# Existence of a Unique Solution and the Hyers–Ulam-H-Fox Stability of the Conformable Fractional Differential Equation by Matrix-Valued Fuzzy Controllers

Authors: Zahra Eidinejad; Reza Saadati; Tofigh Allahviranloo; Farzad Kiani; Samad Noeiaghdam; Unai Fernandez-Gamiz
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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:(1)Dωafϖ−ϑ=ϱfϖ−ϑ+Kϖ−ϑ,fϖ−ϑ,ϖ,ϑ∈a,b,0<ω<1,fa=fa,where Dωaf is called the conformable fractional derivative (CFD) with a lower index a of the function f and T=a,b, K∈CT×ℝ,ℝ[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 mappingf:a,∞⟶ℝ, the CFD starting from a of order ω is defined by(2)Dωafϖ−ϑ=limi⟶0fϖ−ϑ+iϖ−ϑ−a1−ω−fϖ−ϑi,ϖ−ϑ>a,0<ω<1.
If ona,bDωafϖ−ϑ exists, then Dωafa=limϖ−ϑ⟶a+Dωafϖ−ϑ.Remark 1.
For a finite givenDωafϖ−ϑ0, f is ω–differentiable at ϖ−ϑ0. If f∈C1a,∞,ℝ, then Dωafϖ−ϑ=ϖ−ϑ−a1−ωf′ϖ−ϑ.Definition 2.
For a mappingfϖ−ϑ, the Hadamard fractional integral with the order 0<ω<1 and parameter s∈ℝ is defined by(3)Iϖ−ϑω,saHfϖ−ϑ=1Γω∫aϖ−ϑƛtslogtƛω−1fƛƛdƛ,where ϖ−ϑ∈a,b and a≤b in ℝ.Lemma 1.
Letf∈C1a,∞. For the real-valued mapping f and ϖ−ϑ>a,0<ω<1, the following relationship is always established:(4)Iϖ−ϑω,saHDωafϖ−ϑ=fϖ−ϑ−fa.Theorem 1.
By considering the Mittag-Leffler map, we obtain the following equation:(5)Eτϖ−ϑ−aωωϱ=∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ,τ>0.Suppose thatZϖ−ϑ=Eτϖ−ϑ−aω/ωϱ, then we obtain the following equation:(6)DωaZϖ−ϑ=ϱZϖ−ϑ.Proof.
Using Remark1, we have(7)DωaZϖ−ϑ=ϖ−ϑ−a1−ωϖ−ϑZϖ−ϑbϖ−ϑ=ϖ−ϑ−a1−ωϱϖ−ϑ−aω−1∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ=ƛ∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ=ϱZϖ−ϑ.
Next, we study the mappingf.Theorem 2.
If for equation (1), the mapping f∈CT,ℝ is a solution, thus we have(8)fϖ−ϑ=fa∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ+1Γω∫aϖ−ϑ∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ−aω/ω−ςϱΓ1+τςϱƛtslogtƛω−1Kƛ,fƛƛdƛ.Proof.
For any solution of (1), it should be as follows:(9)fϖ−ϑ=∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱcϖ−ϑ,where cϖ−ϑ is an unknown continuously differentiable function. From (9) and Remark 1, we get(10)Dωafϖ−ϑ=Dωa∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱcϖ−ϑ=ϖ−ϑ−a1−ω∑ς=0∞ϖ−ϑ−aω/ωkϱΓ1+τςϱϖ−ϑ−aω−1cϖ−ϑ+∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱϖ−ϑ−a1−ωc′ϖ−ϑ=ϱcϖ−ϑ∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ+∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱDωacϖ−ϑ=ϱfϖ−ϑ+∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱDωacϖ−ϑ.
As a result, we obtain the following equation:(11)Dωacϖ−ϑ=∑ς=0∞ϖ−ϑ−aω/ω−ςϱΓ1+τςϱKϖ−ϑ,fϖ−ϑ.
From (11) and Lemma 1, we get(12)cϖ−ϑ=ca+1Γω∫aϖ−ϑ∑ς=0∞ƛ−ccω/ω−ςϱΓ1+τςϱƛtslogtƛω−1Kƛ,fƛƛdƛ,where ca=fa=fa.
By using (9) and (12), the desired result is obtained. □Definition 3.
A mappingf∈C1T,ℝ is said to be the solution of (1) if f satisfies Dωafϖ−ϑ=ϱfϖ−ϑ+Kϖ−ϑ,fϖ−ϑ,ϖ−ϑ∈a,d and fa=fa. Thus, we obtain the following equation:(13)fϖ−ϑ=fa∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ+1Γω∫aϖ−ϑ∑ς=0∞ϖ−ϑ−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ−aω/ω−ςϱΓ1+τςϱƛtslogtƛω−1Kƛ,fƛƛdƛ.Definition 4 (see [12]).
The multivariate Mittag-Leffler (MM-L) function is defined by the following series representation:(14)Eα1,…,αm,βu1,…,um=∑ς=0∞∑ς1+⋯+ςm=ςςς1,…,ςmu1ς1⋯umςmΓα1ς1+⋯+αmςm+β,where αi,β>0 for i=1,2,…,m.Definition 5 (see [13–15]).
