Some stability properties related to initial time difference for Caputo fractional differential equations

Definition 1

([15]) We say the function m(t) ∈ C^q([t₀, T], ℝⁿ) if it is differentiable and the Caputo derivative t0CDqm(t) exists and satisfies (2.1) for t ∈ [t₀, T].

Remark 2.1

If m(t) ∈ C^q([t₀, T], ℝⁿ), then t0CD+qm(t)=t0CDqm(t).

Lemma 3.1

([7]) Let the function x(t) ∈ C^q(ℝ₊, ℝⁿ), a ≥ 0, be a solution of the initial value problem for FrDE

Then the functionx∼ (t) = x(t+η) satisfies the initial value problem for the FrDE

where b ≥ 0, η = a − b.

The relation between (3.1) and (3.2) is given by the following result:

Corollary 3.1

([7]) For any solution x(t) = x(t;τ₀, y₀) of (3.2) the functionx∼(t) = x(t+η) is a solution of IVP for FrDE

where η = τ₀ − t₀.

Remark 3.1

In the autonomous case, i.e. f(t, x) ≡ F(x), from Corollary 1 it follows that we can study only the case of zero initial time and zero lower bound of the fractional derivative. At the same time changing the initial time of the IVP for nonautonomous case leads to a change of the lower limit of the fractional derivative and to a change of the equation.

The main goal is to compare the behavior of two solutions of Caputo fractional differential equations with different initial data, both initial time τ₀ ≠ t₀ and initial points y₀ ≠ x₀. In real life situations it may not be possible to keep measurements with the expected initial time. So, when we study the influence of parameters, sometimes we need to consider two solutions which have not only different initial points, but also different initial times. The stability with respect to initial time difference (ITD) gives us an opportunity to compare solutions of FrDE when both initial time and position are different. We will study the Lipschitz stability with ITD and its connection with Mittag-Leffler stability with ITD of the system of Caputo fractional differential equations.

Definition 2

The given solution x^∗(t) = x(t;t₀, x₀) of FrDE (3.1) is called:

1. Lipschitz stable with initial time difference (ITD), if there exist M ≥ 1, δ = δ (t₀) > 0 and σ = σ (t₀) > 0 such that for any initial value y₀ ∈ ℝⁿ: ||y₀ − x₀|| < δ and any initial time τ₀ ∈ ℝ₊: |τ₀ − t₀| < σ the inequality ||y(t+η; τ₀, y₀) − x^∗(t)|| ≤ M||y₀ − x₀|| for t ≥ t₀ holds, where η = τ₀ − t₀ and y(t;τ₀, y₀) is a solution of (3.2).

2. eventually Lipschitz stable with ITD, if there exist M ≥ 1, δ = δ (t₀) > 0, T = T(t₀) > 0 and σ = σ (t₀) > 0 such that for any initial value y₀ ∈ ℝⁿ: ||y₀ − x₀|| < δ and any initial time τ₀ ∈ ℝ₊: |τ₀ − t₀| < σ the inequality ||y(t+η ;τ₀, y₀) − x^∗(t)|| ≤ M||y₀ − x₀|| for t ≥ t₀ + T holds, where η = τ₀ − t₀ and y(t;τ₀, y₀) is a solution of (3.2)

Definition 3

The system of FrDE (3.1) is called:

1. uniformly Lipschitz stable with ITD, if there exist constants M ≥ 1 and δ, σ > 0 such that for any initial values x₀, y₀ ∈ ℝⁿ and any initial times t₀, τ₀ ∈ ℝ₊ the inequalities || y₀ − x₀|| < δ and |τ₀ − t₀| < σ imply ||y(t+η ;τ₀, y₀) − x(t;t₀, x₀)|| ≤ M ||y₀ − x₀|| for t ≥ t₀ where η = τ₀ − t₀ and x(t;t₀, x₀), y(t;τ₀, y₀) are solution of (3.1), (3.2), respectively;

2. globally uniformly Lipschitz stable with ITD, if there exist constants M ≥ 1, σ > 0 such that for any initial values x₀, y₀ ∈ ℝⁿ and any initial times t₀, τ₀ ∈ ℝ₊ the inequalities ||y₀ − x₀|| < ∞ and |τ₀ − t₀| < σ imply ||y(t+η ;τ₀, y₀) − x(t;t₀, x₀)|| ≤ M|| y₀ − x₀|| for t ≥ t₀ where η = τ₀ − t₀ and x(t;t₀, x₀), y(t;τ₀, y₀) are solution of (3.1), (3.2), respectively.

