Existence and stability of impulsive coupled system of fractional integrodifferential equations

Abstract: In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained bymeans of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.


Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order, but with this definition, many interesting questions will arise; for example, if the first derivative of a function gives you the slope of the function, what is the geometrical meaning of half derivative? In half order, which operator must be used twice to obtain the first derivative? The early history of these questions goes back to the birth of fractional calculus in 1695 when Gottfried Wilhelm Leibniz suggested the possibility of fractional derivatives for the first time [1].
Fractional differential equations (FDEs) have recently gained much importance and attention. It is the extension of classical calculus. FDEs as well as fractional integrodifferential equations appear naturally as generalizations to existing models with integer derivatives and they also present new models for many applications in physics, control theory, chemistry, biology, electrical circuits, mechanics, signal and image processing, heat conduction, computer analysis and economics etc., reader is referred to [2][3][4][5][6][7]. For example, in the last three fields, some important considerations such as modeling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here.
The study of coupled system (CS) of FDEs has also attracted some attention. Because mathematical models of various phenomena in the field of biology, physics and psychology etc. are in the form of CS of differential equations (DEs). For the study of CS of FDEs, we refer the reader to [26][27][28][29][30][31].
Another important class of DEs is known as impulsive differential equations (IDEs). This class plays the role of an effective mathematical tools for those evolution processes that are subject to abrupt changes in their states. There are many physical phenomena that exhibit impulsive behavior such as the maintenance of a species through periodic stocking or harvesting, mechanical systems subject to impacts, the thrust impulse maneuver of a spacecraft and the function of heart, we recommended [32][33][34][35][36][37][38][39][40][41] for more details on the theory of IDEs. It is well known that in the evolution processes the impulsive phenomena can be found in many situations. For example, change of the valve shutter speed in its transition from open to closed state [42], operation of a damper subjected to the percussive effects [43], disturbances in cellular neural networks [44,45], relaxational oscillations of the electro mechanical systems [46], percussive systems with vibrations [47], using the radial acceleration, control of the satellite orbit [48], dynamic of system with automatic regulation [48], fluctuations of pendulum systems in the case of external impulsive effects [49], price fluctuations in commodity markets [50] and so on.
In this manuscript, we study four different types of Ulam stability for implicit FDEs with impulses and initial conditions, which are HU stability, generalized HU stability, HU-Rassias stability and generalized HU-Rassias stability.
. . , m}. The rest of this paper is arranged as follows: In section 2, we present some basic notions needed to prove our main results. In section 3, we setup some adequate conditions for the EUSs, by applying some standard fixed point principles to the proposed system (1.1) and (1.2), respectively. In section 4, we setup applicable results under which the solution of the considered problems (1.1) and (1.2), respectively, fulfills the conditions of different kinds of Ulam stability. The establish results are illustrated by examples in section 5.

Supplementary results
The following definitions and lemmas are adopted from [18].
Definition 2.1. The integral of a function u ∈ L 1 (J, R) of order r ∈ (0, ∞) is defined by where ρ = [r] + 1 in which [r] represents the integer part of r. Lemma 2.3. For r > 0, the following result hold: where n = [r] + 1.
is given by where n = [r] + 1.

Existence and uniqueness
Here we present our result about the existence of at least one solution to considered problem (1.1).
Theorem 3.1. Let 0 < r ≤ 1 and α ∈ M be a continuous function, then a function x ∈ M is solution to problem if and only if x satisfies Proof. Let x be the solution of problem (3.1), then using Lemma 2.4, for each t ∈ J 0 , we have Using the initial condition x(0) = h(x), we get from (3.3) Similarly, for t ∈ J 1 , we have we get Putting for x(t 1 ), (3.5) implies Generalizing in this way, for t ∈ J i , we have Using (3.4) and (3.6), we obtain (3.2). Conversely, let x be the solution of integral equation (3.2), then the rth order derivative of (3.2) will lead us to the first equation in (3.1). Further, it is easy to obtain the initial and impulsive conditions of (3.1).

