Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Abstract

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

1. Introduction

From the last few years, fractional differential equations () theory has gained significant attraction and importance. It arises naturally in various models in areas such as control theory, biology, nonlinear waves of earthquake, mechanics, signal processing, modeling the seepage flow in porous media, and in fluid dynamics, memory mechanism and hereditary properties of materials. Some recent existence and uniqueness () results of solutions for with initial as well as boundary conditions can be found in [1–8]. In fact, are the effective tools in real-world problems that motivate many researchers to work in this field, see [6, 9–20] and references cited therein.

Another important aspect in the qualitative theory of differential equations (), which is exclusively studied for integer-order , is Hyers–Ulam () stability and its various types. This stability concept was originated in 1940 from the question of Ulam [21], which was answered by Hyers [22]. Many researchers extended and generalized Hyers’s results in which the work of Rassias [23] is considered to be the first notable contribution. Many researchers studied and –Rassias stability of various functional equations, see [24–44] and references cited therein. This field got notorious attention when mathematicians started studying the stability for the solution of differential equations, initiated by Obloza [45, 46]. Motivated by the work of Obloza, various classes of integer-order ordinary differential equations were investigated [39, 47]. The idea was then extended for nonintegral-order differential equations; for some recent work, we refer to [48, 49]. As far as we know, only few researchers studied the different kinds of Ulam’s type stabilities for the coupled system of . For details, see [38, 50–52].

Nowadays, both Riemann–Liouville-type () and Caputo-type derivatives are introduced generally, and the impact of applying it in mathematical physics and equations associated with probability is exposed. The fractional integral that generalizes both - and Hadamard-type integrals into a single form was initiated by Katugampola [53]. Later on, in [54], new fractional derivative that generalizes the two derivatives was introduced by Katugampola.

Motivated by the work [54, 55], in this paper, we study the and stability of the following implicit switched coupled system of involving the Katugampola–Caputo () fractional derivative:where is a positive real number and and are the Katugampola–Caputo fractional derivatives of and , respectively. The functions are closed and bounded. Also, are nonlocal continuous functions.

2. Preliminaries

In this portion, we introduce some notions and preliminaries. Suppose denotes the Banach space () of continuous functions from into , defined by , endowed with norms as ; indeed, these are s’ under these norms, and hence, their product is also with , where are in .

Definition 1. (see [5]). Let , and the arbitrary order integral in the sense for a function iswhere the integral on the right-hand side (RHS) is pointwise defined on .

Definition 2. (see [54]). left-sided noninteger-order integral of the function on a closed interval of order is defined asThe corresponding fractional derivative to the above integral is given by

Definition 3. (see [5]). The noninteger-order derivative in the Caputo sense of on closed interval iswhere . In particular,Moreover, the integral on the RHS is pointwise defined on .

Lemma 1 (see [56]). Let , for , and the only one solution of has the formula , where , , .

Lemma 2 (see [56]). Let , for and , for some , , .

Definition 4. (see [57]). Consider . Let be two operators, and the operator systemis called stable if we can find constants such that, for each and for each solution of the inequalitieshold, then there exists a solution of system (7), which satisfies the inequalities

Lemma 3 (see [57], Theorem 4). Consider with operators such thatIf the spectral radius ofis less than one, then the fixed points corresponding to operational system (10) are stable.

3. Existence and Uniqueness of the Solution

In this section, we prove of system (1). We consider the following assumptions:

: suppose are continuous, and for all with , there exist such that

: there exist such thatfor all .

: suppose is continuous, and for all , there exist such that

: let with such thatwhere .

: let be continuous, and for all with , there exist such that

Theorem 1. Let and be continuous, and the solution ofis equivalent to

Proof. ConsiderApplying integral , we getUsing , we haveSimilarly,For the upcoming result, here, we define an operator.
: choose , with , , , , , , , and .Construct on as

Theorem 2. Let the conditions from to hold; then, system (1) has only one solution.

Proof. For any ,From (24), we haveTherefore,Furthermore, (24) givesIn a similar way, we getTherefore, (25) givesSo, .
Now, for and , we havewhereSince , therefore,Similarly, we getUsing (31), we getHere, , withThus, is a contraction. For the continuity and compactness of , take a sequence in with approaching to as approaches to in . So,implies , as . That is why is continuous on . Now, for the uniform boundedness of on , consider , and by using (31), we haveThus, is uniformly bounded on . Now, for the equicontinuity of the operator , take from with and . Since is bounded on , we can take . Thus, as . Using the same approach with , we haveCombining these inequalities, we get as ; hence, is relatively compact on . Thus, by Arzela–Ascolli theorem, is compact and continuous, so (1) has a unique solution.

Theorem 3. Under conditions to with , system (1) has a unique solution.

Proof. We define the operator bywhereFor and , we getHence, is a contraction, and by the assumption that , (1) has only one solution.

4. Stability

Now, we are analyzing different kinds of stabilities such as stability of the proposed system given in (1).

Theorem 4. Let the hypothesis from to hold true with the conditions , and if the matrix , then (1) is stable.

Proof. Proceeding from Theorem 3, for any and , we haveSimilarly,whereWriting together the above inequalities, we haveFrom this, we getwhere . As given, converges to 0; therefore, (1) is stable.

Example 1. Considerwhere . Set functions asTaking from and , we haveIn particular, let we take , and from the inequalities in (51), .
Thus, we have . Therefore, by Theorem 3, (5.1) has a unique solution. After calculations, we getUpon calculations, we obtained the eigenvalues 0.81675 and 0.4433, which show that is converging to zero. Using Theorem 4, the solution of (5.1) is stable.

5. Conclusion

In this manuscript, we used theorem and Banach contraction principle to achieve the sufficient conditions for of solutions to a nonlocal implicit switched system. With the help of assumptions, we proved the stability result for the couple system given in (1).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the China Postdoctoral Science Foundation (Grant no. 2019M652348), the Technology Research Foundation of Chongqing Educational Committee (Grant no. KJQN201900539), and the Natural Science Foundation of Chongqing Normal University (Grant no. 16XYY24).