According to a standard notation, the Foxℋ function is defined as(15)ℋτ,ωm,ng=12πi∫ℒℋτ,ωm,negede,where ℒ is a suitable path in the complex plane ℂ to be disposed later. ge=explogg+iargg and(16)ℋτ,ωm,ne=VeWeXeYe,(17)Ve=∏j=1mΓsj−ψje,We=∏j=1nΓ1−rj+ϕje,(18)Xe=∏j=m+1qΓ1−sj+ψje,Ye=∏j=n+1pΓrj−ϕje,with 0≤n≤p, 1≤m≤q, rj,sj∈ℂ, ϕj,ψj∈ℝ+. An empty product, when it occurs, is taken to be one, so we get(19)n=0⟷We=1,m=q⟷Xe=1.n=p⟷Ye=1.
Theℋ function is, in general, multivalued, but it can be made one-valued on the Riemann surface of logg by choosing a proper branch. We also note that when α and β are equal to 1, we obtain G functions Gτ,ωm,ng. The above integral representation of ℋ 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).(20)ℋτ,ωm,ng=ℋτ,ωm,ng|rj,αjj=1,…,psj,βjj=1,…,p.
Here, we assumeF1=0,p, F2=0,+∞, F3=0,1, F4=0,+∞ , F5=0,1F5°=0,1, and F6=0,+∞.
Assume that(21)diagMnF5=r1⋱rn=diagr1,…,rn,r1,…,rn∈F5,in which(22)r=diagr1,…,rn,s=diags1,…,sn∈diagMnF5,r≤¯s⇔ri≤sifor everyi=1,…,n.
Also,r≺s denotes that r≺¯s and r≠s; r≪s for every i=1,…,n. We define a=diaga,…,a in diagMnF5 where a∈F5. Note that, diag1,…,1 is 1 and diag0,…,0 is 0.Definition 6 (see [16–18]).
A mapping⊛:diagMnF5×diagMnF5⟶diagMnF5 is called a GTN if the following conditions are met:(a)
∀r∈diagMnF5r⊛1=r (boundary condition)(b)
∀r,s∈diagMnF52r⊛s=s⊛r (commutativity)(c)
∀r,s,c∈diagMnF53r⊛s⊛c=r⊛s⊛c (associativity)(d)
∀r1,s2,s1,s2∈diagMnF54r1≺¯r2ands1≺¯s2implies thatr1⊛s1≺¯r2⊛s2 (monotonicity)
If for everyr,s∈diagMnF5 and each sequences rk and sk converging to r and s, we get(23)limkrk⊛sk=r⊛r,and we conclude that the continuity of ⊛ on diagMnF5 (CGTN).(1)
Define⊛M:diagMnF5×diagMnF5⟶diagMnF5, such that(24)r⊛Ms=diagr1,…,rn⊛Mdiags1,…,sn=diagminr1,s1,…,minrn,sn,then⊛M is CGTN (minimum CGTN).(2)
Define⊛P:diagMnF5×diagMnF5⟶diagMnF5, such that(25)r⊛Ps=diagr1,…,rn⊛Pdiags1,…,sn=diagr1.s1,…,rn.sn,then⊛P is CGTN (product CGTN).(3)
Define⊛L:diagMnF5×diagMnF5⟶diagMnF5, such that(26)r⊛Ls=diagr1,…,rn⊛Ldiags1,…,sn=diagmaxr1+s1−1,0,…,maxrn+sn−1,0,then⊛P is CGTN (Lukasiewicz CGTN).
Numerical examples of CGTN are as follows:(27)diag47,79,59,314,43,67⊛Mdiag35,518,611,57,1321,23=diag47,518,611,314,1321,23or4779593144367⊛M3551861157132123=47518611314132123,diag47,79,59,314,43,67⊛Pdiag35,518,611,57,1321,23=diag1235,2562,1033,1598,5263,47or4779593144367⊛P3551861157132123=1235356210331598526347,diag47,79,59,314,43,67⊛Ldiag35,518,611,57,1321,23=diag635,118,1099,0,5663,1121or4779593144367⊛L3551861157132123=6351181099056631121.We get the following equations:(28)diag47,79,59,314,43,67⊛Mdiag35,518,611,57,1321,23≽diag47,79,59,314,43,67⊛Pdiag35,518,611,57,1321,23≽diag47,79,59,314,43,67⊛Ldiag35,518,611,57,1321,23.
By considering the matrix-valued fuzzy function (MVFF)G:F1k×F2⟶diagMnF3, then we have following conclusions:(i)
It is a left continuous and increasing function.(ii)
limy⟶+∞Gϖ1−ϑ1,…,ϖk−ϑk,y=1 for any ϖ1−ϑ1,…,ϖk−ϑk∈F1 and y∈F2.(iii)
For MVFFsG and W, the relation “≺” is defined as follows:(29)G≾W⇔Gϖ1−ϑ1,…,ϖk−ϑk,y≤¯Wϖ1−ϑ1,…,ϖk−ϑk,y,∀y∈F2andϖ1−ϑ1,…,ϖk−ϑk∈F1.Definition 7.
