13)

Definition 2.1. An Euler solution is any curved arc 𝑥=𝑥(𝑡) which is the uniform limit of Euler curved arcs 𝑥_𝜋, corresponding to some sequence 𝜋_𝑗 such that 𝜋_𝑗→0, which means the convergence of the diameter 𝜇_𝜋𝑗→0 as 𝑗→∞.

Now, we can give the following result on existence of an Euler solution of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative for (2.8).

Theorem 2.2. Assume that
(i)‖𝑓(𝑡,𝑥)‖≤𝑔(𝑡,‖𝑥‖),(𝑡,𝑥)∈[𝑡₀,𝑇]×ℝ^𝑛, where 𝑔∈𝐶[[𝑡₀,𝑇]×ℝ₊,ℝ₊],𝑔(𝑡,𝑢) is nondecreasing in (𝑡,𝑢);(ii)The maximal solution 𝑟(𝑡)=𝑟(𝑡,𝑡₀,𝑢⁰) of the fractional-order scalar differential equation with Reimann-Liouville's derivative 𝐷^𝑞𝑢=𝑔(𝑡,𝑢),𝑢(𝑡)(𝑡−𝑡₀)^1−𝑞∣_𝑡=𝑡0=𝑢⁰≥0for𝑡≥𝑡₀,𝑡₀∈ℝ₊ (2.14) exists on [𝑡₀,𝑇].
Then    (a)there exists at least one Euler solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥⁰) to the initial value problem (2.8), which satisfies a Hölder condition;(b)any Euler solution 𝑥(𝑡) of (2.8) satisfies the relation ‖𝑥(𝑡)−𝑥⁰(𝑡)‖≤𝑟(𝑡,𝑡₀,𝑢⁰)−𝑢⁰,𝑡∈[𝑡₀,𝑇], (2.15) where 𝑢⁰=‖𝑥⁰‖ and 𝑥⁰(𝑡)=𝑥⁰(𝑡−𝑡₀)^𝑞−1/Γ(𝑞).

For proof of Theorem 2.2, please see in [6].

If 𝑓(𝑡,𝑥) in (2.8) is assumed to be continuous, then 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥⁰), an Euler solution, is actually a solution of the initial value problem (2.8).

Theorem 2.3. Under the assumptions of Theorem 2.2 and if we suppose that 𝑓∈𝐶[[𝑡₀,𝑡₀+𝑇]×ℝ^𝑛,ℝ^𝑛], then 𝑥(𝑡) is a solution of initial value problem (2.8).

For proof Theorem 2.3, please see in [6].

2.3. Fractional-Order Differential Equations with Caputo’s Derivative

Consider the initial value problems of the fractional-order differential equations with Caputo's derivative

^𝑐𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥(𝑡₀)=𝑥₀for𝑡≥𝑡₀,𝑡₀∈ℝ₊,_𝑐𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥(𝜏₀)=𝑦₀for𝑡≥𝜏₀,𝜏₀∈ℝ₊, (2.16)(2.17) where 𝑥₀=lim_𝑡→𝑡0𝐷^𝑞−1𝑥(𝑡) and 𝑦₀=lim_𝑡→𝜏0𝐷^𝑞−1𝑥(𝑡) exist, and the perturbed system of initial value problem of the fractional-order differential equation with Caputo’s derivative of (2.17)

^𝑐𝐷^𝑞𝑦=𝐹(𝑡,𝑦),𝑦(𝜏₀)=𝑦₀for𝑡≥𝜏₀≥𝑡₀, (2.18) where 𝑦₀=lim_𝑡→𝜏0𝐷^𝑞−1𝑦(𝑡), exists, and 𝑓,𝐹∈𝐶[[𝑡₀,𝜏₀+𝑇]×ℝ^𝑛,ℝ^𝑛]; satisfy a local Lipschitz condition on the set ℝ₊×𝑆𝜌,𝑆𝜌=[𝑥∈ℝ^𝑛∶‖𝑥‖<𝜌<∞] and 𝑓(𝑡,0)=0 for 𝑡≥0. In particular, 𝐹(𝑡,𝑦)=𝑓(𝑡,𝑦)+𝑅(𝑡,𝑦), we have a special case of (2.18) and 𝑅(𝑡,𝑦) is said to be perturbation term.

Corollary 2.4. Let 0<𝑞<1, and 𝑓∶(𝑡₀,𝑡₀+𝑇]×𝑆𝜌→ℝ a function such that 𝑓(𝑡,𝑥)∈𝐿(𝑡₀,𝑡₀+𝑇) for any 𝑥∈𝑆𝜌. If 𝑥(𝑡)∈𝐿(𝑡₀,𝑡₀+𝑇), then 𝑥(𝑡) satisfies a.e. the initial value problems of the fractional-order differential equations with Reimann-Liouville's derivative (2.19) if, and only if, 𝑥(𝑡) satisfies a.e. the Volterra fractional-order integral equation (2.20).

For proof of Corollary 2.4, please see in [2].

