Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

: In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate deﬁnition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is deﬁned and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufﬁcient conditions for the deﬁned stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.


Introduction
Fractional differential equations have attracted considerable attention due to their many applications in science and engineering (see the monographs [1][2][3][4] and the references therein). The main advantage of fractional derivatives is that they can describe the properties of heredity and memory of many materials. There are various types of fractional derivatives known in the literature. One of the most important properties of the solutions is stability. There are various types of stability that describe different properties of the solutions. One of them is Lipschitz stability, defined and studied for ordinary differential equations in [5]. Later, this type of stability was studied for various types of differential equations and problems, such as nonlinear differential systems [6][7][8], impulsive differential equations with delays [9], fractional differential systems [10], Caputo fractional differential equations with non-instantaneous impulses [11], a piecewise linear Schrödinger potential [12], a hyperbolic inverse problem [13], the electrical impedance tomography problem [14], the radiative transport equation [15] and neural networks with non-instantaneous impulses [16].
In this paper we define and study Lipschitz stability for Riemann-Liouville (RL) fractional differential equations with non-instantaneous impulses. We will initially introduce the statement of the problem.
There are mainly two types of impulses involved in differential equations: instantaneous impulses (known as impulses), where time of action is negligibly small comparatively with the whole duration of the process and non-instantaneous impulses that start their actions abruptly and continue to act on a finite interval.
The presence of the RL fractional derivative leads to two particular types of initial conditions that are equivalent (see the classical book [2]): integral form of the initial condition weighted form of the initial condition .
Following the ideas of the impulses in ordinary differential equations, i.e., after the impulse the differential equation is the same with a new initial condition, the integral form and weighted form of the impulsive conditions can be defined.
In this paper we will use the integral form of both the initial condition and the impulsive conditions. Keeping in mind the above description, in this paper we will study the initial value problem (IVP) for the following system of nonlinear RL fractional differential equations with non-instantaneous impulses (NIRLFDE) of fractional order q ∈ (0, 1): , for i = 1, 2, . . . , where x ∈ R n and RL 0 D q t x(t) is the Riemann-Liouville fractional derivative.

Remark 2. The equality lim
could be replaced by , which is an impulsive condition for ordinary differential equations with impulses (see the book [17]).
Note that the solutions of the IVP for the NIRLFDE of Equation (1) have singularities at each point t i , i = 0, 1, 2, . . . . This requires stability properties to be studied at intervals excluding these points. In this paper we will define a new type of Lipschitz stability for NIRLFDE of the type in Equation (1), which is an appropriate generalization of the classical Lipschitz stability introduced in [5]. It is called generalized Lipschitz stability in time. This type of stability is connected with the singularity of the solution at both the initial time point and the stop time points of impulses. In connection with this we consider an interval excluding these time points. We use Lyapunov functions and two types of derivatives of these Lyapunov functions among the studied RL fractional equation with non-instantaneous impulses. Several sufficient conditions for Lipschitz stability in time are obtained. Some examples illustrating the theoretical results and comparing the application of both fractional derivatives of Lyapunov functions are given.
We will use the following sets: The main contributions of the paper can be summarized as follows: for a nonlinear system with RL fractional derivatives of order q ∈ (0, 1) and noninstantaneous impulses we define in an appropriate way both the initial condition and the non-instantaneous impulsive conditions; -generalized Lipschitz stability in time of the zero solution of a system of nonlinear RL fractional differential equations with non-instantaneous impulses is defined; -two types of derivatives of Lyapunov functions among the RL fractional differential equations with non-instantaneous impulses are applied; -comparison results with Lyapunov functions, scalar RL fractional equations with non-instantaneous impulses and both types of derivatives of Lyapunov functions are proved; -sufficient conditions for generalized Lipschitz stability in time are obtained by the application of both types of derivatives of Lyapunov functions.

Preliminaries
In this section we will give the definitions of fractional derivatives used in the paper (see, for example, [1][2][3]). These definitions are given for scalar functions but they also are easily generalized to the vector case by taking fractional derivatives component-wisely. Throughout the paper we will assume q ∈ (0, 1). -Riemann-Liouville (RL) fractional integral: where Γ(.) denotes the Gamma function; -Riemann-Liouville fractional derivative: The Grünwald-Letnikov fractional derivative is given by and the Grünwald-Letnikov fractional Dini derivative by where q C r = q(q−1  [18]). Let m ∈ C 1−q ([a, a + T), R). Suppose that for an arbitrary t 1 ∈ (a, a + T), we have m(t 1 ) = 0 and m(t) < 0 for a ≤ t < t 1 . Then it follows that RL a D q t m(t)| t=t 1 ≥ 0. The practical definition of the initial condition as well as the impulsive conditions of fractional differential equations with RL derivatives is based on the following result: ). Let q ∈ (0, 1) and a, T > 0, m : [a, a + T] → R be a Lebesgue measurable function.
(a) If there exists a.e. a limit lim t→a+ [(t − a) 1−q m(t)] = c ∈ R, then there also exists a limit a I 1−q t m(t)| t=a := lim

Remark 7.
According to Proposition 2 the initial condition and the impulsive conditions in Equation (1) could be replaced by the equalities 0 I We introduce the assumptions: . Let ρ > 0 and J ⊂ R + , 0 ∈ J be an interval. Defining the classes: We will generalize Lipschitz stability ( [5]) to systems of RL fractional differential equations with non-instantaneous impulses. In our further considerations below we will assume the existence of the solution of the IVP for the NIRLFDE of Equation (1) and we will denote it by