Let⊛ be a CGTN, J be a vector space, and N:Jk×F2⟶diagMnF3 be a matrix-valued fuzzy set (MVFS). Triple J,Y,⊛ is called a matrix-valued fuzzy k-normed space (MVFkN-space) if
(MVFkN1)Nc1−c1′,…,ck−ckk′,y=1 if and only if c1−c1′,…,ck−ck′ are linearly dependent and y∈F2;
(MVFkN2)Nϰc1−c1′,…,ck−ck′,y=Nc1−c1′,…,ck−ck′,y/ϰ for all c1−c1′,…,ck−ck′∈J and ϰ∈ℂ with ϰ≠0;
(MVFkN3)Nc0+c1−c1′,…,ck−ck′,y+z≽Nc0,c2−c2′,…,ck−ck′,N⊛Nc1−c1′,c2−c2′,…,ck−ck′,z for all c1−c1′,…,ck−ck′∈J and any y,z∈F2;
(MVFkN4)limy⟶+∞Nϖ1−ϑ1,…,ϖk−ϑk,y=1 for any y∈F2.
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 functionHτ,ωm,n:F1k×F2⟶diagMnF3τ,ω∈F2 as a control function in the MVFkN-spaces as follows:(30)Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky=diagHτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p.
For an MVF H-Fox functionHτ,ωm,n, we have(1)
It is a left continuous and increasing function for positive values.(2)
limy⟶+∞Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk/y=1.(3)
ForHτ,ωm,n and also, for the matrix-valued fuzzy function Φτ,ω, we have(31)Φτ,ω≾Hτ,ωm,n⇔Φτ,ω−ϖ1−ϑ1,…,ϖk−ϑky≤¯Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky,and also, we have(1)
Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk/y≻0.(2)
We can easily show that fory∈F2, Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk/y=1⇔ϖ1−ϑ1,…,ϖk−ϑk=0.(3)
We show that(32)Hτ,ωm,n−αϖ1−ϑ1,…,ϖk−ϑky=Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky/α.Then, we have(33)Hτ,ωm,n−αϖ1−ϑ1,…,ϖk−ϑky=diagHτ,ωm,n−−αϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−−αϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−−αϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p=diagHτ,ωm,n−−α‖ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−−α‖ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−−α‖ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p=diagHτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky/α|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky/α|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky/α|aj,αjj=1,…,pbj,βjj=1,…,p=Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky/α.(4)
We show that(34)Hτ,ωm,n−ϖ1−ϑ1+cz1−cp1,…,ϖk−ϑk+czk−cpky+s≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky⊛Hτ,ωm,n−cz1−cp1,…,czk−cpks.
Suppose thatHτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk/y|aj,αjj=1,…,pbj,βjj=1,…,p≤Hτ,ωm,n−cz1−cp1,…,czk−cpk/s|aj,αjj=1,…,pbj,βjj=1,…,p, then we have(35)−ϖ1−ϑ1,…,ϖk−ϑky≤−cz1−cp1,…,czk−cpks⇒ϖ1−ϑ1,…,ϖk−ϑky≥cz1−cp1,…,czk−cpks⇒sϖ1−ϑ1,…,ϖk−ϑky≥cz1−cp1,…,czk−cpk⇒sϖ1−ϑ1,…,ϖk−ϑky+ϖ1−ϑ1,…,ϖk−ϑk≥cz1−cp1,…,czk−cpk+ϖ1−ϑ1,…,ϖk−ϑk≥ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpk⇒ϖ1−ϑ1,…,ϖk−ϑksy+1≥ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpk⇒ϖ1−ϑ1,…,ϖk−ϑks+yy≥ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpk⇒ϖ1−ϑ1,…,ϖk−ϑky≥ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpks+y⇒−ϖ1−ϑ1,…,ϖk−ϑky≤−ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpks+y⇒Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk+cz1−cp1,…,czk−cpks+yaj,αjj=1,…,pbj,βjj=1,…,p≥Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkyaj,αjj=1,…,pbj,βjj=1,…,p.
Therefore, we have(36)diagHτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk+s1−p1,…,sk−pks+y|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk+s1−p1,…,sk−pks+y|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk+s1−p1,…,sk−pks+y|aj,αjj=1,…,pbj,βjj=1,…,p≽diagHτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑky|aj,αjj=1,…,pbj,βjj=1,…,p⊛diagHτ,ωm,n−s1−p1,…,sk−pks|aj,αjj=1,…,pbj,βjj=1,…,p,Hτ,ωm,n−s1−p1,…,sk−pks|aj,αjj=1,…,pbj,βjj=1,…,p,…,Hτ,ωm,n−s1−p1,…,sk−pks|aj,αjj=1,…,pbj,βjj=1,…,p.
Consequently, if(37)Yz1−z1′,…,zk−zk′,y=diagHτ,ωm,n−z1−z1′,…,zk−zk′y,Hτ,ωm,n−z1−z1′,…,zk−zk′y,…,Hτ,ωm,n−z1−z1′,…,zk−zk′y,for y∈F2, then J,Y,⊛M is an MVFkN-space. From now on, we assume ⊛=⊛M.Theorem 3 (see [14, 19]).
