Publicly Available Published by De Gruyter March 13, 2018

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

• Ravi Agarwal , Snezhana Hristova and Donal O’Regan

## Abstract

Lipschitz stability and Mittag-Leffler stability with initial time difference for nonlinear nonautonomous Caputo fractional differential equation are defined and studied using Lyapunov like functions. Some sufficient conditions are obtained. The fractional order extension of comparison principles via scalar fractional differential equations with a parameter is employed. The relation between both types of stability is discussed theoretically and it is illustrated with examples.

## 1 Introduction

Fractional calculus is the theory of integrals and derivatives of arbitrary non-integer order. The subject is as old as classical calculus and goes back to the 17-th century. It was realized that various processes with anomalous dynamics in science and engineering can be formulated mathematically using fractional differential operators because of its memory and hereditary properties [14], [21]. The qualitative theory of fractional differential equations (FrDE) has received a lot of attention. One of the main problems in the qualitative theory of differential equations is stability of the solutions. Some stability concepts were presented and studied by applying various methods such as the first and second method of Lyapunov ([1], [3], [16], [23]). One type of stability, useful in real world problems, is the so called Lipschitz stability. In 1986, F.M. Dannan, S. Elaydi ([13]) introduced the concept of Lipschitz stability for nonlinear ordinary differential equations. They mention that uniform Lipschitz stability lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation and uniform stability in variation on the other side. Furthermore, uniform Lipschitz stability neither implies asymptotic stability nor is it implied by it (see also [11]).

Recently, fractional calculus was used for the stability analysis of FrDE. However, there are several difficulties in applying Lyapunov’s technique to stability analysis of FrDE which is connected with the type of derivative of the Lyapunov function:

1. continuously differentiable Lyapunov functions: the Caputo derivative of Lyapunov functions of unknown solution is applied (see, for example, [8, 19, 20]). In this case the chain rule in fractional calculus can cause trouble in application.

2. continuous Lyapunov functions: the Dini derivative of Lyapunov functions in the case of ordinary derivative is extended to the fractional Dini derivative of the Lyapunov function (see, for example, [17, 18, 22]). This derivative does not have a memory and it is independent on the initial time and this differs from the idea of fractional calculus.

In real life situations it may be impossible to have only a change in the space variable and to keep the initial time unchanged. However, this situation requires introducing and studying a new generalization of the classical concept of stability which involves a change in both the initial time and the initial values. This type of stability generalizes known stability in the literature (see, for example [2]). Recently Lyapunov functions are applied to study some types of stability with respect to initial time difference for FrDE by an application of a new type of Caputo fractional Dini derivative of the Lyapunov function ([5, 6, 7]).

In this paper Lipschitz stability and Mittag-Leffler stability for nonlinear nonautonomous Caputo fractional differential equations are studied using the Caputo fractional Dini derivative of the Lyapunov functions relative to initial data difference ([5, 6, 7]) along the given FrDE. The Lipschitz, uniformly Lipschitz, globally uniformly Lipschitz stability and Mittag-Leffler stability are appropriately defined relative to initial time difference for fractional differential equations. Several sufficient conditions for Lipschitz stability and Mittag-Leffler stability with initial data difference for nonlinear fractional differential equations via Lyapunov functions and comparison results for a scalar fractional differential equation with a parameter are obtained.

## 2 Notes on Fractional Calculus

Fractional calculus generalizes the derivatives and the integrals of a function to a non-integer order [14, 21] and there are several definitions of fractional derivatives and fractional integrals.

In many applications in science and engineering, the fractional order q is often less than 1, so we restrict q ∈ (0, 1) everywhere in the paper.

The most widely used definitions for fractional derivatives are the Riemann-Liouville (e.g., in calculus), the Caputo (e.g., in physics and numerical integration), and the Grunwald-Letnikov (e.g., in signal processing, engineering, and control) ones:

1. The Riemann-Liouville (RL) fractional derivative of order q ∈ (0, 1) is given by

t0RLDtqm(t)=1Γ1qddtt0ttsqm(s)ds,tt0,

where Γ (.) denotes the Gamma function. The above definition of the fractional differentiation of Riemann-Liouville type leads to a conflict between the well-established mathematical theory of differential equations, such as the initial problem of the fractional differential equation, and the nonzero problem related to the Riemann-Liouville derivative of a constant.