Definition 4

The solution x^∗(t) = x(t;t₀, x₀) of FrDE (3.1) is called Mittag-Leffler stable with ITD, if there exist λ > 0, C > 0, σ = σ (t₀) > 0 and constants a, b > 0, such that for any initial time τ₀ ∈ ℝ₊: |τ₀ − t₀| < σ the inequality || y(t+η ;τ₀, y₀) − x^∗(t)|| ≤ C||y₀ − x₀||^a{E_q(−λ(t − t₀)^q)}^b for t ≥ t₀ holds, where η = τ₀ − t₀ and y(t;τ₀, y₀) is a solution of (3.2).

Note the concept of any type of stability with ITD is meaningful only in the case of non-autonomous systems (see Remark 3.1).

Remark 3.2

The concept of Lipschitz stability with ITD defined in Definition 2 generalizes Lipschitz stability for the zero solution of fractional equations in the literature [22] if x^∗(t) ≡ 0 and τ₀ = t₀.

Remark 3.3

In the case when x^∗(t) ≡ 0 and τ₀ = t₀ the concept of Mittag-Leffler stability with ITD defined in Definition 4 generalizes the definition of Mittag-Leffler stability in [20].

Remark 3.4

In the special case of C ≥ 1 and a = 1 the Mittag-Leffler stability with ITD (Definition 4) implies the Lipschitz stability with ITD of x^∗(t) = x(t;t₀, x₀) of FrDE (3.1) (Definition 2).

Let J ⊂ ℝ₊, λ > 0. In our further consideration we will use the following sets:

Remark 3.5

The function a(u) = K₁u, K₁ ∈ (0, 1] is from the class 𝓚(ℝ₊) with P_a(u) ≡ u and from the class 𝓜(ℝ₊). The function b(u) = K₂u², K₂ > 0 is from the class 𝓜([0, 1]).

We will use comparison results for the IVP for the scalar fractional differential equation with a parameter of the type

where u, u₀ ∈ ℝ, g: ℝ₊ × ℝ × B_H → ℝ, g(t, 0, 0) ≡ 0, η ∈ B_H is a parameter and H > 0 is a given number. We denote the solution of the IVP for the scalar FrDE (3.6) by u(t;t₀, u₀, η) ∈ C^q([t₀, ∞), ℝ). In the case of non-uniqueness of the solution we will assume the existence of a maximal one.

Definition 5

The zero solution of the scalar FrDE (3.6) with a parameter η is said to be

1. Lipschitz stable with respect to a parameter, if for any t₀ ∈ ℝ₊ there exists M ≥ 1, δ = δ (t₀) > 0 and σ = σ (t₀) > 0 such that for any u₀ ∈ ℝ : |u₀| < δ and any |η| ≤ σ the inequality |u(t)| ≤ M|u₀| for t ≥ t₀, where u(t) = u(t;t₀, u₀, η) is a solution of (3.6);

2. eventually Lipschitz stable with respect to a parameter, if for any t₀ ∈ ℝ₊ there exists M ≥ 1, δ = δ (t₀) > 0, T = T(t₀) > 0 and σ = σ (t₀) > 0 such that for any u₀ ∈ ℝ : |u₀| < δ and any |η| ≤ σ the inequality |u(t)| ≤ M|u₀| for t ≥ t₀+T, where u(t) = u(t;t₀, u₀, η) is a solution of (3.6).

