Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

This study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.


Introduction
The topic of fractional differential equations (FDEs) has attracted the interest of researchers from various disciplines thanks to it being considered a useful gizmo in modeling the dynamics of various physical systems and their applications in many fields of applied sciences, engineering and technical sciences, etc. For further details, we refer the readers to [2][3][4][5].
On the other hand, the study of coupled systems involving (FDEs) is additionally important as intrinsically systems occur in various problems of applied nature. For a few theoretical works on coupled systems of (FDEs), we refer to a series of papers [27][28][29][30][31].
© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
The topic of stability of systems is one among the foremost important qualitative characteristics of a solution. But as far as we know, this is often the primary work with regard to a tripled system of weighted fractional differential equations with time delay.
Tripled fractional boundary systems may be a generalization of coupled fractional systems as they are governed by three associated differential equations with three conditions [32,33].
By a solution of system (1.1), it is meant that there is a sequence ς = (ς 1 , ς 2 , ς 3 ) satisfying system (1.1) on (0, b]. The major contribution of this paper is to derive equivalent fractional integral equations to the (TSWFDEs) and to establish the existence of a unique solution and Ulam-Hyers-Mittag-Leffler stability results for (TSWFDEs) with respect to Chebyshev and Bielecki norms with time delay. The Picard operator method and the Banach fixed point theorem are the important tools used to prove our main result. To the best of our observation, there is no analytical literature on studying the existence of tripled systems of fractional differential equations (TSWFDEs). This paper is the first work to study the existence of a unique solution and an Ulam-Hyers-Mittag-Leffler stability result for (TSWFDEs) with respect to Chebyshev and Bielecki norms with time delay. This paper is systematized as follows: In Sect. 2, we render the rudimentary definitions and prove some lemmas that are applied throughout this paper, also we present the concepts of some fixed point theorems. In Sect. 3, we prove the existence of unique solutions and (UHML) stability results of system (1.1) under Chebyshev and Bielecki norms. In Sect. 4, we give a pertinent example to illustrate our results. Concluding remarks about our results are given in the last section.

Preliminaries
In this part, we give important definitions and auxiliary lemmas that are pertinent to our main results. Let respectively. Clearly, E and are the Banach spaces with the above norms. Let E * := E × E × E and * := × × be the product spaces with the norms respectively.
, and ϕ is a strictly increasing function on [0, b].
In this paper, due to κ ∈ (0, 1), then = ς} is the fixed point set of T, and the sequence (T n (ς 0 )) n∈N converges to ς * for all ς 0 ∈ X.
be an ordered metric space, and let T : if and only if there exists a function wη i ∈ such that

Lemma 2.12
satisfies the following integral inequality: Then, in the light of Remark 2.9 and Lemma 2.11, we have Thus, we have Let us consider the continuous operator G :

Main results
In this section, we prove the existence of a unique solution and a (UHML) stability result for system (1.1) with respect to Chebyshev and Bielecki norms with time delay. For our analysis, the following hypotheses should be satisfied.
Proof Define a closed ball set as and In order to examine the existence of a unique solution by means of the Banach fixed point theorem, we only prove that the operator G(ς) defined by (2.5) has a fixed point in . For this purpose, we split the proof into the following steps.
Step (3): G(ς) is a contraction in P ζ . In this step, we will show that the operator G(ς) is a contraction mapping on P ζ with respect to the norm (ς 1 , Case (1): It follows that Case (2): For (ς 1 , ς 2 , ς 3 ), (v 1 , v 2 , v 3 ) ∈ P ζ , κ ∈ (0, b] and by (H 2 ), we obtain It follows that Thus, in light of the above cases, for all κ ∈ [-r, b], we get Thus, the operator G(ς 1 , ς 2 , ς 3 ) is a contraction mapping on with respect to the norm So, by the above steps and the Banach fixed point theorem, we deduce that system (1.1) has a unique solution in .
For i = 1, 2, 3, we get Next, we prove that the solution (y * 1 , y * 2 , y * 3 ) is increasing. Let σ = max{σ 1 , σ 2 , σ 3 }, where σ i := min s∈(0,b] [y * i (s) + y * i (h(s))] ∈ R + , i = 1, 2, 3, then for all 0 ≤ κ 1 < κ 2 ≤ b, we have Therefore y * i is increasing for all i = 1, 2, 3, and consequently (y * 1 , y * 2 , y * 3 ) is increasing too. Due to h(κ) ≤ κ, we get y * i (h(κ)) ≤ y * i (κ) and hence In particular, if where U i is an increasing Picard operator. As a result, we get It follows that As a result, we get Hence, equation ( Clearly, B is a Banach space with the following Bielecki norm: then system (1.1) has a unique solution (ς 1 , Proof In order to prove the uniqueness of solution, by means of the Banach fixed point theorem, we only prove that the operator G(ς 1 , ς 2 , ς 3 ) defined by (2.5) has a fixed point in with respect to Bielecki's norm. For this purpose, we divided the proof into the following steps.
Step (1): The continuity of a function f i implies that the operator G(ς 1 , ς 2 , ς 3 ) is continuous too.
Step (3): G(ς 1 , ς 2 , ς 3 ) is a contraction. In this step, we need only to prove that the operator G(ς 1 , ς 2 , ς 3 ) is a contraction mapping on with respect to the Bielecki norm B.
The proof of UHML stability is just like in Theorem 3.1, so we omit it here.

Conclusion
We have obtained the existence of unique solutions and Ulam-Hyers-Mittag-Leffler stability results for the solution of a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. [1] with respect to Chebyshev and Bielecki norms and time delay based on the reduction of fractional differential equations to integral equations. We employed the Picard operator method and fixed point theorems to obtain our results. To the best of our observation, there is no analytical literature on studying the existence of tripled systems of fractional differential equations. This paper is the first work to study existence of a unique solution and an Ulam-Hyers-Mittag-Leffler stability result for (TSWFDEs) with respect to Chebyshev and Bielecki norms with time delay. We trust the reported results here will have a positive impact on the event of further applications in engineering and applied sciences.