We consider theF4-valued metric space A,d. For f,h∈A, we consider the self-map Π on A such that(38)dΠf,Ξh≤νdh,f,where ν<1 is a Lipschitz constant. Let f∈A. Therefore, we have following two ways:(i)
dΠef,Πe+1f=∞,∀e∈ℕor(ii)
we can finde0∈ℕ such that: dΠef,Πe+1f<∞,∀e≥e0
If condition (ii) is true for us, then we have following conclusions:(1)
The fixed pointh∗ of Π is the convergence point of the sequence Πef(2)
In the setℋ∗=h∈A∣dΠe0f,h<∞,h∗ is the unique fixed point of Π(3)
1−νdh,h∗≤dh,Πh for every h∈ADefinition 8.
Let functionHτ,ωm,n be an MVF function. Equation (1) is said to be Hyers–Ulam-H-Fox stable, and if hϖ1−ϑ1,…,hϖk−ϑk is a given differentiable function, we obtain the following equation:(39)NDωahϖ1−ϑ1−ϱhϖ1−ϑ1−Kϖ1−ϑ1,hϖ1−ϑ1,…,Dωahϖk−ϑk−ϱhϖk−ϑk−Kϖk−ϑk,hϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ,for ϖ1−ϑ1,…,ϖk−ϑk∈F1, and we can find a solution fϖ1−ϑ1…,fϖk−ϑk of (1) such that for some η>0, which is as follows:(40)Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/η.Remark 2.
Leth be a solution of inequality (39). Then, h is a solution of the following integral inequality:(41)Nhϖ1−ϑ1−ha∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ−1Γω∫aϖ−ϑ∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱKƛ1−ƛ1′,hϖ1−ϑ1ƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1dƛ1−ƛ1′ƛ1−ƛ1′,…,hϖk−ϑk−ha∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ−1Γω∫aϖ−ϑ∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱKƛk−ƛk′,hϖk−ϑkƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′,ℵ≽Hτ,ωm,n|1Γω∫aϖ−ϑ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1dƛ1−ƛ1′ƛ1−ƛ1′,…,1Γω∫aϖ−ϑ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′|1ℵ/Ϝ∑ς=0∞b−aω/ωςϱ/Γ1+τςϱ,for every ϖ1−ϑ1,…,ϖk−ϑk∈F1 and ℵ∈F2.


## 3. Hyers–Ulam-H-Fox Stability for the Conformable Fractional Differential Equation

Now, we use the fixed-point method based on Theorem3 to show that equation (1) is Hyers–Ulam-H-Fox stable [14] in the MVFkB-space J,N,⊛ with MVFF Hτ,ωm,n [20–22].We set the setℋ as follows:(42)ℋ=f:F1⟶J,fis differentiable,and we consider the mapping d:ℋ×ℋ⟶F4 as(43)df,h=infℜ∈F6:Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/ℜ,∀h,f∈ℋ,ϖ1−ϑ1,…,ϖk−ϑk∈F1,ℵ∈F2.Theorem 4.
ℋ
,
d is a complete F4-valued metric space.Proof.
We havedh,f=0 if and only if h=f. Assume that dh,f=0, then we have(44)infℜ∈F6:Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/ℜ,∀h,f∈ℋ,ϖ1−ϑ1,…,ϖk−ϑk∈F1,ℵ∈F2=0,so(45)Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/ℜ,for all ℜ∈F6. We assume ℜ to zero in the above inequality, and we get(46)Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ=1.
Thus,hϖ1−ϑ1,…,ϖk−ϑk=fϖ1−ϑ1,…,ϖk−ϑk for every ϖ1−ϑ1,…,ϖk−ϑk∈F1 and vice versa. Also, we have dh,f=df,h for every h,f∈ℋ. Now, let dh,f=α1∈F2 and dh,f=α2∈F2. Then, we have(47)Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/α1,Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/α2,for every ℵ∈F2. Then, we have(48)Nhϖ1−ϑ1−pϖ1−ϑ1,…,hϖk−ϑk−pϖk−ϑk,α1+α2ℵ≽Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,α1ℵ⊛Nfϖ1−ϑ1−pϖ1−ϑ1,…,fϖk−ϑk−pϖk−ϑk,α2ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ⊛Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ=Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ,so dh,p≤α1+α2. Thus, dh,p≤dh,f+df,p. To show the completeness of ℋ,d, we suppose that hkk is a Cauchy sequence in ℋ,d. Let ϖ1−ϑ1,…,ϖk−ϑk∈F1. Assume that γ∈F2 and θ∈F5° are arbitrary and consider ℵ∈F2 such that Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk/ℵ≻1−θ. For αℵ<γ, we choose k0∈ℕ such that(49)dhk,hℓ<α∀k,ℓ≥k0.
Then, we have(50)Nhkϖ1−ϑ1−hℓϖ1−ϑ1,…,hkϖk−ϑk−hℓxk−yk,γ≽Nhkϖ1−ϑ1−hℓϖ1−ϑ1,…,hkϖk−ϑk−hℓϖk−ϑk,αℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ≻1−θ.So we have the following equation:(51)Nhϖ1−ϑ1−hϖ1−ϑ1,…,hϖk−ϑk−hϖk−ϑk,γ≻1−θ,which implies that the sequence hkϖ1−ϑ1,hkϖ2−ϑ2,…,hkϖk−ϑkk is Cauchy in a complete space J,N,⊛ on a compact set F1. Then, it is uniformly convergent to the mapping h:F1⟶J. By uniform convergence property, we conclude that h is differentiable, i.e., an element of ℋ, and then, ℋ,d 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.