Definition 6

The scalar FrDE (3.6) with a parameter η is said to be

1. uniformly Lipschitz stable w.r.t. a parameter, if there exist constants M ≥ 1 and δ, σ > 0 such that for any t₀ ∈ ℝ₊ and for any |η| ≤ σ the inequality | u₀| < δ implies |u(t)| ≤ M|u₀| for t ≥ t₀;

2. globally uniformly Lipschitz stable w.r.t. a parameter, if there exist constants M ≥ 1, σ > 0 such that for any t₀ ∈ ℝ₊ and for any |η| ≤ σ the inequality | u₀| < ∞ implies |u(t)| ≤ M|u₀| for t ≥ t₀.

Remark 3.6

Note that similar to Definition 2 and Definition 6, respectively, we can define eventually uniform Lipschitz stability with ITD, eventually globally uniformly Lipschitz stability with ITD for (3.1) and eventually uniform Lipschitz stable w.r.t. a parameter and eventually globally uniform Lipschitz stability w.r.t. a parameter for (3.6).

Definition 7

The FrDE (3.6) is called Mittag-Leffler stable with respect to a parameter, if there exist constants λ, C, H, a, b > 0, such that for any η ∈ B_H the inequality || u(t;t₀, u₀, η)|| ≤ C|u₀|^a{E_q(−λ(t − t₀)^q)}^b for t ≥ t₀ holds, where u(t;t₀, u₀, η) is a solution of (3.6).

Example 1

Consider the IVP forthe scalar FrDE with a parameter

where c > 0 and α > 0 are constants, η ∈ ℝ is a parameter, t₀ ∈ ℝ₊ is an arbitrary number.

The above IVP is a special case of (3.6) with g(t, u, η) = (−α + cη)u, g(t, 0, 0) ≡ 0 and it has a unique solution for any η ∈ ℝ defined by

Consider the positive constants H<αc and λ = α − cH. Then for η ∈ B_H we have −α+cη ≤ −α+ cH = −λ and the following estimate is true

i.e. the scalar FrDE (3.6) is Mittag-Leffler stable with respect to a parameter with a = b = C = 1. □

We introduce the class Λ of Lyapunov-like functions which will be used to investigate the stability properties with ITD for the system FrDE (3.1).

Definition 8

Let I ⊂ ℝ₊ and Δ ⊂ ℝⁿ. We will say that the function V(t, x): I × Δ → ℝ₊ belongs to the class Λ (I, Δ) if V(t, x) is continuous and locally Lipschitzian with respect to its second argument in I × Δ.

The application of Lyapunov functions for stability analysis with ITD requires an appropriate definition of the derivative of Lyapunov like function along the given nonlinear Caputo fractional differential equation. In [5, 6, 7], based on Eq. (2.4), we introduced the generalized Caputo fractional Dini derivative w.r.t. ITD of the function V(t, x) ∈ Λ ([t₀, ∞), ℝⁿ) along the system of FrDE (3.1) for t > t₀, η ∈ ℝ: t + η ≥ 0 and x, y, x₀, y₀ ∈ ℝⁿ by the equality

The generalized Caputo fractional Dini derivative w.r.t. ITD (3.7) was applied to study stability ([7]), practical stability ([5]), strict stability ([6]).

Example 2

We give some examples of Lyapunov functions and their generalized Caputo fractional Dini derivative w.r.t. ITD.

1. Lyapunov functions which do not depend on the time variable, i.e. V(t, x) ≡ V(x) for x ∈ ℝ. Then for any x, y, x₀, y₀ ∈ ℝ, t > t₀, and η ∈ ℝ: t+η ≥ 0 the generalized Caputo fractional Dini derivative w.r.t. ITD is

   t0CD(3.1)qV(t,x,y,η,x0,y0)=limh→0+sup1hq(V(y−x)−V(y−x−hq(f(t+η,y)−f(t,x)))hq)+(V(y−x)−V(y0−x0))(t−t0)−qΓ(1−q).(3.8)

   Special case. Let V(x) = x². Then

   t0CD(3.1)qV(t,x,y,η,x0,y0)=2(y−x)(f(t+η,y)−f(t,x))+((y−x)2−(y0−x0)2)(t−t0)−qΓ(1−q).(3.9)