2. The Caputo fractional derivative of order q ∈ (0, 1) is defined by

t0CDtqm(t)=1Γ1qt0ttsqm(s)ds,tt0.(2.1)

The Caputo and Riemann-Liouville formulations coincide when the initial conditions are zero. Also, the RL derivative is meaningful under weaker smoothness requirements.

The properties of the Caputo derivative are more similar to those of ordinary derivatives, such as the constant’s property. Also, the initial conditions of fractional differential equations with the Caputo derivative has a clear physical meaning and the Caputo derivative is extensively used in real life applications.

3. The Grünwald-Letnikov fractional derivative of order q ∈ (0, 1) is given by

t0GLDtqm(t)=limh0+1hqr=0[tt0h]1rqrmtrh,tt0

and Grünwald-Letnikov fractional Dini derivative by

t0GLD+qm(t)=limh0+sup1hqr=0[tt0h]1rqrmtrh,tt0,(2.2)

where qr=q(q1)...(qr+1)r! and [tt0h] denotes the integer part of the fraction tt0h.

For a wide class of functions, the definitions of Grünwald-Letnikov fractional derivative and the Riemann-Liouville fractional derivative are equivalent (for example, if the functions are sufficiently smooth). This allows us to use Grünwald-Letnikov fractional derivative for the formulation of the problem and for proving theoretical results, and then one can turn to the Riemann-Liouville fractional derivative for applied problems.

The relation between both Caputo fractional derivative and Grünwald-Letnikov fractional derivative is:

t0CDqm(t)=t0RLDq[m(t)m(t0)]=t0GLDq[m(t)m(t0)].(2.3)

Using (2.2) we define the Caputo fractional Dini derivative of a function as

t0CD+qm(t)=limh0+sup1hq[m(t)m(t0)r=1[tt0h]1r+1qr(mtrhm(t0))].(2.4)

## Definition 1

([15]) We say the function m(t) ∈ Cq([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) ∈ Cq([t0, T], ℝn), then t0CD+qm(t)=t0CDqm(t).

## 3 Statement of the problem

Let t0 ∈ ℝ+ be an arbitrary initial time and consider the following initial value problem (IVP) for the system of fractional differential equations (FrDE) with a Caputo derivative

t0CDqx(t)=f(t,x(t)) for t>t0,x(t0)=x0,(3.1)

where 0 < q < 1, x0 ∈ ℝn.

Let τ0 ∈ ℝ+, τ0t0, be an initial time and consider the following IVP for FrDE

τ0CDqx(t)=f(t,x(t)) for t>τ0,x(τ0)=y0,(3.2)

where y0 ∈ ℝn.

We will assume in the paper that the function fC[ℝ+ × ℝn, ℝn] is such that for any initial data (t0, x0) ∈ ℝ+ × ℝn the corresponding IVP for FrDE (3.1) has a solution x(t; t0, x0) ∈ Cq([t0, ∞), ℝn). Note some sufficient conditions for global existence of solutions of (3.1) are given in [10, 9, 17].

We will make use of the following result:

## Lemma 3.1

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

aCDqx(t)=f(t,x(t))fort>a,x(a)=x0.(3.3)

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

bCDqx(t)=f(t+η,x(t))fort>b, x(b)=x0.(3.4)

where b ≥ 0, η = ab.

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

t0CDqx(t)=f(t+η,x(t))fort>t0,x(t0)=y0,(3.5)

where η = τ0t0.

## 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 τ0t0 and initial points y0x0. 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: ||y0x0|| < δ and any initial time τ0 ∈ ℝ+: |τ0t0| < σ the inequality ||y(t+η; τ0, y0) − x(t)|| ≤ M||y0x0|| for tt0 holds, where η = τ0t0 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: ||y0x0|| < δ and any initial time τ0 ∈ ℝ+: |τ0t0| < σ the inequality ||y(t+η ;τ0, y0) − x(t)|| ≤ M||y0x0|| for tt0 + T holds, where η = τ0t0 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 || y0x0|| < δ and |τ0t0| < σ imply ||y(t+η ;τ0, y0) − x(t;t0, x0)|| ≤ M ||y0x0|| for tt0 where η = τ0t0 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 ||y0x0|| < ∞ and |τ0t0| < σ imply ||y(t+η ;τ0, y0) − x(t;t0, x0)|| ≤ M|| y0x0|| for tt0 where η = τ0t0 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+: |τ0t0| < σ the inequality || y(t+η ;τ0, y0) − x(t)|| ≤ C||y0x0||a{Eq(−λ(tt0)q)}b for tt0 holds, where η = τ0t0 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:

K(J)={aC[J,R+]:a(0)=0,a(r) is strictly increasing inJ and there existsa functionPaC(R+,R+) such thatPa(α)1forα1and a1(αr)rPa(α)for α1,r0};M(J)={aC[J,R+]:a(0)=0,a(r) is strictly increasing inJand a(r)Karfor some constantKa>0};Sλ={xRn:||x||λ},Bλ={uR:|u|λ}.