LetJ,N,⊛ be an MVFkB-space and consider the constant coefficient ϱ, E, and ℘. Then, we have ℶ=℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ϱ⋯ϖk−ϑk−aω/ωςkϱ/Γς1⋯ςk+τςϱE<1.Suppose that the following conditions hold:(1)
For continuous functionK:F1×J⟶J, we obtain(52)NKϖ1−ϑ1,fϖ1−ϑ1−Kϖ1−ϑ1,hϖ1−ϑ1,…,Kϖk−ϑk,fϖk−ϑk−Kϖk−ϑk,hϖk−ϑk,ℵ≽Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖ1−ϑ1,ℵℜ,(2)
MVFFHτ,ωm,n satisfying the following equation:(53)Hτ,ωm,n|1Γω∫aϖ1−ϑ1∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1dƛ1−ƛ1′ƛ1−ƛ1′,…,∫aϖk−ϑk∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′|1ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/E.Leth:F1⟶J be a differentiable function satisfying the following equation:(54)NDωahϖ1−ϑ1−ϱhϖ1−ϑ1−kϖ1−ϑ1,hϖ1−ϑ1,…,Dωahϖk−ϑk−ƛhϖk−ϑk−Kϖk−ϑk,hϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/Ϝ.Then, there is a unique solutionf:F1⟶J for (1) such that(55)Yhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ1−ℶ/E∑ς=0∞b−aω/ωςϱ/Γ1+τςϱϜ,for every ϖ1−ϑ1,…,ϖk−ϑk∈F1 and ℵ∈F2.Proof.
We set(56)ℋ≔f:F1⟶J,fis differentiable,and introduce the F4-valued metric on ℋ as(57)infℜ∈F6:Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/ℜ,∀f,h∈ℋ,ϖ1−ϑ1,…,ϖk−ϑk∈F1,ℵ∈F2=0.
By Theorem4, we have ℋ,d that is a complete F4-valued metric space.Step 1.
We defineΠ from ℋ to ℋ by the following equation:(58)Πfϖi−ϑi=fa∑ς=0∞ϖi−ϑi−aω/ωςϱΓ1+τςϱ+1Γω∫aϖi−ϑi∑ς=0∞ϖi−ϑi−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛi−ƛi′−aω/ω−kϱΓ1+τςϱƛi−ƛi′tslogtƛi−ƛi′ω−1Kƛi−ƛi′,fƛi−ƛi′ƛi−ƛi′dƛi−ƛi′,for ϖi−ϑi∈F1i=1,2,…,k, and we show Π is a strictly contractive mapping.
Letf,h∈ℋ and consider the coefficient ℜfh∈F4 with df,h≤ℜfh; thus, we have(59)Nfϖ1−ϑ1−hϖ1−ϑ1,…,fϖk−ϑk−hϖk−ϑk,ℜfhℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ,for all f,h∈ℋ,ϖ1−ϑ1,…,ϖk−ϑk∈F1 and ℵ∈F2. Applying (MVFkN2) and (MVFkN3), we imply that(60)NΠfϖ1−ϑ1−Πhϖ1−ϑ1,…,Πfϖk−ϑk−Πhϖk−ϑk,ℜfhℵ=Nfa∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ+1Γω∫aϖ1−ϑ1∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛi′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1kƛ1−ƛ1′,fƛ1−ƛ1′ƛ1−ƛ1′dƛ1−ƛ1′−ha∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ+1Γω∫aϖ1−ϑ1∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1Kƛ1−ƛ1′,hƛ1−ƛ1′ƛ1−ƛ1′dƛ1−ƛ1′,⋯,fa∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ+1Γω∫aϖk−ϑk∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1Kƛk−ƛk′,fƛk−ƛk′ƛk−ƛk′dƛk−ƛk′−ha∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ+1Γω∫aϖk−ϑk∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1Kƛk−ƛk′,hƛk−ƛk′ƛk−ƛk′dƛk−ƛk′,ℜfhℵ≽N1Γω∫aϖ1−ϑ1∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1Kƛ1−ƛ1′,fƛ1−ƛ1′−Kƛ1−ƛ1′,hƛ1−ƛ1′ƛ1−ƛ1′dƛ1−ƛ1′,…,1Γω∫aϖk−ϑk∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1Kƛk−ƛk′,fƛk−ƛk′−Kƛk−ƛk′,hƛk−ƛk′ƛk−ƛk′dƛk−ƛk′,ℜfhℵ≽N1Γω∫aϖ1−ϑ1∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1fƛ1−ƛ1′−hƛ1−ƛ1′ƛ1−ƛ1′dƛ1−ƛ1′…,1Γζ∫aϖk−ϑk∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1fƛk−ƛk′−hƛk−ƛk′ƛk−ƛk′dƛk−ƛk′,ℜfhμ℘≽Hτ,ωm,n|∑ς=0∞ϖ1−ϑ1−aω/ωςƛΓ1+αςϱ1Γω∫aϖ1−ϑ1∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1dƛ1−ƛ1′ƛ1−ƛ1′,…,∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ1Γω∫aϖk−ϑk∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′|1ℵ/℘≽Hτ,ωm,n|∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ,…,∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ1Γω∫aϖ1−ϑ1∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱ,…,∫aϖk−ϑk∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′|1ℵ/℘≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ϱ⋯ϖk−ϑk−aω/ωςkϱ/Γς1⋯ςk+τςϱE,which implies that(61)dΠf,Πh≤℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ϱ⋯ϖk−ϑk−aω/ωςkϱΓς1⋯ςk+τςϱEℜfh,so(62)dΠf,Πh≤℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ϱ⋯ϖk−ϑk−aω/ωςkϱΓς1⋯ςk+τςϱEdf,h,where 0<℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−ccω/ως1ϱ⋯ϖk−ϑk−ccω/ωςkϱ/Γς1⋯ςk+τςϱE<1; therefore, Π is a contraction mapping.Step 2.