We assume that we have sufficient conditions to the existence and uniqueness of solutions through (𝑡₀,𝑥₀) and (𝜏₀,𝑦₀). If 𝑓∈𝐶[[𝑡₀,𝑡₀+𝑇]×ℝ^𝑛,ℝ^𝑛] and 𝑥(𝑡) is the solution of

𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥(𝑡)(𝑡−𝑡₀)^1−𝑞∣_𝑡=𝑡0=𝑥⁰for𝑡≥𝑡₀,𝑡₀∈ℝ₊, (2.19) where 𝐷^𝑞𝑥 is the Reimann-Liouville fractional-order derivative of 𝑥 as in (2.2), then it also satisfies the Volterra fractional-order integral equation

𝑥𝑥(𝑡)=⁰(𝑡−𝑡₀)^𝑞−1+1Γ(𝑞)Γ(𝑞)^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡₀≤𝑡≤𝑡₀+𝑇 (2.20) that is, every solution of (2.20) is also a solution of (2.19); for detail please see [2].

We will only give the basic existence and uniqueness result with the Lipschitz condition by using contraction mapping theorem and a weighted norm with Mittag-Leffler function in [6].

Theorem 2.5. Assume that(i)𝑓∈𝐶[𝑅,ℝ^𝑛] and bounded by 𝑀 on 𝑅, where 𝑅=[(𝑡,𝑥)∶𝑡₀≤𝑡≤𝑡₀+𝑇,‖𝑥−𝑥₀‖≤𝑏];(ii)‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖≤𝐿‖𝑥−𝑦‖,𝐿>0,(𝑡,𝑥)∈𝑅, where the inequalities are componentwise. Then there exists a unique solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥₀) on [𝑡₀,𝑡₀+𝛼] for the initial value problem of the fractional-order differential equation with Caputo's derivative of (2.16), where 𝛼=min[𝑇,(𝑏Γ(𝑞+1)/𝑀)^1/𝑞].

For proof of Theorem 2.5, please see in [6].

2.4. Stability Criteria with ITD and Lyapunov-Like Function

Before we can establish our comparison theorem and Lyapunov stability criteria for initial time difference, we need to introduce the following definitions of ITD stability and Lyapunov-like functions.

Definition 2.6. The solution 𝑦(𝑡,𝜏₀,𝑦₀) of the initial value problems of fractional-order differential equation with Caputo’s derivative of (2.18) through (𝜏₀,𝑦₀) is said to be initial time difference stable with respect to the solution 𝑥(𝑡,𝜏₀,𝑥₀)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the initial value problems of fractional-order differential equation with Caputo's derivative of (2.16) for 𝑡≥𝜏₀,𝜏₀∈ℝ₊ and 𝜂=𝜏₀−𝑡₀ if and only if given any 𝜖>0 there exist 𝛿₁=𝛿₁(𝜖,𝜏₀)>0 and 𝛿₂=𝛿₂(𝜖,𝜏₀)>0, such that ‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀)‖<𝜖,whenever‖𝑦₀−𝑥₀‖<𝛿₁,|𝜏₀−𝑡₀|<𝛿₂for𝑡≥𝜏₀. (2.21) If 𝛿₁,𝛿₂ are independent of 𝜏₀, then the solution 𝑦(𝑡,𝜏₀,𝑦₀) of the initial value problems of fractional-order differential equation with Caputo’s derivative of (2.18) is initial time difference uniformly stable with respect to the fractional solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀). If the solution of initial value problems of fractional-order differential equation with Caputo’s derivative of 𝑦(𝑡,𝜏₀,𝑦₀) of the fractional system (2.18) through (𝜏₀,𝑦₀) is initial time difference stable and there exist 𝛾₁(𝜏₀)>0 and 𝛾₂(𝜏₀)>0 such that lim_𝑡→∞‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀)‖=0 (2.22) for all 𝑦(𝑡,𝜏₀,𝑦₀) and 𝑥(𝑡−𝜂,𝑡₀,𝑥₀) with ‖𝑦₀−𝑥₀‖<𝛾₁ and |𝜏₀−𝑡₀|<𝛾₂ for 𝑡≥𝜏₀, then it is said to be initial time difference asymptotically stable with respect to the fractional solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀). It is initial time difference uniformly asymptotically stable with respect to the fractional solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀) if 𝛾₁ and 𝛾₂ are independent of 𝜏₀.

Definition 2.7. A function 𝜙(𝑟) is said to belong to the class 𝒦 if 𝜙∈𝐶[(0,𝜌),ℝ₊],𝜙(0)=0, and 𝜙(𝑟) is strictly monotone increasing in 𝑟.

Definition 2.8. For any Lyapunov-like function 𝑉(𝑡,𝑥)∈𝐶[ℝ₊×ℝ^𝑛,ℝ₊], we define the fractional-order Dini derivatives in Caputo’s sense ^𝑐𝐷^𝑞₊𝑉(𝑡,𝑥) and ^𝑐𝐷^𝑞_{_}𝑉(𝑡,𝑥) as follows ^𝑐𝐷^𝑞₊𝑉(𝑡,𝑥)=lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑥)−𝑉(𝑡−ℎ,𝑥−ℎ^𝑞𝑓(𝑡,𝑥))],_𝑐𝐷^𝑞_{_}𝑉(𝑡,𝑥)=lim_ℎ→0−1infℎ^𝑞[𝑉(𝑡,𝑥)−𝑉(𝑡−ℎ,𝑥−ℎ^𝑞𝑓(𝑡,𝑥))] (2.23)(2.24) for (𝑡,𝑥)∈ℝ₊×ℝ^𝑛.