2. Let V(t, x) ≡ V(x) for x ∈ ℝ. Let x(t) ∈ C^q([t₀, ∞), ℝ) and y(t) ∈ C^q([τ₀, ∞), ℝ) be solutions of (3.1) and (3.2), respectively. Then for t > t₀ and η = τ₀ − t₀ we have t+η = t+τ₀ − t₀ > τ₀ ≥ 0 and the generalized Caputo fractional Dini derivative w.r.t. ITD is

   t0CD(3.1)qV(t,x(t),y(t+η),η,x0,y0)=limh→0+sup1hq(V(y(t+η)−x(t))−V(y(t+η)−x(t)−hq(f(t+η,y(t+η))−f(t,x(t)))))+(V(y(t+η)−x(t))−V(y0−x0))(t−t0)−qΓ(1−q).(3.10)

3. Let V(t, x) = m²(t)x² for x ∈ ℝ where m ∈ C¹(ℝ₊, ℝ). Then for any x, y, x₀, y₀ ∈ ℝ, t > t₀, and η ∈ ℝ: t+η ≥ 0 the generalized Caputo fractional Dini derivative w.r.t. ITD is

   t0CD(3.1)qV(t,x,y,η,x0,y0)=2(y−x)m2(t)(f(t+η,y)−f(t,x))+(y−x)2(t0GLD+qm2(t))−(y0−x0)2m2(t0)(t−t0)−qΓ(1−q)=2(y−x)m2(t)(f(t+η,y)−f(t,x))+(y−x)2(t0RLDtqm2(t))−(y0−x0)2m2(t0)(t−t0)−qΓ(1−q).(3.11)

   Special case. Let m(t)=tq2,q∈(0,1),t0=0. Applying t0RLDtqtk=Γ(k+1)Γ(k−q+1)tk−q,k>−1,t>0 we get

   t0CD(3.1)qV(t,x,y,η,x0,y0)=2(y−x)tq(f(t+η,y)−f(t,x))+(y−x)2Γ(1+q).

4. Let V(t, x) = m²(t)x² for x ∈ ℝ where m ∈ C¹(ℝ₊, ℝ). Let x(t) ∈ C^q([t₀, ∞), ℝ) and y(t) ∈ C^q([τ₀, ∞), ℝ) be solutions of (3.1) and (3.2), respectively. Then for t > t₀ and η = τ₀ − t₀ we have t+η = t+τ₀ − t₀ > τ₀ ≥ 0 and the generalized Caputo fractional Dini derivative w.r.t. ITD is

   t0CD(3.1)qV(t,x(t),y(t+η),η,x0,y0)=2(y(t+η)−x(t))m2(t)(f(t+η,y(t+η))−f(t,x(t)))+(y(t+η)−x(t))2(t0RLDtqm2(t))−(y0−x0)2m2(t0)(t−t0)−qΓ(1−q).(3.12)

5. Let V(t,x1,x2)=m12(t)x12+m22(t)x22 for x₁, x₂ ∈ ℝ where m₁, m₂ ∈ C¹(ℝ₊, ℝ). Then for any x, y, x₀, y₀ ∈ ℝ², t > t₀, and η ∈ ℝ: t+η ≥ 0 the generalized Caputo fractional Dini derivative w.r.t. ITD is

   t0CD(3.1)qV(t,x,y,η,x0,y0)=2(y1−x1)m12(t)(f1(t+η,y1,y2)−f1(t,x1,x2))+2(y2−x2)m22(t)(f2(t+η,y1,y2)−f2(t,x1,x2))+(y1−x1)2(t0RLDtqm12(t))+(y2−x2)2(t0RLDtqm22(t))−(m12(t0)(y1(0)−x1(0))2+m22(t0)(y2(0)−x2(0))2)(t−t0)−qΓ(1−q).(3.13)

Remark 3.7

Note in some papers (see, for example [17, 18]) the derivative of Lyapunov functions with respect to system (3.1) is defined by

This derivative is called a fractional derivative of Lyapunov functions in Caputo’s sense of order q with respect to system (3.1). This operator has no memory, which is different than the fractional derivative and it is independent on the initial time and it is not equivalent to the Caputo fractional derivative. In the general case if x(t) is a solution of (3.1) then the inequality

holds, where the operator ^cD^q is defined by (3.14) and the operator  t0cDtq is defined by (2.4).

