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

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.


Introduction
From the last few years, fractional differential equations (FDES) 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 (EU) results of solutions for FDES with initial as well as boundary conditions can be found in [1][2][3][4][5][6][7][8]. In fact, FDES are the effective tools in real-world problems that motivate many researchers to work in this field, see [6,[9][10][11][12][13][14][15][16][17][18][19][20] and references cited therein.
Another important aspect in the qualitative theory of differential equations (DES), which is exclusively studied for integer-order DES, is Hyers-Ulam (HU) stability and its various types. is 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 HU and HU-Rassias stability of various functional equations, see  and references cited therein. is field got notorious attention when mathematicians started studying the HU 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]. e 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 FDES. For details, see [38,[50][51][52].
Nowadays, both Riemann-Liouville-type (RL) and Caputo-type derivatives are introduced generally, and the impact of applying it in mathematical physics and equations associated with probability is exposed. e fractional integral that generalizes both RLand 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 EU and HU stability of the following implicit switched coupled system of FDE involving the Katugampola-Caputo (KC) fractional derivative: where σ is a positive real number and σ c D α u(σ) and σ c D α v(σ) are the Katugampola-Caputo fractional derivatives of u(σ) and v(σ), respectively. e functions f, g: J × X × X × ⟶ X are closed and bounded. Also, ψ and φ are nonlocal continuous functions.
Definition 1 (see [5]). Let α > 0, and the arbitrary order integral in the RL sense for a function p: J ⟶ R is where the integral on the right-hand side (RHS) is pointwise defined on R + .
Definition 2 (see [54]). KC left-sided noninteger-order integral ρ I α a + f of the function f on a closed interval [a, b] of order α is defined as e corresponding KC fractional derivative to the above integral is given by Definition 3 (see [5]). e noninteger-order derivative in the Caputo sense of p on closed interval [a, b] is where n − 1 � [α]. In particular, Moreover, the integral on the RHS is pointwise defined on R + .

Existence and Uniqueness of the Solution
where σ ∈ J.
(H 5 ): let ψ, φ: J × X × X ⟶ X be continuous, and for all Applying integral ρ I α , we get Using Similarly, For the upcoming result, here, we define an operator.

HU Stability
Now, we are analyzing different kinds of stabilities such as HU stability of the proposed system given in (1). Proof. Proceeding from eorem 3, for any (μ, ]), (μ * , ] * ) ∈ X and ς ∈ J, we have

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

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.