Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.


Introduction
Impulsive differential equations (IDEs) are applied in many fields, such as mechanical engineering, biology, and medical science.Generally speaking, there are two classes of impulsive equations.One is composed of instantaneous IDEs, for which the duration of the impulsive perturbation is very short compared to the entire evolution process, see for example, References [1,2].The other class is composed of non-instantaneous IDEs, for which the impulsive action starts at a fixed point, and remains active over a period of time that may be related to the previous state.
Recently, Dishlieva [34] studied a class of instantaneous IDEs with random impulsive effects and established sufficient conditions to ensure continuous dependence of the solutions.Motivated by Reference [34], we investigate the continuous dependence of the solutions of the following first-order nonlinear differential equations with random non-instantaneous impulsive effects: and also of the fractional-order random non-instantaneous IDEs: where C D α s i ,t denotes the classical Caputo fractional derivative of order α, by changing the lower limit s i , as in Reference [35].The random impulse and junction points, t i and s i , respectively, satisfy represent the right and left limits of x(t) at t = t i , respectively.In addition, we set The piecewise continuous solutions of Equations ( 1) and ( 2) have been represented in Reference [14] [Equations ( 5) and (7), therein].
We also introduce the following related original and perturbed equations without impulses: and: Denote any solution of Equations ( 1) or (2) by x(•; 0, , where J = [0, ∞) and C((t k , t k+1 ], R) denotes the space of all continuous functions from (t k , t k+1 ] into R. Additionally, denote any solution of Equations ( 3) or (5) by X(•; s i , x s i ) ∈ C([s i , t i+1 ], R) .For the interval (t i , s i ], we denote its solutions by X(•; t i , x(t + i )).Then, the following relationship is valid: The main objective of this article is to present the continuous dependence of solutions with respect to the initial condition when random impulse and junction points are incorporated in Equations ( 1) and (2).We will take notice of the fact that the location and the number of the impulse points and junction points are not determined in a finite time interval, and so, we can assume that the impulse and junction points are random.
The main contributions of this paper are two folds.We extend the concept and results in Reference [34] to random non-instantaneous impulsive cases by imposing different conditions on the nonlinear term.We also extend the continuous dependence of solutions of first-order non-instantaneous impulsive equations to fractional-order non-instantaneous impulsive equations.
The rest of this paper is organized as follows.Section 2 gives the relevant definitions and notions for the continuous dependence of solutions and contains the main results.Section 3 gives two examples to demonstrate the application of our results.In Section 4, conclusions are drawn.
[H 3 ] There exists a positive constant and for all x, y ∈ R.
[H 4 ] The solutions of Equations ( 3) or ( 5) depend uniformly and continuously on the initial point.
[H 5 ] The functions h i (t, x), i ∈ N + , are uniformly bounded, i.e., for any i ∈ N + , there exists an Then the solution of Equation ( 1) depends continuously on the initial point (0, x 0 ) at the random impulse and junction points.
Proof.Let ε and T be two arbitrary positive constants and Ω = {t 1 , s 1 , t 2 , s 2 , • • •} be an arbitrary set of impulse and junction points.Note that if t i → ∞ as i → ∞, then for any selection of the set of impulse points and junction points, there exists k ∈ N such that s k < T ≤ t k+1 , i.e., there exists at most k impulse points and at most k junction points belonging to the interval [0, T].Without loss of generality, we assume that T = t k+1 .
Proof.Let ε and T be two arbitrary positive constants like in Theorem 3. We divide the proof into two cases.
Owing to 0 < δ k,k+1 < ε, one can get: Case 2. From the interval (t k , s k ], the expression of the solutions are given by: Assume that there exists a δ kk < δ k,k+1 , where Similar to the above procedure, for the interval (s k−1 , t k ], one has: For t = t k , we get: and then: Therefore, (24) becomes: From the above, one can deduce that for the interval (t 1 , s 1 ], there exists a δ 11 < δ 12 , and if For [0, t 1 ], there is δ 01 < Thus, for t = t 1 , we have: and then:  1) depends continuously on the initial point (0, x 0 ) at the random impulse and junction points provided that 2M ≤ | x s i − x s i |.
Proof.We divide proof into several steps.
Step 1.According to [H 4 ], assume there is a δ k,k+1 = δ k,k+1 (ε, T), where 0 Step 2. For the interval (t k , s k ], according to [H 5 ], we obtain: For t = s k , we get: Step 3. In this step, we check the continuity of the solutions in the interval (s k−1 , t k ].From Equation (25), we assume there exists a δ k−1,k depending on ε, T, δ k,k+1 .For brevity, we denote Step 4. For (t k−1 , s k−1 ], we get: Similar to the above steps, we have the following general results.

Numerical Examples
Let ε and T be two arbitrary positive constants.

Example 7. Consider:
Then the solution of Equation ( 27) can be analytically determined, namely: ] all hold.Therefore, all the assumptions in Theorem 3 are satisfied.
From Definition 2, choosing δ = ε e aT , we find that the solution of Equation ( 27) without impulses satisfies uniform and continuous dependence on the initial point.Choosing 2M ≤ ε e aT , then h i (t, x) are uniformly bounded.Thus, the conditions [H 1 ], [H 4 ], and [H 5 ] hold, and all the assumptions of Theorem 5 are satisfied.

Conclusions
In this paper, we presented the continuous dependence of the solutions to first order non-instantaneous IDEs with random impulse and junction points.Then, we extended the results to study the same problem for fractional order cases.The backward checking approach [34] (from the last subinterval to the first subinterval) is extended to differential and algebra equations and is used to prove the main results.The approach is different from Reference [13].

Figure 1 .
Figure 1.The blue line denotes the solution of Equation (27) and the red line denotes the corresponding perturbation problem.

3 2 , a = 1 5 , ε = 1 2 )
that the solution of Equation (29) without impulses satisfies uniform and continuous dependence on the initial point.Choosing 2M ≤ h i (t, x) are uniformly bounded.Thus, the conditions [H 1 ], [H 4 ], and [H 5 ] hold, and all the assumptions in Remark 6 are satisfied.The solutions of Equation (29) and the corresponding perturbation problem (with x 0 = 1, x 0 = 1.3, T = are shown in Figure 2.

Figure 2 .
Figure 2. The blue line denotes the solution of Equation (29) and the red line denotes the corresponding perturbation problem.