## Remark 3.5

The function a(u) = K1u, K1 ∈ (0, 1] is from the class 𝓚(ℝ+) with Pa(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

t0CDqu(t)=gt,u(t),η for t>t0,u(t0)=u0(3.6)

where u, u0 ∈ ℝ, g: ℝ+ × ℝ × BH → ℝ, g(t, 0, 0) ≡ 0, ηBH 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) ∈ Cq([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 tt0, 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 tt0+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 tt0;

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 tt0.

## 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 ηBH the inequality || u(t;t0, u0, η)|| ≤ C|u0|a{Eq(−λ(tt0)q)}b for tt0 holds, where u(t;t0, u0, η) is a solution of (3.6).

## Example 1

Consider the IVP forthe scalar FrDE with a parameter

t0CDqu(t)=(α+cη)u(t),u(t0)=u0,

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 + )u, g(t, 0, 0) ≡ 0 and it has a unique solution for any η ∈ ℝ defined by

u(t;t0,u0,η)=u0Eq((α+cη)(tt0)q),tt0.

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

|u(t;t0,u0,η)|=|u0|Eq((α+cη)(tt0)q)|u0|Eq(λ(tt0)q),

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

t0CD(3.1)qV(t,x,y,η,x0,y0)=limh0+sup1hq[V(t,yx)Vt0,y0x0r=1[tt0h]1r+1qr(V(trh,yxhq(f(t+η,y)f(t,x)))V(t0,y0x0))].(3.7)

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)=limh0+sup1hq(V(yx)V(yxhq(f(t+η,y)f(t,x)))hq)+(V(yx)V(y0x0))(tt0)qΓ(1q).(3.8)

Special case. Let V(x) = x2. Then

t0CD(3.1)qV(t,x,y,η,x0,y0)=2(yx)(f(t+η,y)f(t,x))+((yx)2(y0x0)2)(tt0)qΓ(1q).(3.9)
2. Let V(t, x) ≡ V(x) for x ∈ ℝ. Let x(t) ∈ Cq([t0, ∞), ℝ) and y(t) ∈ Cq([τ0, ∞), ℝ) be solutions of (3.1) and (3.2), respectively. Then for t > t0 and η = τ0t0 we have t+η = t+τ0t0 > 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)=limh0+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(y0x0))(tt0)qΓ(1q).(3.10)
3. Let V(t, x) = m2(t)x2 for x ∈ ℝ where mC1(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(yx)m2(t)(f(t+η,y)f(t,x))+(yx)2(t0GLD+qm2(t))(y0x0)2m2(t0)(tt0)qΓ(1q)=2(yx)m2(t)(f(t+η,y)f(t,x))+(yx)2(t0RLDtqm2(t))(y0x0)2m2(t0)(tt0)qΓ(1q).(3.11)

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

t0CD(3.1)qV(t,x,y,η,x0,y0)=2(yx)tq(f(t+η,y)f(t,x))+(yx)2Γ(1+q).
4. Let V(t, x) = m2(t)x2 for x ∈ ℝ where mC1(ℝ+, ℝ). Let x(t) ∈ Cq([t0, ∞), ℝ) and y(t) ∈ Cq([τ0, ∞), ℝ) be solutions of (3.1) and (3.2), respectively. Then for t > t0 and η = τ0t0 we have t+η = t+τ0t0 > 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))(y0x0)2m2(t0)(tt0)qΓ(1q).(3.12)
5. Let V(t,x1,x2)=m12(t)x12+m22(t)x22 for x1, x2 ∈ ℝ where m1, m2C1(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(y1x1)m12(t)(f1(t+η,y1,y2)f1(t,x1,x2))+2(y2x2)m22(t)(f2(t+η,y1,y2)f2(t,x1,x2))+(y1x1)2(t0RLDtqm12(t))+(y2x2)2(t0RLDtqm22(t))(m12(t0)(y1(0)x1(0))2+m22(t0)(y2(0)x2(0))2)(tt0)qΓ(1q).(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

cDqV(t,x)=limh0+sup1hq[V(t,x)V(th,xhqf(t,x))].(3.14)

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

cDqV(t,x(t))t0cDtqV(t,x(t))(3.15)

holds, where the operator cDq is defined by (3.14) and the operator t0cDtq is defined by (2.4).