Definition 2.9. For a real-valued function 𝑉(𝑡,𝑥)∈𝐶[ℝ₊×ℝ^𝑛,ℝ₊], we define the generalized fractional-order derivatives (Dini-like derivatives) in Caputo's sense ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥) and ^𝑐_∗𝐷^𝑞_{_}𝑉(𝑡,𝑦−𝑥) as follows ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥)=lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑦−𝑥)−𝑉(𝑡−ℎ,𝑦−𝑥−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))],_𝑐∗𝐷^𝑞_{_}𝑉(𝑡,𝑦−𝑥)=lim_ℎ→0−1infℎ^𝑞[𝑉(𝑡,𝑦−𝑥)−𝑉(𝑡−ℎ,𝑦−𝑥−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))] (2.25)(2.26) for (𝑡,𝑥)∈ℝ₊×ℝ^𝑛.

3. Comparing Fractional Stability with Fractional (ITD) Stability

3.1. Fractional Classical Notion of Stability

Let 𝑥(𝑡,𝑡₀,𝑥₀) and 𝑥(𝑡,𝑡₀,𝑦₀) be any solutions of the initial value problems of fractional-order differential equations with Caputo’s derivative of (2.16) and of (3.1), respectively,

^𝑐𝐷^𝑞𝑥=𝑓(𝑡,𝑥),𝑥(𝑡₀)=𝑦₀for𝑡≥𝑡₀,𝑡₀∈ℝ₊, (3.1) where 𝑓∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛].

Assume that 𝑓(𝑡,0)=0,𝑡∈ℝ₊ so that 𝑥=0 is a null solution of fractional-order differential equation with Caputo’s derivative of (3.1) through (𝑡₀,0). Now, we can state the well-known definitions concerning the stability of the null solution.

Definition 3.1. The null solution 𝑥=0 of fractional-order differential equation with Caputo’s derivative of (3.1) is said to be stable if and only if for each 𝜖>0 and for all 𝑡₀∈ℝ₊, there exists a positive function 𝛿=𝛿(𝜖,𝑡₀) that is continuous in 𝑡₀ for each 𝜖 such that ‖𝑥₀‖≤𝛿implies‖𝑥(𝑡,𝑡₀,𝑥₀)‖<𝜖for𝑡≥𝑡₀. (3.2) If 𝛿 is independent of 𝑡₀, then the null solution 𝑥=0 of initial value problems of fractional-order differential equation with Caputo’s derivative of (3.1) is said to be uniformly stable.

Definition 3.2. The solution 𝑥(𝑡,𝑡₀,𝑦₀) of initial value problems of fractional-order differential equation with Caputo’s derivative of (3.1) through (𝑡₀,𝑦₀) is said to be stable with respect to the solution 𝑥(𝑡,𝑡₀,𝑥₀) of fractional-order differential equation with Caputo’s derivative of (3.1) for 𝑡≥𝑡₀∈ℝ₊ if and only if given any 𝜖>0 there exists a positive function 𝛿=𝛿(𝜖,𝑡₀) that is continuous in 𝑡₀ for each 𝜖 such that ‖𝑦₀−𝑥₀‖≤𝛿implies‖𝑥(𝑡,𝑡₀,𝑦₀)−𝑥(𝑡,𝑡₀,𝑥₀)‖<𝜖for𝑡≥𝑡₀. (3.3) If 𝛿 is independent of 𝑡₀ , then the solution of the fractional-order differential equation with Caputo’s derivative of (3.1) is uniformly stable with respect to the solution 𝑥(𝑡,𝑡₀,𝑥₀) of (2.16).
We remark that for the purpose of studying the classical stability of a given solution 𝑥(𝑡,𝑡₀,𝑦₀) of the initial value problem of fractional-order differential equation with Caputo's derivative of (3.1), it is convenient to make a change of variable. Let 𝑥(𝑡,𝑡₀,𝑥₀) and 𝑥(𝑡,𝑡₀,𝑦₀) be the unique solutions of the fractional-order differential equations with Caputo’s derivative (2.16) and (3.1), respectively, and set 𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)=𝑥(𝑡,𝑡₀,𝑦₀)−𝑥(𝑡,𝑡₀,𝑥₀) for 𝑡≥𝑡₀. Then
^𝑐𝐷^𝑞𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)=^c𝐷^𝑞𝑥(𝑡,𝑡₀,𝑦₀)−^𝑐𝐷^𝑞𝑥(𝑡,𝑡₀,𝑥₀),_𝑐𝐷^𝑞𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)=𝑓(𝑡,𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)+𝑥(𝑡,𝑡₀,𝑥₀))−𝑓(𝑡,𝑥(𝑡,𝑡₀,𝑥₀)),_𝑐𝐷^𝑞𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)=𝑓(𝑡,𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)). (3.4)(3.5) It is easy to observe that 𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)≡0 is a solution of the transformed initial value problems of the fractional differential equation with Caputo’s derivative if 𝑦₀−𝑥₀=0 which implies 𝑓(𝑡,0)=0. Since ^𝑐𝐷^𝑞𝑧(𝑡)=0 and 𝑧(𝑡,𝑡₀,0)=0 is the null solution, the solution of 𝑥(𝑡,𝑡₀,𝑥₀) initial value problems of the fractional-order differential equation with Caputo’s derivative of (2.16) corresponds to the identically null solution of ^𝑐𝐷^𝑞𝑧=𝑓(𝑡,𝑧), where 𝑓(𝑡,𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀))=𝑓(𝑡,𝑧(𝑡,𝑡₀,𝑦₀−𝑥₀)+𝑥(𝑡,𝑡₀,𝑥₀))−𝑓(𝑡,𝑥(𝑡,𝑡₀,𝑥₀)). Hence, we can always assume, without any loss of generality, that 𝑥(𝑡,𝑡₀,𝑥₀)≡0 is the null solution of the given fractional-order differential equation with Caputo’s derivative of (3.1) and we can limit our study of stability to that of the null solution [2, 6, 8]. However, it is impossible to do the same transformation for fractional (ITD) stability which we deal with it.

