Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays

In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results.


Introduction
Many papers in the literature study stability of solutions of differential equations via Lyapunov functions.One type of stability, useful in real world problems, is the so-called Lipschitz stability and Dannan and Elaydi [1] introduced the notion of Lipschitz stability for ordinary differential equations.As noted in [1], this type of stability is important only for nonlinear problems since it coincides with uniform stability in linear systems.Based on theoretical results for Lipschitz stability in [1], the dynamic behavior of a spacecraft when a single magnetic torque-rod is used for achieving a pure spin condition is studied in [2].Recently, stability properties of delay fractional differential equations without any type of impulse are considered and we refer the reader to [3] and the references therein.
In this paper, we study the Lipschitz stability for a nonlinear system of non-instantaneous impulsive fractional differential equations with state dependent delay (NIFrDDE).The impulses start abruptly at some points and their actions continue on given finite intervals.Non-instantaneous impulsive differential equations were introduced by Hernandez and O'Regan in 2013 (see, for example, [4]).The systematic description of solutions of both ordinary and Caputo fractional differential equations with non-instantaneous impulse and without delays is given in the monograph [5].In addition, some results for non-instantaneous fractional equations without any type of delay are presented in [6][7][8].In [9], Caputo fractional differential equations with time varying delays is considered (we note that the model had no impulses).However, in this paper, for the first time, we consider together 1.
state dependent delays (note a special case is time varying delays); and 3. models with non-instantaneous impulses.
There are two different approaches in the literature for the interpretation of the solution of fractional differential equations with impulses (for more details, see [6] and Chapter 2 of the book [5]).In the first interpretation, the lower limit of the fractional derivative is one and the same on the whole interval of study and at each point of jump we consider a boundary value problem defined by the impulsive function.In the second interpretation, the lower limit of the fractional derivative changes at each time of jump with the idea of considering an initial value problem at each jump point.
In this paper, we use the second approach to study Lipschitz stability properties of nonlinear non-instantaneous impulsive delay differential equations.The delays are bounded and depend on both the time and the state.Note several stability properties are studied in the literature for Caputo fractional differential equations (for example, see [10] (without delays), [3] (with delays and no impulses), and [11] (with multiple discrete delays without impulses)).Our study is based on Lyapunov functions and the Razumikhin technique.A brief overview in the literature of different types of derivatives of Lyapunov functions among the studied fractional differential equation is given.Several sufficient conditions for uniform Lipschitz stability and global uniform Lipschitz stability are obtained by an application of these derivatives.Some examples illustrating the results are given.

Notes on Fractional Calculus
We give the main definition of fractional derivatives used in the literature (see, for example, [12][13][14]).We give these definitions for scalar functions.Throughout the paper, we assume q ∈ (0, 1).
-Riemann-Liouville (RL) fractional derivative : Note that for a constant m the equality C t 0 D q t m = 0 holds.However, for any given t * ,we denote The Grünwald-Letnikov fractional derivative is given by and the Grünwald-Letnikov fractional Dini derivative by where q C r = q(q−1)...(q−r+1) r! and [ t−t 0 h ] denotes the integer part of the fraction t−t 0 h .From the relation between the Caputo fractional derivative and the Grünwald-Letnikov fractional derivative using Equation (1), we define the Caputo fractional Dini derivative of a function as i.e., The fractional derivatives for scalar functions could be easily generalized to the vector case by taking fractional derivatives with the same fractional order for all components.

Statement of the Problem and Basic Definitions
Let the positive constant r be given and the points {t i } ∞ 1 , {s i } ∞ 1 be such that 0 < s i < t i < s i+1 , i = 1, 2, . . . .Let t 0 ≥ 0 be the given initial time.Without loss of generality, we can assume t 0 ∈ [0, s 1 ).
Consider the space PC 0 of all functions y : [−r, 0] → R n , which are piecewise continuous endowed with the norm ||y|| The intervals (t i , s i+1 ), i = 0, 1, 2, . . .are the intervals on which the fractional differential equations are given and on the intervals (s i , t i ), i = 1, 2, . . . the impulsive conditions are given.
The Caputo fractional derivative has a memory and it depends significantly on its lower derivative.This property as well as the meaning of impulses in the differential equation lead to two basic approaches to Caputo fractional differential equations with non-instantaneous impulses: -Unchangeable lower limit of the Caputo fractional derivative: the lower limit of the fractional derivative is equal to the initial time t 0 on the whole interval of consideration.-Changeable lower limit of the Caputo fractional derivative: the lower limit of the fractional derivative is equal to the left end t i on the interval (t i , s i+1 ), i = 0, 1, 2, . . .without impulses.
In this paper, we study the case of changeable lower limit of the Caputo fractional derivative.Consider the initial value problem (IVP) for a nonlinear system of non-instantaneous impulsive fractional differential equations with state dependent delay (NIFrDDE) with q ∈ (0, 1): where x ∈ R n , C t i D q t x(t) denotes the Caputo fractional derivative with lower limit t i for the state x(t), the functions f : the history of the state from time t − r up to the present time t.Note that for any t ≥ 0 we let x ρ(t,x t ) = x(ρ(t, x(t + s))), s ∈ [−r, 0], i.e., the function ρ determines the state-dependent delay.Note, the integer order differential equations with non-instantaneous impulses and state dependent delay are studied in [15].
Let P C[t 0 , ∞) be the space of all functions y : [t 0 − r, ∞) → R n which are piecewise continuous on [t 0 − r, ∞) with points of discontinuity s i , i = 1, 2, . . ., the limits y(s i − 0) = lim t→s i , t<s i y(t) = y(s i ) and y(s i +) = lim t→s i , t>s i y(t) exist, for any t ∈ (t i , s i ] the Caputo fractional derivative C t i D q t y(t), i = 0, 1, . . ., exists and it is endowed with the norm ||y|| P We introduce the assumptions:  1) iff it satisfies the following integral-algebraic equation