## 4 Main Results

First we recall the following comparison results giving us the relationship between Lyapunov functions, system FrDE (3.1) and the scalar FrDE (3.6).

## 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) ∈ Cq([t0, t0+θ], ℝn), y(t) ∈ Cq([τ0, τ0+θ], ℝn) and y(t+η) − x(t) ∈ Δ for [t0, t0+θ] where t0, τ0 ∈ ℝ+: η = τ0t0, Δ ⊂ ℝn, θ > 0 is a given number.

2. The function GC[[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) ∈ Cq([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, y0x0) ≤ 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, y0x0) ≤ u0implies V(t, y(t+η) − x(t)) ≤ u(t) for tt0where 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

t0CD(3.1)qV(t,x(t),y(t+η),η,x0,y0)γV(t,y(t+η)x(t))+Cη,tt0

holds where C, γ ∈ ℝ are constants.

ThenV(t,y(t+η)x(t))[V(t0,y0x0+1γCη]Eq(γ(tt0)q)1γCηfor tt0.

## Proof

Consider the IVP for the scalar FrDE t0CDqu = γ u+, u(t0) = V(t0, y0x0). Denote v = u + 1γ. Then t0CDqv = γ v, v(t0) = V(t0, y0x0)+ 1γ. Then v(t) = [V(t0, y0x0)+ 1γ]Eq(γ(tt0)q) for tt0. Therefore, u(t) = [V(t0, y0x0)+ 1γ]Eq(γ (tt0)q) − 1γ for tt0. 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) ∈ Cq([t0, ∞), ℝn) is a solution of system of FrDE (3.1), where t0 ∈ ℝ+, x0 ∈ ℝnare given points.

2. The function gC[[t0, ∞) × ℝ × BH, ℝ], g(t, 0, 0) ≡ 0 and for any parameter ηBH the IVP for the scalar FrDE (10) has a solution u(t;t0, u0, η) ∈ Cq([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 Sρ 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 ηBH the inequality

t0CD(3.1)qV(t,x(t),y,η,x0,y0)g(t,V(t,yx(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

|u(t;t0,u0,η)|M|u0| for tt0(4.3)

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

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

V(t0,x)<δ1forx<δ2.(4.4)

Without loss of generalization we can assume δ2δ1. The function V(t, x) is Lipschitz on Sρ and

|V(t0,x)|=|V(t0,x)V(t0,0)|Lxforx<ρ.(4.5)

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

b1(αr)rPb(α)forα1.(4.6)

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

Now let y0 ∈ ℝn and τ0 ∈ ℝ+ be such that ∥y0x0∥ < 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, y0x0). Then from the choice of y0 and Eq. (4.4) it follows that u0 = V(t0, y0x0) < δ1. Therefore, applying (4.3) and (4.5) we obtain the inequality

|u(t)|M|u0|=MV(t0,y0x0)ML||y0x0||fortt0(4.7)

holds where u(t) = u(t; t0, u0, η) ∈ Cq([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

V(t,y(t+η)x(t))u(t) for tt0.(4.8)

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

b(y(t+η)xt)V(t,y(t+η)x(t))u(t)ML||y0x0||<M1||y0x0||.(4.9)

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

y(t+η)xtb1(M1||y0x0||)||y0x0||Pb(M1)=M2||y0x0||,tt0.

□

## 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 tt0 + T.

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

The proof of Theorem 4.2 is similar to the one in Theorem 4.1, so we omit it.

## Theorem 4.3

Let the following conditions be satisfied:

1. The function gC[[t0, ∞) × ℝ × BH, ℝ], g(t, 0, 0) ≡ 0 and for any parameterηBHthe IVP for the scalar FrDE (3.6) has a solutionu(t; t0, u0, η) ∈ Cq([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 : yxS(λ), y0x0S(λ) andηBHthe inequality

t0CD(3.1)qV(t,x,y,η,x0,y0)g(t,V(t,yx),η)for tt0

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

|u(t;t0,U0,η)|M|U0| for tt0,(4.10)

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 PbC(ℝ+, ℝ+) and a positive constant Ka such that b−1(αr) ≤ αPb(r) and a(r) ≤ Kar for r ≥ 0, ]ga ≥ 1. Choose M1 ≥ 1 such that M1 > Pb(M)Ka. Now let δ = minδ1,λM1,δ1Ka and initial points x0, y0 ∈ ℝn and τ0, t0 ∈ ℝ+ be such that

y0x0<δ and |η|<σ(4.11)

where η = τ0t0. 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 ∥y0x0∥ < δλM1λ, i.e. y0x0S(λ).