We will show thatdΠh,h<∞.
Leth∈ℋ, we have(63)NΠhϖ1−ϑ1−hϖ1−ϑ1,…,Πϖk−ϑk−hϖk−ϑk,ℵ=Nha∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ+1Γω∫aϖ1−ϑ1∑ς=0∞ϖ1−ϑ1−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1Kƛ1−ƛ1′,hƛ1−ƛ1′ƛ1−ƛ1′dƛ1−ƛ1′−hϖ1−ϑ1,…,ha∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ+1Γω∫aϖk−ϑk∑ς=0∞ϖk−ϑk−aω/ωςϱΓ1+τςϱ∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1Kƛk−ƛk′,hƛk−ƛk′ƛk−ƛk′dƛk−ƛk′−hϖk−ϑk,ℵ≽Hτ,ωm,n|∑ς=0∞b−aω/ωςϱΓ1+τςϱ1Γβ∫aϖ1−ϑ1∑ς=0∞ƛ1−ƛ1′−aω/ω−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′ω−1dƛ1−ƛ1′ƛ1−ƛ1′,…,1Γω∫aϖk−ϑk∑ς=0∞ƛk−ƛk′−aω/ω−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′|1ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/Ϝ∑ς=0∞b−aω/ωςϱ/Γ1+τςϱE.
Consequently, we obtain the following equation:(64)dΠh,h≤Ϝ∑ς=0∞b−aω/ωςϱΓ1+τςϱE<∞,for every ℵ∈F2 and E<1. Then, we have dΠh,h<∞.
Therefore, all the conditions of Theorem1 hold. Then, we have(1)
The sequenceΠef converges to a fixed point such as f.(2)
The unique elementf is in the set ℋ∗=h∈ℋ:dΠh,f<∞ and is the unique fixed point of Π, which means Πf=f or equivalently as shown in the following equation:(65)fϖi−ϑi=fa∑ς=0∞ϖi−ϑi−aω/ωςϱΓ1+τςϱ+1Γω∫aϖi−ϑi∑ς=0∞ϖi−ϑi−aω/ωςϱΓ1+τςƛ∑ς=0∞ϖi−ϑi−aω/ω−ςϱΓ1+τςϱƛi−ƛi′tslogtƛi−ƛi′ω−1Kƛi−ƛi′,fƛi−ƛi′ƛi−ƛi′dƛi−ƛi′,wherei=1,…,k.Sincef is a differentiable function, by the CFD and according to (65) and Lemma 1, we have(66)Dωafϖ−ϑ=ϱfϖ−ϑ+Kϖ−ϑ,fϖ−ϑ.(3)
Using inequality (64), we get(67)dw,f≤11−ℶdΠh,h≤Ϝ∑ς=0∞d−aω/ωςϱ/Γ1+τςϱE1−ℶ.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:(68)ϵ=Ϝ∑ς=0∞b−aω/ωςϱ/Γ1+τςϱE1−ℶ,and let g be another differentiable function satisfying equation (66), and this means that the following equation holds:(69)Dωagϖ−ϑ=ϱfϖ−ϑ+Kϖ−ϑ,gϖ−ϑ.
We are ready to prove thatg is a fixed point of Π and g∈ℋ∗. Using equation (69), we get Πg=g. Now, we show that dΠh,g<∞. Let h∈ℋ, dh,g<ϵ, and from equation (69), we get(70)1ℵ/℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ñ⋯ϖk−ϑk−aω/ωςkñ/Γς1⋯ςk+τςñò≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ñ⋯ϖk−ϑk−aω/ωςkñ/Γς1⋯ςk+τςñòE,Then, we have(71)dΠh,g≤℘∑ς=0∞∑ς1⋯ςk=ς∞ςς1,…,ςkϖ1−ϑ1−aω/ως1ñ⋯ϖk−ϑk−aωωςkñ/Γς1⋯ςk+τςñEϵ<∞.


## 4. Example

Now, we provide numerical examples according to the results obtained.Example 1.