Lemma 4.1

([7]) Assume the following conditions are satisfied:

1. The functions x^∗(t) = x(t;t₀, x₀) and y(t) = y(t;τ₀, y₀) are solutions of systems of FrDE (3.1) and (3.2) respectively, x^∗(t) ∈ C^q([t₀, t₀+θ], ℝⁿ), y(t) ∈ C^q([τ₀, τ₀+θ], ℝⁿ) and y(t+η^∗) − x^∗(t) ∈ Δ for [t₀, t₀+θ] where t₀, τ₀ ∈ ℝ₊: η^∗ = τ₀ − t₀, Δ ⊂ ℝⁿ, θ > 0 is a given number.

2. The function G ∈ C[[t₀, t₀+θ] × ℝ, ℝ] be such that for any ϵ ∈ [0, H] and v₀ ∈ ℝ the scalar FrDE

   t0CDqu=G(t,u)+ϵfort>0,u(t0)=v0(4.1)

   has a solution u(t;t₀, v₀, ϵ) ∈ C^q([t₀, t₀+θ], ℝ) where H, Θ > 0 are given constants.

3. The function V ∈ Λ ([t₀, t₀+θ], Δ) and for t ∈ (t₀, t₀+θ] the inequality

   t0CD(3.1)qV(t,x∗(t),y(t+η∗),η∗,x0,y0)≤G(t,V(t,y(t+η∗)−x∗(t)))

   holds.

Then V(t₀, y₀ − x₀) ≤ u₀implies V(t, y(t+η^∗) − x^∗(t)) ≤ u^∗(t) for t ∈ [t₀, t₀+θ] where u^∗(t) = u(t;t₀, u₀, 0) is the maximal solution of IVP for the scalar FrDE (4.1) with v₀ = u₀and ϵ =0.

Corollary 4.1

([7]) Let the conditions of Lemma 4.1 be satisfied for θ = ∞.

Then V(t₀, y₀ − x₀) ≤ u₀implies V(t, y(t+η^∗) − x^∗(t)) ≤ u^∗(t) for t ≥ t₀where u^∗(t) = u(t;t₀, u₀, 0) is the maximal solution of IVP for scalar FrDE (2.4) with v₀ = u₀and ϵ = 0.

Corollary 4.2

Let condition 1 of Lemma 4.1 be satisfied and the inequality

holds where C, γ ∈ ℝ are constants.

ThenV(t,y(t+η∗)−x∗(t))≤[V(t0,y0−x0+1γCη∗]Eq(γ(t−t0)q)−1γCη∗for t ≥ t₀.

Proof

Consider the IVP for the scalar FrDE t0CDqu = γ u+Cη^∗, u(t₀) = V(t₀, y₀ − x₀). Denote v = u + 1γCη^∗. Then t0CDqv = γ v, v(t₀) = V(t₀, y₀ − x₀)+ 1γCη^∗. Then v(t) = [V(t₀, y₀ − x₀)+ 1γCη^∗]E_q(γ(t − t₀)^q) for t ≥ t₀. Therefore, u(t) = [V(t₀, y₀ − x₀)+ 1γCη^∗]E_q(γ (t − t₀)^q) − 1γCη^∗ for t ≥ t₀. Applying Corollary 4.1 we obtain the claim in Corollary 4.2. □

We will obtain sufficient conditions for several types of stability such as Lipschitz (uniform Lipschitz) stability, globally uniformly Lipschitz stability, Mittag-Leffler stability with (ITD) by using continuous Lyapunov-like functions from the Λ class and the generalized Caputo fractional Dini derivative defined by (3.7). Lipshitz stability for fractional equations is studied in [22] using the derivative defined by (3.14) derivative and applying (3.15) as an equality instead of an inequality (see Remark 3.7).