3.2. New Notion of Fractional (ITD) Stability

Let 𝑥(𝑡,𝜏₀,𝑦₀) be a fractional solution of (2.17) and 𝑥(𝑡,𝜏₀,𝑥₀)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of initial value problems of the fractional-order differential equations with Caputo’s derivative of the system (2.16) for 𝑡≥𝜏₀≥0. Let us make a transformation similar to that in (3.5). Set 𝑧(𝑡,𝜏₀,𝑦₀−𝑥₀)=𝑥(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀) for 𝑡≥𝜏₀. Then

^𝑐𝐷^𝑞𝑧(𝑡,𝜏₀,𝑦₀−𝑥₀)=^𝑐𝐷^𝑞𝑥(𝑡,𝜏₀,𝑦₀)−^𝑐𝐷^𝑞𝑥(𝑡−𝜂,𝑡₀,𝑥₀),_𝑐𝐷^𝑞𝑧(𝑡,𝜏₀,𝑦₀−𝑥₀)=𝑓(𝜂;𝑡,𝑧(𝑡,𝜏₀,𝑦₀−𝑥₀)). (3.6) One can observe that even if 𝑦₀=𝑥₀,𝑧(𝑡,𝜏₀,0) is not zero and is not the null solution of the initial value problems of transformed fractional-order differential equation with Caputo’s derivative and the solution. 𝑥(𝑡−𝜂,𝑡₀,𝑥₀) does not correspond to the identically zero solution of ^𝑐𝐷^𝑞𝑧=𝑓(𝜂;𝑡,𝑧). Therefore, we cannot use stability properties of the fractional-order differential equation with Caputo's derivative of null solution in order to find fractional (ITD) stability properties using this approach.

4. A Fractional Comparison Result

In our earlier work and in the work of others [4–6], the differences between the classical notion of fractional stability and fractional ITD stability did not allow the use of the behavior of the null solution in our fractional ITD stability analysis. The main result presented in this section resolves those difficulties with a new approach that allows the use of the fractional stability of the null solution of the comparison system to predict the Caputo’s fractional stability properties of 𝑦(𝑡,𝜏₀,𝑦₀) the solution of fractional-order differential equation with Caputo’s derivative of (2.18) with respect to 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the fractional-order differential equation with Caputo’s derivative of (2.16).

Let 0<𝑞<1 and 𝑝+𝑞=1. The function space is denoted by 𝐶_𝑝[[𝜏₀,𝑇],ℝ] as follows:

𝐶_𝑝[[𝜏₀,𝑇],ℝ]=[𝑢∈𝐶[(𝜏₀,𝑇],ℝ],(𝑡−𝜏₀)^1−𝑞𝑢(𝑡)∈𝐶[[𝜏₀,𝑇],ℝ]]. (4.1) The Riemann-Liouville fractional derivative is defined by

𝐷^𝑞𝑥(𝑡)=𝑥(𝜏₀)(𝑡−𝜏₀)^−𝑞+Γ(1−𝑞)^𝑡_𝜏0(𝑡−𝑠)^−𝑞𝑑Γ(1−𝑞)𝑑𝑠𝑥(𝑠)𝑑𝑠. (4.2) Now, we will prove the following comparison result.

Theorem 4.1. Assume that 𝑚∈𝐶_𝑝[[𝜏₀,𝑇],ℝ₊] is locally Hölder continuous, 𝑔∈𝐶_𝑝[[𝜏₀,𝑇]×ℝ₊,ℝ] and 𝐷^𝑞𝑚(𝑡)≤𝑔(𝑡,𝑚(𝑡)),𝜏₀≤𝑡≤𝑇. (4.3) Let 𝑟(𝑡) be the maximal solution of the initial value problem of fractional-order scalar differential equation with Riemann-Liouville's derivative 𝐷^𝑞𝑢(𝑡)=𝑔(𝑡,𝑢(𝑡)),𝑢(𝑡)(𝑡−𝜏₀)^1−𝑞∣_𝑡=𝜏0=𝑢⁰≥0 (4.4) existing on [𝜏₀,𝑇] such that 𝑚⁰≤𝑢⁰, where 𝑚⁰=𝑚(𝑡)(𝑡−𝜏₀)^1−𝑞∣_𝑡=𝜏0. Then we have 𝑚(𝑡)≤𝑟(𝑡),𝜏₀≤𝑡≤𝑇. (4.5)

