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 [3–6]. 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 [7–9].
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 [13–15]). 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 [16–18]). 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 that then is CGTN (minimum CGTN).(2)Define , such that then is CGTN (product CGTN).(3)Define , such that then 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 that Then, we have(4)We show thatSuppose that , then we haveTherefore, we have Consequently, iffor , then is an MVFkN-space. From now on, we assume .
Theorem 3 (see [14, 19]). We consider the -valued metric space . For , we consider the self-map on such thatwhere is a Lipschitz constant. Let . Therefore, we have following two ways:(i) or(ii)we can find such that: If condition (ii) is true for us, then we have following conclusions:(1)The fixed point of is the convergence point of the sequence (2)In the set is the unique fixed point of (3) for every
Definition 8. Let function be an MVF function. Equation (1) is said to be Hyers–Ulam-H-Fox stable, and if is a given differentiable function, we obtain the following equation:for , and we can find a solution of (1) such that for some , which is as follows:
Remark 2. Let be a solution of inequality (39). Then, is a solution of the following integral inequality:for every and .
3. Hyers–Ulam-H-Fox Stability for the Conformable Fractional Differential Equation
Now, we use the fixed-point method based on Theorem 3 to show that equation (1) is Hyers–Ulam-H-Fox stable [14] in the MVFkB-space with MVFF [20–22].
We set the set as follows:and we consider the mapping as
Theorem 4. is a complete -valued metric space.
Proof. We have if and only if . Assume that , then we havesofor all . We assume to zero in the above inequality, and we getThus, for every and vice versa. Also, we have for every . Now, let and . Then, we havefor every . Then, we haveso . Thus, . To show the completeness of , we suppose that is a Cauchy sequence in . Let . Assume that and are arbitrary and consider such that . For , we choose such thatThen, we haveSo we have the following equation:which implies that the sequence is Cauchy in a complete space on a compact set . Then, it is uniformly convergent to the mapping . By uniform convergence property, we conclude that is differentiable, i.e., an element of , and then, is complete. □
Now, we can investigate Hyers–Ulam-H-Fox stability and get an approximation for the solution of conformable FDE (1). In [23–48], there are new stability problems that one can prove them by our method.
Theorem 5. Let be an MVFkB-space and consider the constant coefficient , , and . Then, we have
Suppose that the following conditions hold:(1)For continuous function , we obtain(2)MVFF satisfying the following equation:
Let be a differentiable function satisfying the following equation:
Then, there is a unique solution for (1) such thatfor every and .
Proof. We setand introduce the -valued metric on asBy Theorem 4, we have that is a complete -valued metric space.
Step 1. We define from to by the following equation:for , and we show is a strictly contractive mapping.
Let and consider the coefficient with ; thus, we havefor all and . Applying (MVFkN2) and (MVFkN3), we imply thatwhich implies thatsowhere ; therefore, is a contraction mapping.
Step 2. We will show that .
Let , we haveConsequently, we obtain the following equation:for every and . Then, we have .
Therefore, all the conditions of Theorem 1 hold. Then, we have(1)The sequence converges to a fixed point such as .(2)The unique element is in the set and is the unique fixed point of , which means or equivalently as shown in the following equation: where . Since is a differentiable function, by the CFD and according to (65) and Lemma 1, we have(3)Using inequality (64), we get Thus, equation (1) has the Hyers–Ulam-H-Fox stability property.Now, we show the uniqueness of the obtained point. For convenience, we consider the following equation:and let be another differentiable function satisfying equation (66), and this means that the following equation holds:We are ready to prove that is a fixed point of and . Using equation (69), we get . Now, we show that . Let , , and from equation (69), we getThen, we have
4. Example
Now, we provide numerical examples according to the results obtained.
Example 1. We consider the following conformable FDE:where , for . Also, in this equation, . Let for mapping and the MVF control function , we have(1)(2)If be a differentiable function such thatthen is a solution of the inequalityand thus, we can find a unique differentiable function from (72) such that for each , we havefor any . ThereforeIn Figures 1–8, the exact solution of conformable FDE (72) for is demonstrated.
Example 2. We consider the following conformable FDE:where , for . Also, in this equation, . Let for mapping and the MVF control function , we have If be a differentiable function such thatthen is a solution of the inequalityand thus, we can find a unique differentiable function from (79) such that for each , we havefor any . ThereforeFigures 9–11 show the graphs related to the exact solution of conformable FDE (79) for .
5. Conclusion
In this paper, we introduced the H-Fox function as a matrix value fuzzy control function, and by considering the matrix-valued fuzzy k-normed spaces, we investigated the stability of a class of conformable fractional differential equations with a constant coefficient. The alternative fixed-point theorem is used in different spaces. Therefore, we used the Radu–Mihet method, which is derived from the alternative fixed-point theorem, to investigate the existence of a unique solution and the Hyers–Ulam-H-Fox stability for the conformable fractional differential equations in the matrix-valued fuzzy k-normed spaces. The Riemann–Liouville fractional derivative and the Caputo fractional derivative have properties that cause high incompatibility and computational complexity in fractional calculations. To remove these obstacles and overcome these inconsistencies, an adaptable fractional derivative has been introduced, which we use because of these advantages.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Government Of The Basque Country For The ELKARTEK21/10 KK-2021/00014 And ELKARTEK22/85 Research Programs, respectively.