Let u0 = V(t0, y0x0). Then from Condition 2 (i) and the choice of x0, y0 it follows that u0 = V(t0, y0x0) ≤ a(∥y0x0∥) ≤ Kay0x0∥ < Kaδδ1. Then according to inequality (4.10) it follows that

|u(t)|M|u0| for tt0(4.12)

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

y(t+η)xtM1y0x0for tt0.(4.13)

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

y(t+η)xtM1y0x0fort[t0,t1]y(t1+η)xt1=M1y0x0y(t+η)xt>M1y0x0fort(t1,t1+ϵ]

where ϵ > 0 is a small enough number. Then for tt0, t1] the inequality ∥y(t + η) − x(t) ∥ ≤ M1y0x0∥ < M1δλ holds, i.e. y(t + η) − x(t) ∈ S(λ) for t0tt1. Then from Lemma 4.1 applied for θ = t1t0, Δ = S(λ) and G(t, u) = G(t, u, η) we get

V(t,y(t+η)xt)u(t) for t[t0,t1].(4.14)

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

M1y0x0=y(t1+η)xt1)b1(V(t1,y(t1+η)xt1))b1(u(t))b1(Mu0)=b1(MV(t0,y0x0))Pb(M)V(t0,y0x0)Pb(M)a(||y0x0||)Pb(M)Ka||y0x0||<M1||y0x0||.(4.15)

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)∥x2V(t, x) ≤ λ2(t)∥x2for (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 : yxS(λ), y0x0S(λ) andηBHthe inequality

t0CD(3.1)qV(t,x,y,η,x0,y0)(α+Cη)V(t,yx)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 + )u. Let σ = αC and choose the initial times t0, τ0 ∈ ℝ+ : | η| < σ where η = τ0t0. Let the initial points x0, y0 ∈ ℝn : ||x0y0|| < ∞ and u0 = V(t0, y0x0). According to Example 1 the solution of the comparison scalar FrDE (3.6) is u(t; t0, u0, η) = u0Eq((−α + )(tt0)q). Therefore, |u(t; t0, u0, η)| = |u0|Eq((−α +)(tt0)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]), Pb(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

t0CD(3.1)qV(t,x,y,x0,y0)α[V(t,yx)]

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) ∈ Cq([t0, ∞), ℝn) is a solution of system of FrDE (3.1), wheret0 ∈ ℝ+, x0 ∈ ℝnare given points.

2. The function gC[[t0, ∞) × ℝ × BH, ℝ], g(t, 0, 0) ≡ 0 and for any parameterηBHthe IVP for the scalar FrDE (3.6) has a solutionu(t; t0, u0, η) ∈ Cq([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. K1xaV(t, x) ≤ K2xabfor (t, x) ∈ ℝ+ ×ℝn, whereK1, K2, aandbare positive constants;

2. for any y, y0 ∈ ℝnandηBHthe inequality

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

holds.

Then the solutionx(t) of the system of FrDE (3.1) is Mittag-Leffler stable with ITD.

## Proof

Let G(t, u, η) = (−α + )u. Let σ = αC and choose the initial time τ0 ∈ ℝ+ : | η| < αC where η = τ0t0. Let the initial point y0 ∈ ℝn and u0 = V(t0, y0x0). According to Example 1 the solution of the comparison scalar FrDE (3.6) is u(t; t0, u0, η) = u0Eq((−α +)(tt0)q).