We consider the following conformable FDE:(72)D1/21fϖ−ϑ=fϖ−ϑ+15+ϖ−ϑ2f2ϖ−ϑfϖ−ϑ+1cosfϖ−ϑ,ϖ−ϑ∈1,2f1=1,where ω=1/2,ϱ=1,℘=1/8, for ς=1,ϖ1−ϑ1=2,τ=1we haveℶ=0/4618<1. Also, in this equation, Kϖ−ϑ,fϖ−ϑ=1/5+ϖ−ϑ2f2ϖ−ϑ/fϖ−ϑ+1cosfϖ−ϑ. Let for mapping k and the MVF control function Hτ,ωm,n, we have(1)
(73)NKϖ1−ϑ1,fϖ1−ϑ1−Kϖ1−ϑ1,hϖ1−ϑ1,…,Kϖk−ϑk,fϖk−ϑk−Kϖk−ϑk,hϖk−ϑk,ℵ≽Nfϖ1−ϑ1−hϖ1−ϑ1,…,fϖk−ϑk−hϖk−ϑk,ℵ1/8.(2)
(74)Hτ,ωm,n|1Γ1/2∫aϖ1−ϑ1∑ς=0∞2ƛ1−ƛ1′−a1/2−ςϱΓ1+τςϱƛ1−ƛ1′tslogtƛ1−ƛ1′−1/2dƛ1−ƛ1′ƛ1−ƛ1′,…,∫aϖk−ϑk∑ς=0∞2ƛk−ƛk′−a1/2−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′ω−1dƛk−ƛk′ƛk−ƛk′1ℵ|≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/1/2.
Ifh∈C1,2,ℝ be a differentiable function such that(75)ND1/21hϖ1−ϑ1−hϖ1−ϑ1−15+ϖ1−ϑ12h2ϖ1−ϑ1hϖ1−ϑ1+1coshϖ1−ϑ1,…,D1/21hϖk−ϑk−hϖk−ϑk−15+ϖk−ϑk2h2ϖk−ϑkhϖk−ϑk+1coshϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ/Ϝ,then h is a solution of the inequality(76)Nhϖ1−ϑ1−ha∑ς=0∞4ϖ1−ϑ1−a1/4ςΓ1+τς−1Γ1/4∫aϖ1−ϑ1∑ς=0∞4ϖ1−ϑ1−a1/4ςΓ1+τς∑ς=0∞4λ1−λ1′−a1/4−ςΓ1+τςλ1−λ1′+16λ1−λ1′2+6sinhλ1−λ1′λ1−λ1′tslogtλ1−λ1′−3/4dλ1−λ1′λ1−λ1′,…,hϖk−ϑk−fa∑ς=0∞4ϖk−ϑk−a1/4ςΓ1+τς−1Γ1/4∫aϖk−ϑk∑ς=0∞4ϖk−ϑk−a1/4ςΓ1+τς∑ς=0∞4λk−λk′−textgothc1/4−ςΓ1+τςλk−λk′+16λk−λk′2+6sinhλk−λk′λk−λk′tslogtλk−λk′−3/4dλk−λk′λk−λk′,ℵ≽Hτ,ωm,n|1Γ1/4∫aϖ1−ϑ1∑ς=0∞4λ1−λ1′−a1/4−ςΓ1+τςλ1−λ1′tslogtλ1−λ1′−3/4dλ1−λ1′λ1−λ1′,…,1Γ1/4∫aϖk−ϑk∑ς=0∞4λk−λk′−a1/4−ςΓ1+τςλk−λk′tslogtλk−λk′−3/4dλk−λk′λk−λk′|1ℵ/1/3∑ς=0∞4ς/Γ1+τς,and thus, we can find a unique differentiable function f∈C1,2,ℝ from (72) such that for each ϖ−ϑ∈1,2, we have(77)fϖi−ϑi=∑ς=0∞2ϖi−ϑi−11/2ςΓ1+ςτ+1Γ1/2∫1ϖi−ϑi∑ς=0∞2ϖi−ϑi−11/2ςΓ1+ςτ∑ς=0∞2ƛi−ƛi′−11/2−ςΓ1+ςτƛi−ƛi′tslogtƛi−ƛi′−1/21/5+ϖi−ϑi2f2ϖi−ϑi/fϖi−ϑi+1cosfϖi−ϑiƛi−ƛi′dƛi−ƛi′,for any i=1,…,k. Therefore(78)dh,f≤21−ℶ∑ς=0∞2ςΓ1+τςϜ,Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ1−ℶ/2∑ς=0∞2ς/Γ1+τςϜ.
In Figures1–8, the exact solution of conformable FDE (72) for τ=1/2,s=1,t=1/4 is demonstrated.Figure 1
Exact solution of Example1 for ϖ−ϑ∈1,10.Figure 2
Exact solution of Example1 for ϖ−ϑ∈1,10.Figure 3
Exact solution of Example1 for ϖ−ϑ∈−10,10.Figure 4
Exact solution of Example1 for ϖ−ϑ∈−100,100.Figure 5
Exact solution of Example1 for ϖ−ϑ∈−300,300.Figure 6
Exact solution of Example1 for ϖ−ϑ∈−300,300.Figure 7
Exact solution of Example1 for ϖ−ϑ∈−100,100.Figure 8
Exact solution of Example1 for ϖ−ϑ∈−10,10.Example 2.