Theorem 4.1

Let the following conditions be satisfied:

1. The function x^∗(t) = x(t;t₀, x₀) ∈ C^q([t₀, ∞), ℝⁿ) is a solution of system of FrDE (3.1), where t₀ ∈ ℝ₊, x₀ ∈ ℝⁿare given points.

2. The function g ∈ C[[t₀, ∞) × ℝ × B_H, ℝ], g(t, 0, 0) ≡ 0 and for any parameter η ∈ B_H the IVP for the scalar FrDE (10) has a solution u(t;t₀, u₀, η) ∈ C^q([t₀, ∞), ℝ) where H > 0 is a given number.

3. The zero solution of the scalar FrDE (3.6) is Lipschitz stable w.r.t. a parameter.

4. There exists a function V ∈ Λ ([t₀, ∞), ℝⁿ) with Lipschitz constant L in S_ρ such that V(t₀, 0) = 0 and

   1. b(∥x∥) ≤ V(t, x) for (t, x) ∈ ℝ₊ × ℝⁿ, where ρ > 0 is a given number, b ∈ 𝓚(ℝ₊);

   2. for any y, y₀ ∈ ℝⁿand η ∈ B_H the inequality

   t0CD(3.1)qV(t,x∗(t),y,η,x0,y0)≤g(t,V(t,y−x∗(t)),η)fort>t0(4.2)

   holds.

   Then the solution x^∗(t) of the system of FrDE (3.1) is Lipschitz stable with ITD.

Proof

From condition 3 it follows that there exist M ≥ 1, δ₁ = δ₁(t₀), σ = σ (t₀) such that for any u₀ ∈ ℝ : |u₀| < δ₁ and | η | < σ the inequality

holds, where u(t; t₀, u₀, η) is a solution of FrDE (3.6). Without loss of generality we assume σ ≤ H.

Since V(t₀,0) = 0 there exists a δ₂ = δ₂(t₀, δ₁) < ρ such that

Without loss of generalization we can assume δ₂ ≤ δ₁. The function V(t, x) is Lipschitz on S_ρ and

From b ∈ 𝓚(ℝ₊) it follows there exists a function P_b ∈ C(ℝ₊,ℝ₊) such that

Choose M₁ ≥ 1 such that M₁ >ML and let M₂ = P_b(M₁) ≥ 1.

Now let y₀ ∈ ℝⁿ and τ₀ ∈ ℝ₊ be such that ∥y₀ − x₀∥ < δ₂ and | η^∗| < σ where η^∗ = τ₀-t₀. Consider a solution y(t) = y(t; τ₀, y₀) of system of FrDE (3.2) with the chosen initial data (τ₀, y₀). Let u0∗ = V(t₀, y₀ − x₀). Then from the choice of y₀ and Eq. (4.4) it follows that u0∗ = V(t₀, y₀ − x₀) < δ₁. Therefore, applying (4.3) and (4.5) we obtain the inequality

holds where u^∗(t) = u(t; t₀, u0∗, η^∗) ∈ C^q([t₀, ∞), ℝ) is a solution of FrDE (3.6).

Using condition 4 (ii) and applying Lemma 2 with G(t, u) = G(t, u, η^∗), Δ = ℝⁿ, θ = ∞ we get

From inequalities (4.7), (4.8), condition 4 (i) and the Lipschitz property of V(t, x) we get

From the monotonicity property of the function b(r) and inequalities (4.6), (4.9) we have

 □

Corollary 4.3

Let the conditions ofTheorem 1 be satisfied withb(u) = K₁u, K₁ > 0.

Then the solutionx^∗(t) of the system of FrDE (3.1) is Lipschitz stable with ITD.

Proof

The proof is similar to the one in Theorem 1 with M₁ ≥ 1 : M₁ > MLK1 and M₂ = M₁.

Theorem 4.2

Let the conditions1, 2, 4 (i) be satisfied, the zero solution of the scalar FrDE (3.6) is eventually Lipschitz stable w.r.t. a parameter and there existsT = T(t₀) > such that (4.2) is satisfied for t ≥ t₀ + T.