Corollary 3.2.
In light of Theorem 3.1, problem (1.1) has the following solution: For simplicity, we use the following notation: Let M = PC(J, R) be a Banach space endowed with norm If x is a solution of the problem (1.1), then To transform problem (1.1) into a fixed point problem, we define an operator T : For our next results, we put the following hypotheses. Assume that -[A 1 ] there exist constants M 1 > 0 and 0 < N 1 < 1 so that for each t ∈ J and for all u, u ∈ M and w, w ∈ R, the following relation holds Similarly, there exist constants M 2 > 0 and 0 < N 2 < 1 so that for each t ∈ J and for all u, u ∈ M and w, w ∈ R the following relation holds with a * 1 = sup t∈J a 1 (t), b * 1 = sup t∈J b 1 (t), c * 1 = sup t∈J c 1 (t) < 1; Similarly, there exist bounded functions a 2 , b 2 , c 2 ∈ M so that 6 ] for each u ∈ R, the function I i : R → R; i = 1, 2, . . . , m are assumed to be continuous and for constants K, L > 0, the inequality |I i (u(t i ))| ≤ K|u(t)| + L, holds.
The existence of solution for the problem (1.1) is based on Theorem 2.5. Proof. Let the operator T is defined in (3.7). We need to prove that (1.1) has at least one solution.
Let the operator T be continuous. Consider a sequence {xn} so that xn → x ∈ M, t ∈ J, then (3.8) wherexn ,x ∈ M and are given bỹ By utilizing [A 1 ], we have Then Thus, using hypothesis [A 1 ]-[A 3 ] and (3.9), inequality (3.8) implies Since for each t ∈ J, the sequence xn → x as n → ∞, hence by Lebesgue dominated convergence theorem, So T is continuous on J. Now, we have to show that T is bounded in M. So, for any ℘ > 0, there is R E > 0, so that Therefore, we get (3.11) By using (3.11) and [A 4 ]-[A 6 ], (3.10) becomes Thus Similarly, for t ∈ J 0 we can verify that Now we have to show that the operator T is equicontinuous in E. Suppose t 1 , t 2 ∈ J i , i = 1, 2, . . . , m so that 0 < t 1 < t 2 < T and let x ∈ E, then Obviously, the right-hand side of inequality (3.12) tends to zero as t 1 → t 2 . Therefore, Similarly, we can show for t ∈ J 0 . Thus, T is equicontinuous and therefore completely continuous. Finally, we consider a set Ω ⊂ M, which is defined as We need to prove that set Ω is bounded. Suppose x ∈ Ω, so that Then Taking norm on both sides, we have ‖x‖ M ≤ Q. Also, for t ∈ J 0 , we can show that ‖x‖ M ≤ Q. Hence, Ω is bounded. Thus, by Schaefer's fixed point theorem, T has at least one fixed point. So the considered problem (1.1) has at least one solution in M.
The next result is based on Theorem 2.6 and concerned with the uniqueness of solution for (1.1).

Theorem 3.4. If the hypothesis [A 1 ]-[A 3 ] along with the inequality
are satisfied, then problem (1.1) has a unique solution.
Proof. Suppose x, x ∈ M and for t ∈ J i , i = 1, 2, . . . , m, we have wherex,x ∈ M are given bỹ Thus Now taking norm on both sides, we have Similarly, for x, x ∈ M and t ∈ J 0 , we get Hence, T is a contraction. Which implies T has a unique fixed point, so the problem (1.1) has a unique solution.
After this, we consider a CS of nonlinear implicit FDEs with impulsive conditions of (1.2).
Theorem 3.5. The system: has a solution (x, y) if and only if Proof. The proof is similar as given in Theorem 3.1.
Similarly, for t j ∈ J so that t 1 < t 2 < · · · < tn and Proof. If (x, y) is a solution of the system (1.2), then it is a solution of (3.19). Conversely, let (x, y) is a solution of (3.19), then Thus (x, y) is a solution of (1.2).
For convenience, we use the following notations: The system (1.2) can be transformed into a fixed point problem.
Split the operator T into two parts as T = F + G with F = (Fr, Fs) and G = (Gr, Gs), where Gs(y)(t) = g(y) + n ∑︁ j=1 I j (y(t j )).
In the similar manner, we have Therefore, we get (3. Thus Similarly, for t ∈ J 0 we can verify that ‖Try‖ X ≤ η. Hence ‖Tr(x, y)‖ X ≤ η.
Which implies that T(B) ⊆ B. Now we show that G is contraction. For any (x, y), (x, y) ∈ B, we have The contraction of G follows from the assumption that A h + mA I i < 1 and Ag + nA I j < 1.
In the similar manner, we have (3.34) By [H 6 ], for t ∈ J j , j = 1, 2, . . . , n, we have Therefore, we get (3.35) Using (3.35) in (3.34) and after simplification, we get Hence Thus Which implies that F is uniformly bounded on B. Take a bounded subset C of B and (x, y) ∈ C. Then for t 1 , Therefore, we get (3.37) Using (3.37) in (3.36) we see that the right-hand side of (3.36) tends to zero as t 1 → t 2 . Thus Similarly, |Fry(t 1 ) − Fry(t 2 )| → 0 as t 1 → t 2 .