Consider a solution y(t) = y(t; τ0, y0) of system of FrDE (3.2) with initial data (τ0, y0). Using condition 3i and due to Corollary 2 it follows that

K1y(t+η)xtaV(t,y(t+η)x(t))u(t;t0,u0,η)V(t0,y0x0)Eq((α+Cη)(tt0)q)K2||y0x0||abEq((α+Cη)(tt0)q).(4.18)

Thus we get

y(t+η)xt(K2K1)1a||y0x0||bEq((α+Cη)(tt0)q)1a

for tt0 where C = (K2K1)1a,λ = α > 0 with | η| ≤ αC which implies that the solution x(t) of the system of FrDE (3.1) is Mittag-Leffler stable with ITD. □

## Example 3

Consider the following scalar system of FrDE

0CD0.9x(t)=f(t,x(t))for t>0,x(0)=x0(5.1)

where x0 ∈ ℝ and the function f(t, x) = g(t)x, t ≥ 0, x ∈ ℝ with g(t) = 0.01m2(t)2Γ(0.1)t0.9+0.259Γ(0.1)(t)0.12F1 (1, 1.9, 1.1, −t) and m2(t) = 1 + 1(t+1)0.9 ∈ (1, 2) is a decreasing function (see Figure 1), 2F1(a, b; c; z) is the hypergeometric function.

Figure 1

Graph of m2(t).

The equation (5.1) has a zero solution with x0 = 0.

The functions g(t) and respectively f(t, x) are changing their signs for t ≥ 0 (see Figure 2 for the graph of g(t)). It does not allow us to use the quadratic Lyapunov function for obtaining stability properties of the zero solution.

Figure 2

Graph of g(t).

Define the function V(t, x) = m2(t)x2. According to Case d) of Example 1 and Eq.(3.12) the generalized Caputo fractional Dini derivative w.r.t. ITD is

0CD(5.1)0.9V(t,0,y,0,η,y0)=2ym2(t)f(t+η,y)+(y(t))2(0RLDt0.9m2(t))t>0.(5.2)

For the fractional derivative we get

0RLD+0.9(1+1(t+1)0.9)=1Γ(0.1)t0.9+0RLD+0.9(t+1)0.9=0CD+0.9(t+1)0.9+0RLD+q1=9Γ(0.1)t0.12F1(1,1.9,1.1,t)+2Γ(0.1)t0.9.

Then using the functions F(t)=9Γ(0.1)t0.12F1(1,1.9,1.1,t) and t−0.9 are positive decreasing (see Figure 3 for the graph of f(t)), the inequalities m2(t + η) ≤ m2(t) ≤ 2 and 2Γ(0.1)t0.9m2(t)2Γ(0.1)(t+η)0.12Γ(0.1) (t−0.9− (t + 0.1)−0.1) ≤ 0 for η < 0.1 and tT = 0.8 (see Figure 4) we get

0CD(5.1)0.9V(t,0,y+,0,0,y0)=y2(0.02m2(t)m2(t+η)+0.5m2(t)9Γ(0.1)(t+η)0.12F1(1,1.9,1.1,tη)9Γ(0.1)t0.12F1(1,1.9,1.1,t)+2Γ(1q)t0.9m2(t)2Γ(0.1)(t+η)0.9)y2(0.02+9Γ(0.1)(t)0.12F1(1,1.9,1.1,t)9Γ(0.1)t20.1F1(1,1.9,1.1,t)+2Γ(0.1)t0.9m2(t)2Γ(0.1)(t+η)0.9)0.02y2=0.01(2)y20.01m2(t)y2=0.01V(t,y(t)),t0.8.
Figure 3

Graph of F(t).

Figure 4

Graphs of 2Γ(0.1)(t0.9(t+0.1)0.9).

The solution of the comparison FrDE o0CDqu(t) = −0.01u(t) for t > 0, u(0) = u0 > 0 is u(t) = u0Eq(−0.01tq) ≤ u0. Therefore, from Theorem 2 the zero solution of FrDE (5.1) is eventually Lipschitz stable with ITD.

## Acknowledgements

Research was partially supported by Fund MU17-FMI-007, University of Plovdiv “Paisii Hilendarski”.

## References

[1] S. Abbas, M. Benchohra, M.A. Darwish New stabilty results for partial fractional differential inclusions with not instantaneous impulses. Frac. Calc. Appl. Anal. 18, No 1 (2015), 172–191; 10.1515/fca-2015-0012; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Search in Google Scholar

[2] R. Agarwal, S. Hristova, D.O.Regan, Stability with respect to initial time difference for generalized delay differential equations. Electr. J. Diff. Eq. 49 (2015), 1–19.Search in Google Scholar

[3] R. Agarwal, S. Hristova, D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differetial equations. Frac. Calc. Appl. Anal. 19, No 2 (2016), 290–318; 10.1515/fca-2016-0017; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Search in Google Scholar