Proof. In view of the definition of the maximal solution 𝑟(𝑡) of the fractional-order differential equation with Riemann-Liouville’s derivative of (4.4), it is enough to prove that 𝑚(𝑡)<𝑢(𝑡,𝜖),𝜏₀≤𝑡≤𝑇, (4.6) where 𝑢(𝑡,𝜖) is any solution of the initial value problem of fractional-order scalar differential equation with Riemann-Liouville's derivative 𝐷^𝑞𝑢=𝑔(𝑡,𝑢)+𝜖withinitialvalue𝑢⁰+𝜖,𝜖>0. (4.7) Now it follows from (4.7) that 𝐷^𝑞𝑢(𝑡,𝜖)>𝑔(𝑡,𝑢(𝑡,𝜖)). (4.8) Then by applying the comparison result [2, Theorem 2.1 in page 23], we get (4.6) and since lim_𝜖→0𝑢(𝑡,𝜖)=𝑟(𝑡) uniformly on each compact set 𝜏₀≤𝑡≤𝑇₀<𝑇. Hence, 𝑚(𝑡)≤lim_𝜖→0𝑢(𝑡,𝜖)=𝑟(𝑡),𝜏₀≤𝑡≤𝑇. (4.9) The proof is complete.

Theorem 4.2. Assume that 𝑓∈𝐶[[𝜏₀,𝑇]×𝑆(𝜌),ℝ^𝑛] and [𝑥,𝑓(𝑡,𝑥)]₊≤𝑔(𝑡,‖𝑥‖],(𝑡,𝑥)∈[𝜏₀,𝑇]×𝑆(𝜌), where [𝑥,𝑓(𝑡,𝑥)]₊=lim_ℎ→0⁺sup(1/ℎ^𝑞)[‖𝑥‖−‖𝑥−ℎ^𝑞𝑓(𝑡,𝑥)‖] and 𝑔∈𝐶[[𝜏₀,𝑇]×ℝ₊,ℝ],𝑔(𝑡,0)≡0. Then the stability properties of the trivial solution of the comparison initial value problem of fractional-order differential equation with Caputo's derivative ^𝑐𝐷^𝑞𝑢=𝑔(𝑡,𝑢(𝑡)),𝑢(𝜏₀)=𝑢₀≥0 (4.10) imply the corresponding stability results of the solution 𝑥(𝑡,𝜏₀,𝑦₀) of the initial value problem of the fractional-order differential equation with Caputo's derivative of (2.17), respectively.

Proof. Let 𝑢≡0 of (4.10) be stable. Then given 0<𝜖<𝜌 and 𝜏₀≥0, there exists for a 𝛿=𝛿(𝜖,𝜏₀)>0 with the property that 0≤𝑢₀<𝛿implies𝑢(𝑡,𝜏₀,𝑢₀)<𝜖,𝑡≥𝜏₀. (4.11) We claim that the trivial solution of (2.17) is stable for these 𝜖 and 𝛿. If this was false, then there would exists a solution 𝑥(𝑡) of (2.17) and 𝑡₁>𝜏₀ such that ‖𝑥(𝑡₁)‖=𝜖,‖𝑥(𝑡)‖≤𝜖for𝜏₀≤𝑡≤𝑡₁. (4.12) For [𝜏₀,𝑡₁], we set 𝑚(𝑡)=‖𝑥(𝑡)‖ and choose ‖𝑦₀‖≤𝑢₀. Then we have 𝑥(𝑡)−𝑆(𝑥,ℎ,𝑟,𝑞)=ℎ^𝑞𝑓(𝑡,𝑥(𝑡))+𝜖(ℎ^𝑞) (4.13) which shows that 𝑆(𝑥,ℎ,𝑟,𝑞)=𝑥(𝑡)−ℎ^𝑞𝑓(𝑡,𝑥(𝑡))−𝜖(ℎ^𝑞) where 𝜖(ℎ^𝑞)/ℎ^𝑞→0 as ℎ→0⁺. Now, by using the fractional-order Dini derivatives in Caputo’s sense ^𝑐𝐷^𝑞₊𝑚(𝑡) in Definition 2.8 as in (2.23), we have ^𝑐𝐷^𝑞₊𝑚(𝑡)=lim_ℎ→0+1supℎ^𝑞[‖𝑥(𝑡)‖−‖𝑥(𝑡)−ℎ^𝑞𝑓(𝑡,𝑥(𝑡))‖]≤𝑔(𝑡,𝑚(𝑡)) (4.14) for 𝜏₀≤𝑡≤𝑡₁ and 𝑚(𝜏₀)=‖𝑦₀‖.
This yields by comparison Theorem 4.1, the estimate
‖𝑥(𝑡,𝜏₀,𝑦₀)‖≤𝑟(𝑡,𝜏₀,‖𝑦₀‖),𝜏₀≤𝑡≤𝑡₁, (4.15) where 𝑟(𝑡,𝜏₀,𝑢₀) is the maximal solution of (4.4). At 𝑡=𝑡₁, we therefore arrive at the following contradiction 𝜖=‖𝑥(𝑡₁,𝜏₀,𝑦₀)‖≤𝑟(𝑡₁,𝜏₀,‖𝑦₀‖)<𝜖. (4.16) Therefore, this do justify our claim.Hence the trivial solution of initial value problem of the fractional-order differential equation with Caputo’s derivative of (2.17) is stable.
Next suppose that 𝑢≡0 is asymptotically stable. Since this implies by definition of the stability of 𝑢≡0, the stability of the trivial solution of (4.10) is the foregoing argument. This means the inequality (4.15) holds for all 𝜏₀≤𝑡 and hence, it is clear, by hypothesis, that if ‖𝑥₀‖<𝛿₀, then lim_𝑡→∞‖𝑥(𝑡,𝜏₀,𝑥₀)‖=0. The proof is therefore complete.

