Abstract

In this paper, we consider a conformable fractional differential equation with a constant coefficient and obtain an approximation for this equation using the Radu–Mihet method, which is derived from the alternative fixed- point theorem. Considering the matrix-valued fuzzy k-normed spaces and matrix-valued fuzzy H-Fox function as a control function, we investigate the existence of a unique solution and Hyers–Ulam-H-Fox stability for this equation. Finally, by providing numerical examples, we show the application of the obtained results.

1. Introduction

One of the important topics in mathematics and especially in mathematical analysis is fractional calculus. Here, we can refer to the fractional derivatives of Caputo, Riemann–Liouville, Grunwald–Letnikov, Marchaud, or Hadamard as fractional operators. It should be noted that these operators are the result of changes made to ODEs and PDEs over time. Various kinds of fractional derivatives have been discussed by Kilbas in [1] and Butzer et al. in [2]. Derivatives such as the Caputo, Riemann–Liouville, or Hadamard fractional derivatives have complex rules such as the law of chain. The researchers decided to find another derivative to get rid of these complications. Thus, a local fractional derivative containing a limit was proposed instead of a single integral called the consistent fractional derivative. These derivatives have many uses and properties. For example, they are used to extend Newton mechanics [36]. Researchers have recently introduced a new type of derivative that modifies conformable fractional derivatives. They have also studied the method change of the parameters for the conformable fractional differential equations by considering a regular fractional generalization of the Sturm–Liouville eigenvalue problem [79].

In this paper, we consider an MVkFB-space introduced in [10] and consider a modern class of the MVF control function based on the H-Fox functions. Our goal is to obtain an approximation for the conformable fractional differential equation using the alternative fixed-point theorem in MVkFB-spaces. The fuzzy control functions presented in this paper have a dynamic situation and can model new events, such as the COVID-19 disease, as explained in [11]. Using fuzzy controllers, the stability analysis of differential equations and integral equations can be studied.

We consider the following conformable FDE with constant coefficients:where is called the conformable fractional derivative (CFD) with a lower index of the function and , [1, 2].

The paper is organized as follows: In the second section, we present the basic definitions and concepts that are necessary to investigate the main results, and we also introduce the matrix-valued fuzzy H-Fox function as a control function. In the third section, using the alternative FPT, we prove the existence of a unique solution and the Hyers–Ulam-H-Fox stability for the conformable FDE in MVFkN-spaces, and at the end, as an application, we provide a numerical example.

2. Preliminaries

Definition 1. For a mapping , the CFD starting from of order is defined byIf on exists, then .

Remark 1. For a finite given , is –differentiable at . If , then .

Definition 2. For a mapping , the Hadamard fractional integral with the order and parameter is defined bywhere and in .

Lemma 1. Let . For the real-valued mapping and , the following relationship is always established:

Theorem 1. By considering the Mittag-Leffler map, we obtain the following equation:

Suppose that , then we obtain the following equation:

Proof. Using Remark 1, we haveNext, we study the mapping .

Theorem 2. If for equation (1), the mapping is a solution, thus we have

Proof. For any solution of (1), it should be as follows:where is an unknown continuously differentiable function. From (9) and Remark 1, we getAs a result, we obtain the following equation:From (11) and Lemma 1, we getwhere .
By using (9) and (12), the desired result is obtained. □

Definition 3. A mapping is said to be the solution of (1) if satisfies and . Thus, we obtain the following equation:

Definition 4 (see [12]). The multivariate Mittag-Leffler (MM-L) function is defined by the following series representation:where for .

Definition 5 (see [1315]). According to a standard notation, the Fox function is defined aswhere is a suitable path in the complex plane to be disposed later. andwith , , , . An empty product, when it occurs, is taken to be one, so we getThe function is, in general, multivalued, but it can be made one-valued on the Riemann surface of by choosing a proper branch. We also note that when and are equal to 1, we obtain G functions . The above integral representation of functions, by involving products and ratios of Gamma functions, is known to be of the Mellin–Barnes integral type. A compact notation is usually adopted for (15).Here, we assume , , , , , and .
Assume thatin whichAlso, denotes that and ; for every . We define in where . Note that, is and is .

Definition 6 (see [1618]). A mapping is called a GTN if the following conditions are met:(a) (boundary condition)(b) (commutativity)(c) (associativity)(d) (monotonicity)If for every and each sequences and converging to and , we getand we conclude that the continuity of on (CGTN).(1)Define , such thatthen is CGTN (minimum CGTN).(2)Define , such thatthen is CGTN (product CGTN).(3)Define , such thatthen is CGTN (Lukasiewicz CGTN).Numerical examples of CGTN are as follows:We get the following equations:By considering the matrix-valued fuzzy function (MVFF) , then we have following conclusions:(i)It is a left continuous and increasing function.(ii) for any and .(iii)For MVFFs and , the relation “≺” is defined as follows:

Definition 7. Let be a CGTN, be a vector space, and be a matrix-valued fuzzy set (MVFS). Triple is called a matrix-valued fuzzy -normed space (MVFkN-space) if
(MVFkN1)  if and only if are linearly dependent and ;
(MVFkN2)  for all and with ;
(MVFkN3)  for all and any ;
(MVFkN4)  for any .
When an MVFkN-space is complete, we denote it by an MVFkB-space. Using the concept of the H-Fox function, we define an MVF H-Fox function as a control function in the MVFkN-spaces as follows:For an MVF H-Fox function , we have(1)It is a left continuous and increasing function for positive values.(2).(3)For and also, for the matrix-valued fuzzy function , we haveand also, we have(1).(2)We can easily show that for , .(3)We show thatThen, we have(4)We show thatSuppose that , then we have