[4] R.P. Agarwal, D. O’Regan, S. Hristova, Stability of Caputo fractional differential equations by Lyapunov’s functions. Appl. Math. 60, No 6 (2015), 653–676.10.1007/s10492-015-0116-4Search in Google Scholar

[5] R.P. Agarwal, D. O’Regan, S. Hristova, M. Cicek, Practical stability with respect to initial time difference for Caputo fractional differential equations. Commun. Nonl. Sci. Numer. Simul. 42 (2017), 106–120.10.1016/j.cnsns.2016.05.005Search in Google Scholar

[6] R.P. Agarwal, D. O’Regan, S. Hristova, Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions. Georgian Math. J. 24, No 1 (2017), 1–13.10.1515/gmj-2016-0080Search in Google Scholar

[7] R.P. Agarwal, D. O’Regan, S. Hristova, Stability with initial time difference of Caputo fractional differential equations by Lyapunov functions. J. Anal. Appl. 36, No 1 (2017), 49–77.10.4171/ZAA/1579Search in Google Scholar

[8] N. Aguila-Camacho, M.A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simulat. 19 (2014), 2951–2957.10.1016/j.cnsns.2014.01.022Search in Google Scholar

[9] D. Baleanu, O.G. Mustafa, On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59 (2010), 1835–1841.10.1016/j.camwa.2009.08.028Search in Google Scholar

[10] Chung-Sik Sin, Liancun Zheng, Existence and uniqueness of global solutions of Caputo-type fractional differential equations. Frac. Calc. Appl. Anal., 19, No 3 (2015), 765–774; 10.1515/fca-2016-0040; https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml.Search in Google Scholar

[11] S.K. Choi, K.S. Koo, K.H. Lee, Lipschitz stability and exponential asymptotic stability in perturbed systems. J. Korean Math. Soc. 29, No 1 (1992), 175–190.Search in Google Scholar

[12] M. Cicek, C. Yakar, B. Ogur, Stability, Boundedness, and Lagrange stability of fractional differential equations with initial time difference. Sci. World J. 2014 (2014), Art. # 939027.Search in Google Scholar

[13] F.M. Dannan, S. Elaydi, Lipschitz stability of nonlinear systems of differential equations. J. Math. Anal. Appl. 113, No 2 (1986), 562–577.10.1016/0022-247X(86)90325-2Search in Google Scholar

[14] Sh. Das, Functional Fractional Calculus, Springer-Verlag, Berlin-Heidelberg (2011).10.1007/978-3-642-20545-3Search in Google Scholar

[15] J.V. Devi, F.A. Mc Rae, Z. Drici, Variational Lyapunov method for fractional differential equations. Comput. Math. Appl. 64 (2012), 2982–2989.10.1016/j.camwa.2012.01.070Search in Google Scholar

[16] Z. Jiao, Y.Q. Chen, Stability analysis of fractinal order systems with double noncommensurate order for matrix case. Frac. Calc. Appl. Anal. 14, No 3 (2011), 436–453; 10.2478/s13540-011-0027-3; https://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml.Search in Google Scholar

[17] V. Lakshmikantham, S. Leela, J.V. Devi, Theory of Fractional Dynamical Systems. Cambridge Scientific Publishers (2009).Search in Google Scholar

[18] V. Lakshmikantham, S. Leela, M. Sambandham, Lyapunov theory for fractional differential equations. Commun. Appl. Anal. 12, No 4 (2008), 365–376.Search in Google Scholar

[19] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, No 5 (2010), 1810–1821.10.1016/j.camwa.2009.08.019Search in Google Scholar

[20] Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica45 (2009), 1965–1969.10.1016/j.automatica.2009.04.003Search in Google Scholar

[21] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar

[22] I. Stamova, G. Stamov, Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys. 54 (2013), Art. # 043502, 11p.10.1063/1.4798234Search in Google Scholar

[23] D. Wang, A. Xiao, H. Liu Dissipativity and stability analysis for fractional differential equations. Frac. Calc. Appl. Anal. 18, No 6 (2015), 1399–1422; 10.1515/fca-2015-0081; https://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Search in Google Scholar

[24] C. Yakar, Fractional Differential equations in terms of comparison results and Lyapunov stability with initial time difference. Abst. Appl. Anal. 2010 (2010), Art.ID 762857, 16 p.; 10.1155/2010/762857.Search in Google Scholar