5. An Initial Time Difference Fractional Comparison Result

In this section, we have an other comparison result in which the stability properties of the null solution of the comparison system imply the corresponding initial time difference stability properties of the perturbed fractional-order differential system in Caputo’s sense with respect to the unperturbed fractional-order differential system in Caputo’s sense.

Theorem 5.1. Let 𝑓,𝐹∈𝐶[[𝑡₀,𝑇]×ℝ^𝑛,ℝ^𝑛], and let 𝑥𝐺(𝑡,𝑟)=max_{,𝑦∈𝐵(𝑥0;𝑟)}‖𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)‖, (5.1) where 𝐺(𝑡,𝑟)∈𝐶[ℝ₊×ℝ₊,ℝ₊] and 𝐵 is closed ball with center at 𝑥₀ and radius 𝑟. Assume that 𝑟^∗(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖) is the maximal solution of initial value problem of fractional-order differential equation with Caputo’s derivative ^𝑐𝐷^𝑞𝑢=𝐺(𝑡,𝑢),𝑢(𝜏₀)=‖𝑦₀−𝑥₀‖ through (𝜏₀,‖𝑦₀−𝑥₀‖). 𝑥(𝑡,𝜏₀,𝑥₀)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of fractional-order differential equation of (2.16) for 𝑡≥𝜏₀≥0,𝑡₀∈ℝ₊, and 𝜂=𝜏₀−𝑡₀, and 𝑦(𝑡,𝜏₀,𝑦₀) is the solution of fractional-order differential equation (2.18) with Caputo's derivatives. Then ‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀)‖≤𝑟^∗(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖)for𝑡≥𝜏₀. (5.2)

Proof. Let 𝑚(𝑡)=‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡,𝜏₀,𝑥₀)‖ for 𝑡≥𝜏₀. Then by using the fractional-order Dini derivatives in Caputo's sense ^𝑐𝐷^𝑞₊𝑚(𝑡) in Definition 2.8 as in (2.23), we obtain ^𝑐𝐷^𝑞₊𝑚(𝑡)=^𝑐𝐷^𝑞₊[‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡,𝜏₀,𝑥₀)‖]≤‖𝐹(𝑡,𝑦(𝑡,𝜏₀,𝑦₀))−𝑓(𝑡,𝑥(𝑡,𝜏₀,𝑥₀𝑥))‖≤max_{,𝑦∈𝐵(𝑥0;𝑚(𝑡))}‖𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)‖=𝐺(𝑡,𝑚(𝑡)). (5.3) Therefore, ^𝑐𝐷^𝑞₊𝑚(𝑡)≤𝐺(𝑡,𝑚(𝑡)),𝑚(𝜏₀)=‖𝑦₀−𝑥₀‖ and by using a standard result in [6], we get 𝑚(𝑡)≤𝑟^∗(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖).

Now, we can formulate the comparison results via Lyapunov-like functions.

Theorem 5.2. Let 𝑉(𝑡,𝑧)∈𝐶[ℝ₊×ℝ^𝑛,ℝ₊] and 𝑉(𝑡,𝑧) be locally Lipschitzian in 𝑧. Assume that the generalized fractional-order derivatives (Dini-like derivatives) in Caputo’s sense ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥)=lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑦−𝑥)−𝑉(𝑡−ℎ,𝑦−𝑥−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))] (5.4) satisfies ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥)≤𝐺(𝑡,𝑉(𝑡,𝑦−𝑥)) with (𝑡,𝑥),(𝑡,𝑦)∈ℝ₊×ℝ^𝑛, where 𝐺(𝑡,𝑢)∈𝐶[ℝ₊×ℝ₊,ℝ]. Let 𝑟(𝑡)=𝑟(𝑡,𝜏₀,𝑢₀) be the maximal solution of the fractional-order differential equation ^𝑐𝐷^𝑞𝑢=𝐺(𝑡,𝑢),𝑢(𝜏₀)=𝑢₀≥0, for 𝑡≥𝑡₀. If 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the system (2.16) for 𝑡≥𝜏₀≥0, 𝑡₀∈ℝ₊ and 𝜂=𝜏₀−𝑡₀ and 𝑦(𝑡)=𝑦(𝑡,𝜏₀,𝑦₀) is any solution of (2.18) for 𝑡≥𝜏₀ such that 𝑉(𝜏₀,𝑦₀−𝑥₀)≤𝑢₀, then 𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝑟(𝑡) for 𝑡≥𝜏₀.

