Hyers–Ulam–Mittag-Leffler Stability for a System of Fractional Neutral Differential Equations

This article concerns with the existence and uniqueness for a new model of implicit coupled system of neutral fractional differential equations involving Caputo fractional derivatives with respect to the Chebyshev norm. In addition, we prove the Hyers–Ulam–Mittag-Leffler stability for the considered system through the Picard operator. For application of the theory, we add an example at the end. The obtained results can be extended for the Bielecki norm.


Introduction
Models of fractional differential equations (FDEs) have many applications in various fields of engineering and science such as mechanics, electricity, biology, chemistry, physics, and control and signal processing [1,2]. For the different materials and phenomena, the heredity characteristics are well explained by FDEs, and as a result, many research papers and books have been published in this field [3][4][5][6][7]. FDEs in which the highest fractional derivatives (FDs) of unknown term appear both with and without delays are known as neutral FDEs. In the last few years, the study of neutral FDEs have developed dramatically. is is due the fact that the qualitative behavior of the aforesaid equations are quite different from those of nonneutral FDEs. Neutral FDEs also play a key role and has many advantages. For instance, they give more better description over population fluctuations. Also, neutral FDEs with delay appear in models of electrical networks containing lossless transmission lines, for more details see [8]. Different attempts have been made for the investigation of solutions of fractional and neutral FDEs [9][10][11][12][13][14][15][16][17].
Another imperative and more remarkable area of research is committed to the stability analysis of the solutions for the FDEs and ordinary orders. Many targets are achieved in this regard, for some recent work we refer the reader to see [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Niazi et al. [36] investigated the existence, uniqueness (EU), and Hyers-Ulam-Mittag-Leffler (HUML) stability for the following neutral FDEs: where 1 < κ ≤ 2, 0 < σ < 1, c D κ 0 and c D σ 0 are the Caputo derivatives, and g: [0, 1] × C ξ × C ξ ⟶ R is continuous. Here, C ξ denotes the Banach space of all continuous functions If u ω ∈ C ξ , then is defined by As far as we know, results on EU and HUML stability for a coupled system of neutral FDEs have not been investigated by the researchers. Many real-world problems required to be modeled into a coupled system of FDEs as they cannot be described into a single fractional differential equation, see [37] and references cited therein.
Motivated by the abovementioned work, in this article, we study the EU and HUML stability of the following implicit coupled neutral FDEs system involving Caputo FDs. e proposed coupled system (CS) is given by where J � [0, 1], 1 < κ, σ ≤ 2 and c D κ 0 and c D σ 0 represent the Caputo FDs of orders κ and σ, respectively. e set of all is a Banach space denoted by C ξ , and hence their product is also a Banach space. e functions g, h:

Preliminaries
is section is concerned with some notions, definitions, and preliminary results used throughout the article.

Main Results
In this section, we provide results regarding the EU and HUML stability for the solution of the considered system on the compact interval J � [0, 1], using the PO and Henry-Gronwall lemma [41]. Suppose g, h: J × C ξ × C ξ ⟶ R be continuous and C ξ be a Banach space of all the functions where E κ (·) and E σ (·) represent the Mittag-Leffler (ML) function defined by 2 Discrete Dynamics in Nature and Society Definition 5. Extending the definition of Hyers-Ulam (HU) stability for a CS [43], we say that system (12) is en, (u, v) satisfies the following integral inequalities: en, Using the same technique, we can get □ Theorem 3. Let the following assumptions hold: Discrete Dynamics in Nature and Society , en, e solution is equivalent to us, Similarly, on considering we can obtain , 1], R) and C 2 ([− ξ, 1], R), respectively, represent the class of continuous and continuously differentiable functions from [− ξ, 1] to R. e norm is defined in such a way that the norm of each term depends on the derivatives of the fractional order of the other terms of Y ξ . e product Y ξ × Y ξ is also a Banach space with norm ‖(u, v)‖ � ‖u‖ + ‖v‖.
First, we show that K is a contraction mapping. It is clear that For ω ∈ J, we have

Remark
2. If we consider another space with modified Bielecki's norms defined by where 0 < ρ < λ, J � [0, 1], then the results similar to eorem 3 can be obtained for the solution of (4).

Conclusion
We gave sufficient conditions for the EU of the solutions to the nonlinear implicit CS of neutral FDEs. Our main tool was the Banach contraction principle. Likewise under specific conditions, we have found the HUML stability results for the solution of the CS given in (4).

Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.