Example 1. Consider the IVP for the scalar linear NIRLFDE
where y 0 ∈ R, a ∈ R.
The solution of Equation (3) is given by It has singularities at the point t k , k = 0, 1, 2, 3, . . . which are the initial time and the end times of action of the non-instantaneous impulses at which the impulsive condition is switching to the differential equation (in the particular case a = 0.5, y 0 = 1, t k = 2k, s k = k + 1, k = 0, 1, 2, . . . , q = 0.3 the graph of the solution y(t) is given on Figure 1).
Example 1 illustrates that the stability of the solution for non-instantaneous impulsive differential equations in the case of the RL fractional derivative has to be studied on intervals excluding from the right the points t k , k = 0, 1, 2, . . . . In connection with this phenomenon we will define a new type of stability:

Lyapunov Functions and Comparison Results
. , x ∈ ∆, and it is locally Lipschitz with respect to its second argument.
We will use two types of derivatives of Lyapunov functions from the class Λ(R + , ∆) to study the Lipschitz stability of the NIRLFDE of Equation (1) (see Remark 1) : - The RL fractional derivative of the Lyapunov function V ∈ Λ(R + , ∆) among the NIRLFDE of Equation (1) is defined by The Dini fractional derivative of the Lyapunov function V ∈ Λ(R + , ∆) among the NIRLFDE of Equation (1) is defined by: Remark 10. The definition of the Dini fractional derivative of the Lyapunov function V ∈ Λ(R + , ∆) among the NIRLFDE of Equation (1) is similar to the Grünwald-Letnikov fractional Dini derivative in Equation (2). (1). Then for any k = 0, 1, 2, . . . the equality

Remark 11. Let x(t) be a solution of Equation
holds.
We will use as a comparison scalar equation the following equation: , for i = 1, 2, . . . , where We introduce the following conditions: In our study we will use some comparison results with both defined above types of derivatives of Lyapunov functions.

Comparison Result with RL Fractional Derivative of Lyapunov Functions.
Lemma 1. Assume the following conditions are satisfied: holds. (ii) For all i = 0, 1, 2, . . . the inequalities hold. If holds.
If δ ≥ s 0 the inequality in Equation (8) is proved.
If δ 1 ≥ s 1 − t 1 the inequality of Equation (11) is proved. If δ 1 < s 1 − t 1 we assume the inequality of Equation (11) is not true. Then there exists a point t * The inequality of Equation (13) contradicts condition 4(i). Therefore, the inequality of Equation (11) is true for any arbitrary number ε and thus Equation (7) holds for t ∈ (s 1 , t 1 ].

Proof.
The proof is similar to the one in Lemma 1 where instead of the RL fractional derivative of the Lyapunov function we will use the Dini fractional derivative. The main difference between both proofs of Lemma 1 and Lemma 2, respectively, is connected with the inequalities of Equations (10) and (13) for t * ∈ (t 0 , s 0 ] and t * 1 ∈ (t 1 , s 1 ]. We will consider the general case of (t k , s k ], k = 0, 1, 2, . . . , i.e., assume that for a fixed non-zero integer k there exist δ k ∈ (0, t k − s k ) and a point t * For any fixed t ∈ (t k , s k ] we have (see Equation (2)) Denote Therefore, for any r = 1, 2, . . . and h > 0 Thus, by (1 From Equations (15)- (17) and condition 2 of Lemma 2 we get The inequality of Equation (18) contradicts the inequality of Equation (14).

Main Results
We will obtain some sufficient conditions for generalized Lipschitz stability in time by Lyapunov functions and their two fractional derivatives.
3. The zero solution of the scalar comparison Equation (6) is generalized Lipschitz stable in time.
Then the zero solution of the system in Equation (1) is generalized Lipschitz stable in time.
From condition 2(iv) of Theorem 1 for y(t) ≡ x * (t) and y k = x * (t i − 0) we have condition 4(iii) of Lemma 1.

holds.
Then the zero solution of Equation (1) is generalized Lipschitz stable in time.
The proof of Theorem 3 is similar to the one of Theorem 1 where Lemma 2 is applied instead of Lemma 1.

Theorem 4.
Let the conditions of Theorem 1 be satisfied where a(s) = A 2 s p , A 2 > 0, p ≥ 1 in condition 2(ii), the condition 2(i) is replaced by condition 2*(i) of Theorem 2 and the condition 2(v) is replaced by condition 2(v*) of Theorem 3.

Conclusions
A system of nonlinear RL fractional differential equations with non-instantaneous impulses was studied. We studied the case when the lower limit of the RL fractional derivative was changed at each stop point of the action of the impulses. This led to a singularity of the solution at the initial time and the stop time points of impulses and it required appropriate initial conditions as well as non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability was defined and studied. Two types of derivatives of the applied Lyapunov functions among the studied system were applied to obtain sufficient conditions for the defined stability. Some comparison results were obtained.
The study in this paper could be continued in future works in various ways. For example, the obtained theoretical results could be applied to some models described by RL fractional differential equations with non-instantaneous impulses to study the stability properties of the equilibrium. Theoretically, some other types of stability of the solutions for nonlinear NIRLFDE of the type in Equation (1)

Conflicts of Interest:
The authors declare no conflict of interest.