A3. The functions
Definition 2. The functions f , ρ are defined only on the intervals without impulses on which the differential equation is given.
We generalize Lipschitz stability ( [1]) for ordinary differential equations to systems of Caputo fractional non-instantaneous impulsive differential equations with state dependent delay.
Consider the following sets: a(r) is strictly increasing in J, and a(r) ≤ K a r for some constant K a > 0},

Lyapunov Functions and Their Derivatives among Nonlinear Non-Instantaneous Caputo Delay Fractional Differential Equations
One approach to study Lipschitz stability of solutions of Equation ( 1) is based on using Lyapunov-like functions.The first step is to define a Lyapunov function.The second step is to define its derivative among the fractional equation.
We use the class Λ of Lyapunov-like functions, defined and used for impulsive differential equations in [16].Definition 4. Let J ∈ R + be a given interval, and ∆ ⊂ R n be a given set.We say that the function V(t, x) : The function V(t, x) is continuous on J/{s k ∈ J} × ∆ and it is locally Lipschitz with respect to its second argument.-For each s k ∈ J and x ∈ ∆, there exist finite limits In connection with the Caputo fractional derivative, it is necessary to define in an appropriate way the derivative of Lyapunov functions among the studied equation.We give a brief overview of the derivatives of Lyapunov functions among solutions of fractional differential equations known and used in the literature.There are mainly three types of derivatives of Lyapunov functions from the class Λ(J, ∆) used in the literature to study stability properties of solutions of Caputo fractional differential in Equation (1): -First type: the Caputo fractional derivative of the function where x(t) is a solution of Equation ( 1). - where The derivative of Equation ( 4) keeps the concept of fractional derivatives because it has a memory. - among Equation ( 1): Let the initial function ϕ ∈ PC 0 be given and the function φ ∈ PC 0 and t ∈ (t k , s k+1 ) for a non-negative integer k.Then, c or its equivalence The derivative c (1) D q + V(t, φ; t k , ϕ(0)) given by Equation ( 6) depends significantly on both the fractional order q and the initial data (t k , ϕ) of IVP for FrDDE (Equation ( 1)) and it makes this type of derivative close to the idea of the Caputo fractional derivative of a function.
are satisfied.
Remark 5. A derivative of V(t, x) ∈ Λ(J, ∆) among a system of Caputo fractional differential equations without delays was introduced by V. Lakshmikantham et al. [17] in 2009.Later, it was generalized for fractional equations with delays ( [18][19][20]): This definition is a direct generalization of the well known Dini derivative among differential equations with ordinary derivatives.However, for equations with fractional derivatives, it seems strange.It does not depend on the order q of the fractional derivative nor on the initial time t 0 .The operator defined by Equation (9) has no memory, which is typical for the fractional derivative.
In the next example to simplify the calculations and to emphasize the derivatives and their properties, we consider the scalar case, i.e. n = 1.