Proof. Let 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the system fractional-order differential equation (2.16) for 𝑡≥𝜏₀≥0, 𝑡₀∈ℝ₊, and 𝜂=𝜏₀−𝑡₀ and 𝑦(𝑡)=𝑦(𝑡,𝜏₀,𝑦₀) is any solution of fractional-order differential equation (2.18) for 𝑡≥𝜏₀ such that 𝑉(𝜏₀,𝑦₀−𝑥₀)≤𝑢₀ holds. Define 𝑚(𝑡)=𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡)) for 𝑡≥𝜏₀ so that 𝑚(𝑡₀)≤𝑢₀. Then for small enough ℎ>0, we get 𝑚(𝑡)−𝑚(𝑡−ℎ)=𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,𝑦(𝑡−ℎ)−𝑥(𝑡−ℎ)). (5.5) Since 𝑉 is locally Lipschitzian in 𝑥 and fractional-order Dini derivatives in Caputo’s sense ^𝑐𝐷^𝑞₊𝑚(𝑡)𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,𝑆(𝑦−𝑥,ℎ,𝑟,𝑞))≤𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,(𝑦(𝑡)−𝑥(𝑡))−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))+𝑉(𝑡−ℎ,(𝑦(𝑡)−𝑥(𝑡))−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))−𝑉(𝑡−ℎ,𝑆(𝑦−𝑥,ℎ,𝑟,𝑞)), (5.6) where 𝐿>0 is the Lipschitz constant, 𝑆(𝑦−𝑥,ℎ,𝑟,𝑞)=𝑦−𝑥−ℎ^𝑞[(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥))]−[𝜖₁(ℎ^𝑞)−𝜖₂(ℎ^𝑞))] and 𝜖₁ and 𝜖₂ are error terms ^𝑐𝐷^𝑞₊𝑚(𝑡)≤lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,(𝑦(𝑡)−𝑥(𝑡))−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))]+lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡−ℎ,(𝑦(𝑡)−𝑥(𝑡))−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))−𝑉(𝑡−ℎ,𝑆(𝑦−𝑥,ℎ,𝑟,𝑞))],_𝑐𝐷^𝑞₊𝑚(𝑡)≤lim_ℎ→0+1supℎ^𝑞[𝐿(𝜖₁(ℎ^𝑞)−𝜖₂(ℎ^𝑞))]+lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,(𝑦(𝑡)−𝑥(𝑡)))−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥))], (5.7) where lim_ℎ→0⁺sup𝐿[(𝜖₁(ℎ^𝑞)−𝜖₂(ℎ^𝑞))/ℎ^𝑞]→0. Since ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥) is the generalized fractional-order derivatives (Dini-like derivatives) in Caputo’s sense, we have ^𝑐𝐷^𝑞₊𝑚(𝑡)≤^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝐺(𝑡,𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡)))=𝐺(𝑡,𝑉(𝑡,𝑚(𝑡))),_𝑐𝐷^𝑞₊𝑚(𝑡)≤𝐺(𝑡,𝑉(𝑡,𝑚(𝑡))),𝑚(𝜏₀)=𝑉(𝜏₀,𝑦(𝜏₀)−𝑥(𝜏₀))≤𝑢₀. (5.8)

By using Theorem 5.1, this implies

𝑚(𝑡)=𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝑟(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖), (5.9)

where 𝑟 is the maximal solution of ^𝑐𝐷^𝑞𝑢=𝐺(𝑡,𝑢),𝑢(𝜏₀)=𝑢₀≥0 for 𝑡≥𝑡₀.

Now, we present the main comparison result that yields knowledge of initial time difference fractional stability properties if we know the stability properties of the null solution of the fractional comparison system.

Theorem 5.3. Assume that
(i) let 𝑉(𝑡,𝑧)∈𝐶[ℝ₊×ℝ^𝑛,ℝ₊] be locally Lipschitzian in 𝑧, positive definite and decrescent where the fractional-order Dini derivatives in Caputo’s sense ^𝑐𝐷^𝑞₊𝑚(𝑡)
^𝑐𝐷^𝑞₊𝑚(𝑡)≤lim_ℎ→0+1supℎ^𝑞[𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))−𝑉(𝑡−ℎ,(𝑦−𝑥)−ℎ^𝑞(𝐹(𝑡,𝑦)−𝑓(𝑡,𝑥)))] (5.10) satisfies ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑦−𝑥)≤𝐺(𝑡,𝑉(𝑡,𝑦−𝑥)) for (𝑡,𝑥) and (𝑡,𝑦)∈ℝ₊×ℝ^𝑛, where 𝐺(𝑡,𝑢)∈𝐶[ℝ₊×ℝ₊,ℝ] and the generalized fractional-order (Dini-like) derivatives in Caputo's sense ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑥),𝑎(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝑏(‖𝑥‖),(𝑡,𝑥)∈ℝ₊×ℝ^𝑛,𝑎,𝑏∈𝒦. (5.11)
(ii) Let 𝑟(𝑡)=𝑟(𝑡,𝜏₀,𝑢₀) be the maximal solution of the fractional-order differential equation with Caputo’s derivative
^𝑐𝐷^𝑞𝑢=𝐺(𝑡,𝑢),𝑢(𝜏₀)=𝑢₀≥0for𝑡≥𝜏₀. (5.12) Then the stability properties of the null solution of the fractional-order differential system with Caputo’s derivative (5.12) with 𝐺(𝑡,0)=0 imply the corresponding stability properties of 𝑦(𝑡,𝜏₀,𝑦₀) any solution of fractional-order differential system with Caputo’s derivative (2.18) with respect to 𝑥(𝑡,𝜏₀,𝑥₀)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of fractional-order differential system with Caputo’s derivative of (2.16).