Using hypothesis [H 1 ], [H 3 ] and [H 4 ], inequality (3.40) implies
Taking norm on both sides, we have Similarly, for x, x ∈ X and t ∈ J 0 , we get In the same manner, we can obtain Similarly, for y, y ∈ X and t ∈ J 0 , we get So from (3.41) and (3.42), we get Now, suppose x, x ∈ X and for t ∈ J j , j = 1, 2, . . . , n, we have Using [H 2 ], we have Thus Taking norm on both sides, we have Similarly, for x, x ∈ Y and t ∈ J 0 , we get In the same manner, we can obtain Similarly, for y, y ∈ Y and t ∈ J 0 , we get So from (3.45) and (3.46), we get Hence, it follows that Which implies that T is contraction, hence it has a unique fixed point.

Ulam stability results
In this section, we investigate HU stability and its various kinds for problem (1.1). The following definitions are adopted from [7]. For x ∈ M, ϵr > 0, ϕr ≤ 0 and a nondecreasing function ψr ∈ C(J, R+), the following set of inequalities satisfy: and Remark 4.7. A function x ∈ M is a solution of the inequality (4.2) if ∃ a function Φ ∈ M and a sequence Φ i (which depends on x) so that i) |Φ(t)| ≤ ψr(t), |Φ i | ≤ ϕr, ∀ t ∈ J, i = 1, 2, . . . , m;  Proof. Let x ∈ M be any solution of the inequality (4.1) and let x * ∈ M be the solution of the following problem: ∆x * (t i ) = I i (x * (t i )), i = 1, 2, . . . , m.
By Theorem 3.1, for t ∈ J i , we have wherex ∈ C(J, R) is given bỹ Since x is a solution of the inequality (4.1), hence by Remark 4.6, we have By taking norm and simplification, we get From which, we obtain Similarly, for t ∈ J 0 , we have Combining (4.6) and (4.7), for t ∈ J, we have Proof. Let x ∈ M be any solution of the inequality (4.3) and let x * ∈ M be the solution of the problem: From the proof of Theorem 4.10, we have Hence, using [A 1 ]-[A 3 ] and part (i) of Remark 4.8, above inequality, we have Which yields Similarly, for t ∈ J 0 , we have Combining (4.8) and (4.9), for t ∈ J, we have Thus Therefore, problem (1.1) is HU-Rassias stable. Similarly, we can show that it is generalized HU-Rassias stable.
In order to obtain the connections between the HU-Rassias stability concepts we introduce the following hypothesis.

Conclusion
We have presented some existence and uniqueness results for an impulsive initial value problem of coupled fractional integrodifferential systems involving the Caputo type fractional derivative. The proof of the existence results is based on the nonlinear alternative of Schaefer's and Krasnoselskii's fixed point theorem, while the uniqueness of the solution is proved by applying the Banach contraction principle. We have also given the notion of Hyers-Ulam stability for our problem and have given sufficient conditions for EUS and Hyers-Ulam stability. This work provides a base to the study of EUS and different sorts of stabilities for the fractional integrodifferential equations with impulsive initial condition.