Example 1. (Lyapunov function depending directly on the time variable
Case 1. Caputo fractional derivative.Let x be a solution of NIFrDDE (Equation ( 1)).Then, the fractional derivative is difficult to obtain in the general case for any solution of Equation ( 1).In addition, the solution x(t) might not be differentiable on the intervals of impulses.Case 2. Dini fractional derivative.Let φ ∈ PC 0 and t ∈ (t k , s k+1 ) for a non-negative integer k.Then, applying Equation ( 4), we obtain Case 2. Caputo fractional Dini derivative.Let ϕ, φ ∈ PC 0 and t ∈ (t k , s k+1 ) for a non-negative integer k.Then, we use Equation ( 6) and obtain
We use the following comparison scalar fractional differential equation with non-instantaneous impulses: . We obtain some comparison results.Note some comparison results for fractional time delay differential equations are obtained in [18] by applying the derivative defined by Equation ( 9) and substituting it incorrectly as a Caputo fractional derivative (see Remark 5).
We introduce the following conditions: ) is strictly decreasing with respect to its second argument, and for any k = 1, 2, . . . the functions ] and for any k = 1, 2, . . . the function A8.For all k = 1, 2, . . ., the functions In our main results, we use the Lipschitz stability of the zero solution of the scalar comparison non-instantaneous impulsive fractional differential in Equation (10).
Case 1. Suppose for all natural numbers k = 1, 2, . . . the equality holds.Then, the solution of Equation ( 11) is given by The solution of Equation ( 11) is uniformly Lipschitz stable with M 1 = 30 (see Figure 1 for the graph of the solutions with various initial values).
Case 2. Suppose for all natural numbers k = 1, 2, . . . the equality holds.Then, the solution of Equation ( 11) is given by The solution of Equation ( 11) is unbounded (see Figure 2 for the graph of the solution).Therefore, for ψ k (t, u) = u 2t ≤ u the solution is Lipschitz stable but for ψ k (t, u) = u 2t ≥ u it is not (compare with condition (A8)).In our study, we use some comparison results.When the Caputo fractional derivative is used, then the comparison result is: Lemma 2. (Caputo fractional derivative).Assume the following conditions are satisfied: 1.
When the Dini fractional derivative defined by Equation ( 4) or Caputo fractional Dini derivative defined by Equation ( 5) is used then the comparison result is: Assumptions A1-A4 and A6 are satisfied.
Proof.The proof is similar to the one in Lemma 2 where instead of the Caputo fractional derivative of the Lyapunov function, we use the Dini fractional derivative or the Caputo fractional Dini derivative which are less restrictive with respect to the properties of Lyapunov functions (for example, differentiability is not required).We sketch the proof emphasizing the differences with Lemma 2.
. . in Condition 3(i) of Lemma 4. We use induction with respect to the intervals to prove Lemma 4. We prove the inequality in Equation ( 14).

Main Results
Theorem 1. (Caputo fractional derivative) Let the following conditions be satisfied: 1.

There exist a function
For any initial data and any solution x(t) of Equation ( 1) defined on [t 0 , ∞) such that for any τ ∈ For any k = 0, 1, 2, . . .and t ∈ (s k , t k+1 ], y ∈ S ρ the inequality holds.
Then, the zero solution of Equation ( 1) is uniformly Lipschitz stable (uniformly globally Lipschitz stable).
Proof.Let the zero solution of Equation (10) be uniformly Lipschitz stable.Let t 0 ≥ 0 be an arbitrary.Without loss of generality, we assume t 0 ∈ [0, s 1 ).From Condition 3, there exist M ≥ 1, δ 1 > 0 such that for any holds, where u(t; t 0 , u 0 ) is a solution of Equation (10) with the initial data (t 0 , u 0 ).
From the inclusions a ∈ K([0, ρ]) and b ∈ M([0, ρ]), there exist a function q b (u) and a positive constant K a .Without loss of generality, we can assume K a ≥ 1. Choose the constant M 1 such that . Consider the solution y(t) = y(t; t 0 , ϕ) of the system in Equation (1) for the chosen initial data (t 0 , ϕ).
According to Lemma 2, we get From the inequality in Equation ( 20) and Condition 2(i), we obtain The contradiction proves the validity of Equation (19).From the inequality in Equation ( 19) and Condition 2(i), we have Theorem 1.
Case 2. There exists a point Then, as in Case 1 we get y(t) ∈ S ρ for t ∈ [t 0 , T].Let T ∈ (s j , t j+1 ) for a natural number j.According to Condition 2(iii) of Theorem 1, we obtain b(M The contradiction proves this case is not possible.
Case 3.There exists a natural number k such that V(t, y(t)) < b(M . The contradiction proves this case is not possible.
The proof of globally uniformly Lipschitz stability is analogous so we omit it.
If the zero solution of Equation ( 10) is uniformly Lipschitz stable (uniformly globally Lipschitz stable), then the zero solution of Equation ( 1) is uniformly Lipschitz stable (uniformly globally Lipschitz stable).
Proof.The proof is similar to the one in Theorem 1 where M 1 = M A  Assumptions A1-A8 are fulfilled.2.
Then, the zero solution of Equation ( 1) is uniformly Lipschitz stable (uniformly globally Lipschitz stable).
The proof of Theorem 3 is similar to the one in Theorem 1 where Lemma 4 is applied instead of Lemma 2.

Remark 4 .
For any initial data (t k , ϕ) ∈ R + × PC 0 of the IVP for NIFrDDE (Equation (1)) and any function φ ∈ PC 0 and any point t ∈ (t k , s i+1 ) for a non-negative integer k the relations c (1)

Figure 1 .
Figure 1.Example 2. Graph of the solution of Equation (11) with ψ k (t, u) = u 2t for various initial values.