Then the solutionx^∗(t) of the system of FrDE (3.1) is eventually Lipschitz stable with ITD.

Theorem 4.3

Let the following conditions be satisfied:

1. The function g ∈ C[[t₀, ∞) × ℝ × B_H, ℝ], g(t, 0, 0) ≡ 0 and for any parameterη ∈ B_Hthe IVP for the scalar FrDE (3.6) has a solutionu(t; t₀, u₀, η) ∈ C^q([t₀, ∞),ℝ) where H > 0 is a given number.

2. There exists a functionV ∈ Λ (ℝ₊, S(λ)) such that

   1. b(∥x∥) ≤ V(t, x) ≤ a(∥x∥) for (t, x) ∈ ℝ₊ × S(λ),

      whereb ∈ 𝓚([0, λ]), a ∈ 𝓜([0, λ]), λ > 0 is a given number;

   2. for anyt₀ ∈ ℝ₊, x, y, x₀, y₀ ∈ ℝⁿ : y − x ∈ S(λ), y₀ − x₀ ∈ S(λ) andη ∈ B_Hthe inequality

      t0CD(3.1)qV(t,x,y,η,x0,y0)≤g(t,V(t,y−x),η)for t≥t0

      holds.

3. The scalar FrDE (3.6) is uniformly Lipschitz stablew.r.t. a parameter (globally uniformly Lipschitz stablew.r.t. a parameter).

Then the system of FrDE (3.1) is uniformly Lipschitz stable (globally uniformly Lipschitz stable) with ITD.

Proof

Let the scalar FrDE (3.6) be uniformly Lipschitz stable w.r.t. a parameter. According to Definition 6 there exist constants M ≥ 1, δ₁ > 0, σ > 0 such that for any t₀ ∈ ℝ₊ and any | η | < σ the inequality |u₀| < δ₁ implies

where u(t; t₀, u₀, η) is a solution of FrDE (10) with initial data (t₀, U₀). From condition 2 (i) there exist a function P_b ∈ C(ℝ₊, ℝ₊) and a positive constant K_a such that b⁻¹(αr) ≤ αP_b(r) and a(r) ≤ K_ar for r ≥ 0, ]ga ≥ 1. Choose M₁ ≥ 1 such that M₁ > P_b(M)K_a. Now let δ = minδ1,λM1,δ1Ka and initial points x₀, y₀ ∈ ℝⁿ and τ₀, t₀ ∈ ℝ₊ be such that

where η^∗ = τ₀ − t₀. Consider any solutions x(t) = x(t; t₀, x₀) and y(t) = y(t; τ₀, y₀) of system of FrDE (3.1) and (3.2) correspondingly with the chosen initial data (τ₀, y₀) and (t₀, x₀) respectively. From the choices of the constants (4.11) we have ∥y₀ − x₀∥ < δ ≤ λM1 ≤ λ, i.e. y₀ − x₀ ∈ S(λ).

Let u0∗ = V(t₀, y₀ − x₀). Then from Condition 2 (i) and the choice of x₀, y₀ it follows that u0∗ = V(t₀, y₀ − x₀) ≤ a(∥y₀ − x₀∥) ≤ K_a∥y₀ − x₀∥ < K_aδ ≤ δ₁. Then according to inequality (4.10) it follows that

where u^∗(t) = u(t; t₀, u0∗, η^∗) ∈ C^q([t₀, ∞), ℝ) is a solution of FrDE (3.6). We will prove that if inequalities (4.11) holds then

Assume the opposite, i.e. there exists t₁ > t₀ such that

where ϵ > 0 is a small enough number. Then for t ∈ t₀, t₁] the inequality ∥y(t + η^∗) − x^∗(t) ∥ ≤ M₁∥y₀ − x₀∥ < M₁δ ≤ λ holds, i.e. y(t + η^∗) − x(t) ∈ S(λ) for t₀ ≤ t ≤ t₁. Then from Lemma 4.1 applied for θ = t₁ − t₀, Δ = S(λ) and G(t, u) = G(t, u, η^∗) we get

From the choice of t^∗, Condition 2 (i) and inequalities (4.12), (4.14) we obtain

The obtained contradiction proves the validity of (4.13). Therefore, according to Definition 3 the system of FrDE (3.1) is uniformly Lipschitz stable with ITD.