We consider the following conformable FDE:(79)D1/41fϖ−ϑ=fϖ−ϑ+ϖ−ϑ+16ϖ−ϑ2+6sinfϖ−ϑ,ϖ−ϑ∈1,2f1=1,where ω==1/4,ϱ=1,℘=1/100, for ς=2,ϖ1−ϑ1=4,τ=2we haveℶ=0.004274<1. Also, in this equation, kϖ−ϑ,fϖ−ϑ=ϖ−ϑ+1/6ϖ−ϑ2+6sinfϖ−ϑ. Let for mapping k and the MVF control function Hτ,ωm,n, we have(80)NKϖ1−ϑ1,fϖ1−ϑ1−Kϖ1−ϑ1,hϖ1−ϑ1,…,Kϖk−ϑk,fϖk−ϑk−Kϖk−ϑk,hϖk−ϑk,ℵ≽Nfϖ1−ϑ1−hϖ1−ϑ1,…,fϖk−ϑk−hϖk−ϑk,100ℵ.(81)Hτ,ωm,n|1Γ1/4∫aϖ1−ϑ1∑ς=0∞4ƛ1−ƛ1′−a1/4−ςϱΓ1+τςϱƛ1−ƛ1′tslogtξ1−ξ1′−3/4dƛ1−ƛ1′ƛ1−ƛ1′,…,∫aϖk−ϑk∑ς=0∞4ƛk−ƛk′−a1/4−ςϱΓ1+τςϱƛk−ƛk′tslogtƛk−ƛk′−3/4dƛk−ƛk′ƛk−ƛk′1ℵ|≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk4ℵ.
Ifh∈C1,2,ℝ be a differentiable function such that(82)ND1/41hϖ1−ϑ1−hϖ1−ϑ1−ϖ1−ϑ1+16ϖ1−ϑ12+6sinhϖ1−ϑ1,…,D1/41hϖk−ϑk−hϖk−ϑk−ϖk−ϑk+16ϖk−ϑk2+6sinhϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑk3ℵ,then h is a solution of the inequality(83)Nhϖ1−ϑ1−ha∑ς=0∞4ϖ1−ϑ1−a1/4ςΓ1+τς−1Γ1/4∫aϖ1−ϑ1∑ς=0∞4ϖ1−ϑ1−a1/4ςΓ1+τς∑ς=0∞4ƛ1−ƛ1′−a1/4−ςΓ1+τςƛ1−ƛ1′+16ƛ1−ƛ1′2+6sinhƛ1−ƛ1′ƛ1−ƛ1′tslogtƛ1−ƛ1′−3/4dƛ1−ƛ1′ƛ1−ƛ1′,…,hϖk−ϑk−fa∑ς=0∞4ϖk−ϑk−a1/4ςΓ1+τς−1Γ1/4∫aϖk−ϑk∑ς=0∞4ϖk−ϑk−a1/4ςΓ1+τς∑ς=0∞4ƛk−ƛk′−c1/4−ςΓ1+τςƛk−ƛk′+16ƛk−ƛk′2+6sinhƛk−ƛk′ƛk−ƛk′tslogtƛk−ƛk′−3/4dƛk−ƛk′ƛk−ƛk′,ℵ≽Hτ,ωm,n|1Γ1/4∫aϖ1−ϑ1∑ς=0∞4ƛ1−ƛ1′−a1/4−ςΓ1+τςƛ1−ƛ1′tslogtƛ1−ƛ1′−3/4dƛ1−ƛ1′ƛ1−ƛ1′,…,1Γ1/4∫aϖk−ϑk∑ς=0∞4ƛk−ƛk′−a1/4−ςΓ1+τςƛk−ƛk′tslogtƛk−ƛk′−3/4dƛk−ƛk′ƛk−ƛk′|1ℵ/1/3∑ς=0∞4ς/Γ1+τς,and thus, we can find a unique differentiable function f∈C1,2,ℝ from (79) such that for each ϖ−ϑ∈1,2, we have(84)fϖi−ϑi=∑ς=0∞4ϖi−ϑi−11/4ςΓ1+ςτ+1Γ1/4∫1ϖi−ϑi∑ς=0∞4ϖi−ϑi−11/4ςΓ1+ςτ∑ς=0∞2ƛi−ƛi′−11/4−ςΓ1+ςτƛi−ƛi′fslogfƛi−ƛi′−3/4ϖi−ϑi+1/6ϖi−ϑi2+6sinhϖi−ϑiƛi−ƛi′dƛi−ƛi′,for any i=1,…,k. Therefore(85)dh,f≤41−ℶ∑ς=0∞4ςΓ1+τς×13,Nhϖ1−ϑ1−fϖ1−ϑ1,…,hϖk−ϑk−fϖk−ϑk,ℵ≽Hτ,ωm,n−ϖ1−ϑ1,…,ϖk−ϑkℵ1−ℶ/4∑ς=0∞2ς/Γ1+τς×1/3.
Figures9–11 show the graphs related to the exact solution of conformable FDE (79) for τ=2,s=1,t=1/4.Figure 9
Exact solution of Example2 for ϖ−ϑ∈1,10.Figure 10
Exact solution of Example2 for ϖ−ϑ∈1,10.Figure 11
Exact solution of Example2 for ϖ−ϑ∈1,10.


## 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.



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