Existence and data-dependence theorems for fractional impulsive integro-differential system

In this article we have considered a fractional order impulsive integro-differential equation (IDE) in Caputo’s sense for the unique solution and data dependence results. We take help of the Banach fixed point theory and basic literature of fractional calculus. The results are examined with the help of an expressive numerical example for an application of the results.


Introduction
Modeling with the help of fractional order integral and differential operators is very common in the community of engineers and scientists due to the applications. Recently, experts of theory and numeric methods have given interesting tools for the study of fractional order models. In the theoretical aspects of the models, fixed point theorems play a vital role. We suggest the readers for more detail about the fractional calculus and its application to the work in [1][2][3][4][5][6][7]. Among the fractional operators, the Caputo-Fabrizio [8][9][10] and the Atangana-Baleanu fractional differential operators with nonsingular kernel [4][5][6][7][11][12][13][14][15] are recently well studied operators.
Recently, some researchers have focused on the different types of FDEs with impulses for the existence of solutions (EUS). Here, we highlight some of them. Sousa et al. [16] considered the investigation of existence results and Ulam-stability by the help of fixed point approach of an impulsive system. Xu and Liu [17] studied the boundedness criteria for delay impulsive system and provided an application. Zhang and Xiong [18] used some properties of the Mittag-Leffler function with one and two parameters for the existence and stability results. Zhao et al. [19] evaluated fractional order impulsive systems with Dirichlet boundaries by the help of Morse theory for the EUS. Heidarkhani et al. [20] investigated multiple solutions with the help of three critical points approach.
For the application of the IDEs with impulses, we recommend the readers the recent work [21][22][23]. Keeping the importance of the study, we are considering the following im-pulsive IDE for the existence, stability, and numerical solution: functions in the arguments with Ψ (t, x(t))| t=0 = 0. The c a D ϑ is Caputo's differential operator of order ϑ ∈ (0, 1]. We consider the split of the interval [a, b] with respect to t k , δ k such that a < t k < δ k < b for k = 1, 2, 3, . . . , m and assume δ m+1 = b. We consider Ba- is used for the integer part of κ.

Integral form
This section is reserved for the integral form of fractional order impulsive system (1.1) provided that Proof We divide the proof in parts as follows.
Case-I For t ∈ (0, t 1 ], applying the integral operator I ϑ on (2.1), we have , applying the integral operator I ϑ on (2.1), we have where by the help of the impulsive relation we get (2.6) Thus, (2.4) implies Now, using the condition Thus, by the help of (2.3) and (2.8), for t ∈ [0, t 1 ], we have Case-III For t ∈ (t k , δ k ], we have (2.10) This completes the proof.
s, x(s)) ds, while keeping the conditions and order of derivative the same as in the theorem above, we get the following solution for fractional impulsive system (1.1): (2.11)
Step 2. Now, we show that T is a strict contraction. For this, we assume x(t), y(t) ∈ R. And consider the following three cases.
Case I For t ∈ [0, t 1 ], we have Now, for t ∈ (t k , δ k ], where k = 1, 2, 3, . . . , p and x(t) ∈ B, we have Thus, with the help of (3.5)-(3.7), we have that the operator T is a contraction, and by the Banach fixed point theorem T has a unique fixed point. This further implies that fractional impulsive system (1.1) has a unique solution, which accomplishes the proof.

Data dependence
Here, we present data-dependence of the solution of impulsive system (1.1). We follow the results studied given in [12,24].
Proof With the help of Theorem 2.1, we have Finally, for t ∈ [t k , δ k ], we have Thus, by the help of (4.2)-(4.4), xx(t) ≤ ζ .

Conclusion
In this article we have considered a fractional order impulsive IDE (1.1) for the existence of unique solution, data dependence, and stability results. We have used basic results from the fixed point theory and literature for fractional order calculus. The results are examined with the help of an expressive numerical example.