The proof of globally uniformly Lipschitz stable with ITD is analogous, so we omit it. □

Theorem 4.4

Let the following conditions be satisfied:

1. The condition 1 ofTheorem 4.3is satisfied.

2. There exists a function V ∈ Λ (ℝ₊, S(λ)) such that

   1. λ₁(t)∥x∥² ≤ V(t, x) ≤ λ₂(t)∥x∥²for (t, x) ∈ ℝ₊ × S(λ), whereλ₁(t) ≥ A₁ > 1, λ₂(t) ≤ A₂fort ≥ 0, whereA₁, A₂ > 1 are given constants;

   2. for anyt₀ ∈ ℝ₊, x, y, x₀, y₀ ∈ ℝⁿ : y − x ∈ S(λ), y₀ − x₀ ∈ S(λ) andη ∈ B_Hthe inequality

      t0CD(3.1)qV(t,x,y,η,x0,y0)≤(−α+Cη)V(t,y−x)fort>t0(4.16)

      holds, whereλ, H, C, αare given positive numbers.

Then the system of FrDE (3.1) is uniformly globally Lipschitz stable with ITD.

Proof

Let g(t, u, η) = (−α + Cη)u. Let σ = αC and choose the initial times t₀, τ₀ ∈ ℝ₊ : | η^∗| < σ where η^∗ = τ₀ − t₀. Let the initial points x₀, y₀ ∈ ℝⁿ : ||x₀ − y₀|| < ∞ and u₀ = V(t₀, y₀ − x₀). According to Example 1 the solution of the comparison scalar FrDE (3.6) is u(t; t₀, u₀, η) = u₀E_q((−α + Cη)(t − t₀)^q). Therefore, |u(t; t₀, u₀, η)| = |u₀|E_q((−α +Cη)(t − t₀)^q) ≤ |u₀|, i.e. (3.6) is globally Lippshitz stable w.r.t. a parameter with M = 1. Also, Condition 2i of Theorem T3 is satisfied with b(r) = A₁r ∈ 𝓚([0, λ]), P_b(r) = rA1 and a(r) = A₂r ∈ 𝓜([0, λ]). According to Theorem 3 the system of FrDE (3.1) is uniformly globally Lipschitz stable with ITD. □

Corollary 4.4

In the case whenη = 0, i.e. the inequality

holds, the result of Theorem> 4.4 is reduced to the uniformly globally Lipschitz stability of the system of FrDE (3.1).

Theorem 4.5

Let the following conditions be satisfied:

1. The functionx^∗(t) = x(t; t₀, x₀) ∈ C^q([t₀, ∞), ℝⁿ) is a solution of system of FrDE (3.1), wheret₀ ∈ ℝ₊, x₀ ∈ ℝⁿare given points.

2. The function g ∈ C[[t₀, ∞) × ℝ × B_H, ℝ], g(t, 0, 0) ≡ 0 and for any parameterη ∈ B_Hthe IVP for the scalar FrDE (3.6) has a solutionu(t; t₀, u₀, η) ∈ C^q([t₀, ∞), ℝ) where H > 0 is a given number.

3. There exists a function V ∈ Λ ([t₀, ∞), ℝⁿ) such thatV(t₀, 0) = 0 and

   1. K₁∥x∥^a ≤ V(t, x) ≤ K₂∥ x∥ ^abfor (t, x) ∈ ℝ₊ ×ℝⁿ, whereK₁, K₂, aandbare positive constants;

   2. for any y, y₀ ∈ ℝⁿandη ∈ B_Hthe inequality

       t0CD(3.1)qV(t,x∗(t),y,η,x0,y0)≤(−α+Cη)V(t,y−x∗(t)) for t>t0(4.17)

      holds.