Abstract

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.

1. Introduction

Models of fractional differential equations 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 , and as a result, many research papers and books have been published in this field [3–7]. in which the highest fractional derivatives of unknown term appear both with and without delays are known as neutral . In the last few years, the study of neutral have developed dramatically. This is due the fact that the qualitative behavior of the aforesaid equations are quite different from those of nonneutral . Neutral also play a key role and has many advantages. For instance, they give more better description over population fluctuations. Also, neutral 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 [9–17].

Another imperative and more remarkable area of research is committed to the stability analysis of the solutions for the and ordinary orders. Many targets are achieved in this regard, for some recent work we refer the reader to see [18–35]. Niazi et al. [36] investigated the existence, uniqueness (), and Hyers–Ulam–Mittag-Leffler () stability for the following neutral :where , are the Caputo derivatives, and is continuous. Here, denotes the Banach space of all continuous functions with norm

If , then is defined by

As far as we know, results on and stability for a coupled system of neutral have not been investigated by the researchers. Many real-world problems required to be modeled into a coupled system of 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 and stability of the following implicit coupled neutral system involving Caputo . The proposed coupled system () is given bywhere and represent the Caputo of orders respectively. The set of all continuous functions with norms and is a Banach space denoted by , and hence their product is also a Banach space. The functions are continuous. If are continuous, then , and the functions are presented by .

2. Preliminaries

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

Definition 1. (see [38]). The order Riemann–Liouville integral for iswhere the integral is point-wise convergent.

Definition 2. (see [38]). Suppose be a given function on , the order Caputo derivative of is stated bywith . Furthermore, if domain of is and , thenwhere .

Theorem 1 (see [39]). Let , then , which implies has the formulawhere , , .

Definition 3. (see [40]). For a metric space , an operator is said to be Picard operator if there is so that(a) where .(b)The sequence has the limit .

Definition 4. (see [40]). For an ordered metric space , if is increasing with , then , implies and implies .

Lemma 1 (see [41]). Let . If , is increasing and there is so thatThen,

Lemma 2 (see [42]). For and , we have

3. Main Results

In this section, we provide results regarding the and stability for the solution of the considered system on the compact interval , using the and Henry–Gronwall lemma [41]. Suppose be continuous and be a Banach space of all the functions which are continuous with norms and . Consider the systemwhere and . For , we have the following inequalities:where and represent the Mittag-Leffler function defined by

Definition 5. Extending the definition of Hyers–Ulam stability for a [43], we say that system (12) is stable with respect to if there is so that for every and each solution of the inequalities (13) there is a unique solution so that

Remark 1. Functions are, respectively, the solutions of the above inequalities there is so that(1), ;(2)

Theorem 2. For , if are, respectively, the solutions ofThen, satisfies the following integral inequalities:

Proof. From Remark 1, we haveThen,So, we haveUsing the same technique, we can get

Theorem 3. Let the following assumptions hold: , , ;  There are , so that and  

Then,(i)System (12) has a unique solution in ;(ii)roblem (4) is stable.

Proof. (i)Consider The solution is equivalent to Thus, Similarly, on considering we can obtain Let and define norm on by and define norm on by , where and , respectively, represent the class of continuous and continuously differentiable functions from to . The 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 . The product is also a Banach space with norm . Define operators by First, we show that is a contraction mapping. It is clear that For , we have where and . Thus, Also, So, we obtain This shows that is a contraction operator.(ii)Next, let be the approximate solution of (13) and be the unique solutions of system (12), that is,thenFor , we haveAlso, for , we haveNow,Similarly, we can getLet and in view of inequalities (40) and (41) take , where are defined byWe will show that is and consequently need to show that is a contraction. ConsiderMoreover,Thus, we getTherefore, , showing that is a contraction operator. By Definition 3, is a and , thenWe try to show that are increasing, and for this, take and Thus,Similarly, we can getSo that the solution is increasing and . AlsoThus, by using Gronwall’s lemma we havewhere . In the same waywhere . Particularly, if and , then by (40) and (41), we have and ; therefore, by Gronwall’s lemma we obtainThus, from the above we haveHence, it follows that (4) is stable.