Proof. Let 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the fractional-order differential system with Caputo's derivative of (2.16) for 𝑡≥𝑡₀≥0, 𝑡₀∈ℝ₊, and 𝜂=𝜏₀−𝑡₀ and 𝑦(𝑡)=𝑦(𝑡,𝜏₀,𝑦₀) is any solution of fractional-order differential system with Caputo's derivative of (2.18) for 𝑡≥𝑡₀ such that 𝑉(𝜏₀,𝑦₀−𝑥₀) holds. If we define 𝑚(𝑡)=𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡)) for 𝑡≥𝜏₀ so that 𝑚(𝜏₀)≤𝑢₀, then Theorems 5.1 and 5.2 imply that ^𝑐𝐷^𝑞₊𝑚(𝑡)≤𝐺(𝑉(𝑡,𝑚(𝑡))),𝑚(𝜏₀)=𝑉(𝜏₀,𝑦(𝜏₀)−𝑥(𝜏₀))≤𝑢₀, (5.13) where the fractional-order Dini derivatives in Caputo’s sense ^𝑐𝐷^𝑞₊𝑚(𝑡) and the generalized fractional-order (Dini-like) derivatives in Caputo's sense ^𝑐_∗𝐷^𝑞₊𝑉(𝑡,𝑥) have been used. Thus, 𝑚(𝑡)=𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝑟(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖), where 𝑟 is the maximal solution of the fractional comparison system of (5.12) with Caputo's derivative. Let the null solution of the fractional comparison system of (5.12) with Caputo's derivative be stable. Given any 𝜖>0, since 𝑉 is positive definite and 𝑎∈𝒦 by (i) we have 𝑎(‖𝑦−𝑥‖)≤𝑉(𝑡,𝑦−𝑥)for(𝑡,𝑥),(𝑡,𝑦)∈ℝ₊×ℝ^𝑛,𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝑟(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖)<𝑎(𝜖)for𝑡≥𝜏₀providedthat‖𝑦₀−𝑥₀‖<𝛿(𝜖,𝜏₀). (5.14) Hence 𝑉(𝑡,𝑦(𝑡)−𝑥(𝑡))≤𝑟(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖)<𝑎(𝜖). (5.15) Since 𝑎⁻¹ exists, we have ‖𝑦(𝑡)−𝑥(𝑡)‖<𝜖 and ‖𝑦₀−𝑥₀‖<𝛿(𝜖,𝜏₀). Therefore, 𝑦(𝑡,𝜏₀,𝑦₀) the solution of fractional-order differential equation of (2.18) with Caputo's derivative is stable with respect to 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the fractional-order differential equation of (2.16) with Caputo's derivative. If the null solution of the fractional comparison system of (5.12) with Caputo's derivative is asymptotically stable, then 𝑎(‖𝑦(𝑡)−𝑥(𝑡)‖)≤𝑟(𝑡,𝜏₀,‖𝑦₀−𝑥₀‖)holdsfor𝑡≥𝜏₀,‖𝑦₀−𝑥₀‖smallenough. (5.16) That implies lim_𝑡→∞‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀)‖=0 (5.17) since 𝑎∈𝒦. Hence, 𝑦(𝑡,𝜏₀,𝑦₀) the solution of fractional differential equation (2.18) is asymptotically stable with respect to 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the system (2.16). Since 𝑉 is decrescent and 𝑏∈𝒦 by (i) we have 𝑉(𝑡,𝑦−𝑥)≤𝑏(‖(𝑦−𝑥)‖)for(𝑡,𝑥),(𝑡,𝑦)∈ℝ₊×ℝ^𝑛 (5.18) and the choice of 𝛿=𝛿(𝜖) is independent of 𝜏₀.
Thus, uniform stability and uniform asymptotic stability of the fractional comparison system of (5.12) with Caputo's derivative imply the corresponding uniform stability and uniform asymptotic stability of the fractional solution of (2.18) in Caputo’s sense with respect to 𝑥(𝑡)=𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the fractional system of (2.16) in Caputo's sense. Hence, 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the fractional system of (2.16) in Caputo’s sense that is uniformly stable and uniformly asymptotically stable.

Acknowledgments

This work has been supported by The Scientific and Technological Research Council of Turkey and Department of Mathematics Yeditepe University.