# Fuzzy Generalized Conformable Fractional Derivative

Authors: Atimad Harir; Said Melliani; Lalla Saadia Chadli
Journal: Advances in Fuzzy Systems
 (2020)

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## Abstract

We give a new definition of fuzzy fractional derivative called fuzzy conformable fractional derivative. Using this definition, we prove some results and we introduce new definition of generalized fuzzy conformable fractional derivative.


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## Body

## 1. Introduction

Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. Their main directions of development have been diversed, and its applications have been varied [1–4].The derivative for fuzzy valued mappings was developed by Puri and Ralescu [5], which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative, Kaleva [6] started to develop a theory for FDE. In [7], a new well-behaved simple fractional derivative called “the conformable fractional derivative” depending just on the basic limit definition of the derivative, namely, for a function f0,∞⟶ℝ the (conformable) fractional derivative of order 0<q≤1 of f at t>0 was defined by(1)Tqft=limε⟶0ft+εt1−q−ftε,and is defined the fractional derivative at 0 as Tqf0=limt⟶0+Tqft. The aim of this paper is to study and generalize the fuzzy conformable fractional derivative.


## 2. Preliminaries

Let us denote byℝℱ=u:ℝ⟶0,1 the class of fuzzy subsets of the real axis satisfying the following properties:(i)
u is normal, i.e., there exists an x0∈ℝ such that ux0=1.(ii)
u is fuzzy convex, i.e., for x,y∈ℝ and 0<λ≤1:(2)uλx+1−λy≥minux,uy.(iii)
u is upper semicontinuous.(iv)
u0=clx∈ℝux>0 is compact.Then,ℝℱ is called the space of fuzzy numbers. Obviously, ℝ⊂ℝℱ. For 0<α≤1 denotes uα=x∈ℝux≥α, then from (i) to (iv) it follows that the α-level sets uα∈PKℝ for all 0≤α≤1 is a closed bounded interval which is denoted by uα=u1α,u2α. By PKℝ we denote the family of all nonempty compact convex subsets of ℝ and define the addition and scalar multiplication in PKℝ as usual.Theorem 1 (see [8]).
Ifu∈ℝℱ, then(i)
uα∈PKℝ for all 0≤α≤1.(ii)
uα2⊂uα1 for all 0≤α1≤α2≤1.(iii)
αk⊂0,1 is a nondecreasing sequence which converges to α, then(3)uα=∩k≥1uαk.
Conversely, ifAα=u1α,u2α;α∈0,1 is a family of closed real intervals verifying (i) and (ii), then Aα defined a fuzzy number u∈ℝℱ such that uα=Aαfor0<α≤1 and u0=∪0<α≤1Aα¯⊂A0.Lemma 1.
Letu,v:X⟶0,1 be the fuzzy sets.
Then,u=v if and only if uα=vα for all α∈0,1 see ([9]).The following arithmetic operations on fuzzy numbers are well known and frequently used below. Ifu,v∈ℝℱ, then(4)u+vα=u1α+v1α,u2α+v2α,λuα=λuα=λu1α,λu2α,λ≥0,λu2α,λu1α,λ<0.Definition 1.
Letu,v∈ℝℱ. If there exists w∈ℝℱ such that u=v+w, then w is called the H-difference of u,v and it is denoted u⊖v.Theorem 2 (see [10]).
(i)
Let us denote(5)0¯=1,t=0,0,t≠0.
Then,0¯∈ℝℱ be a neutral element with respect to +, i.e., u+0¯=0¯+uu∈ℝℱ:(i)
With respect to0¯, none of u∈ℝℱ/ℝ has opposite in ℝℱ.(ii)
For anya,b∈ℝ with a,b≥0 or a,b≤0 and any u∈ℝℱ, we have a+b⋅u=a⋅u+b⋅u, for general a,b∈ℝ the above property does not hold.(iii)
For anyλ∈ℝ and any u,v∈ℝℱ, we have λ⋅u+v=λ⋅u+λ⋅v.(iv)
For anyλ,ν∈ℝ and any u∈ℝℱ, we have λ⋅ν⋅u=λ⋅ν⋅u.Defined:ℝℱ×ℝℱ⟶ℝ+∪0 by the following equation:(6)du,v=supα∈0,1dHuα,vα,for allu,v∈ℝℱ,where dH is the Hausdorff metric:(7)dHuα,vα=maxu1α−v1α,u2α−v2α.It is well known thatℝℱ,d is a complete metric space. We list the following properties of du,v:(8)du+w,v+w=du,v,du,v=dv,u,dku,kv=kdu,v,du,v≤du,w+dw,v,for all u,v,w∈ℝℱ and λ∈ℝ.LetAk be a sequence in PKℝ converging to A. Then, theorem in [6] gives us an expression for the limit.Theorem 3 (see [6]).
IfdAk,A⟶0 as k⟶∞, then(9)A=∩k≥1∪m≥kAm¯.
LetI=0,a⊂ℝ be an interval. We denote by CI,ℝℱ that the space of all continuous fuzzy functions on I is a complete metric space with respect to the metric:(10)hu,v=supt∈Idut,vt.


## 3. The Fuzzy Conformable Fractional Differentiability

Definition 2.
LetF:I⟶ℝℱ be a fuzzy function. qth order “fuzzy conformable fractional derivative” of F is defined by(11)TqFt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε,for all t>0 and q∈0,1. Let Fqt stands for TqFt. Hence,(12)Fqt=limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε.
IfF is q-differentiable in some 0,a and limt⟶0+Fqt,(13)Fq0=limt⟶0+Fqt,and the limits exist (in the metric d).Remark 1.
From the definition, it directly follows that ifF is q-differentiable then the multivalued mapping Fα is q-differentiable for all α∈0,1 and(14)TqFα=Fqtα,where TqFα is denoted from the conformable fractional derivative of Fα of order q. The converse result does not hold, since the existence of Hukuhara difference xα⊖yα,α∈0,1, does not imply the existence of H-difference x⊖y.
However, for the converse result we have the following.Theorem 4.
LetF:I⟶ℝℱ satisfy the assumptions:(i)
For eacht∈I there exists a β>0 such that the H-difference Ft+εt1−q⊖Ft and Ft⊖Ft−εt1−q exists for all 0≤ε<β.(ii)
The set-valued mappingsFα,α∈0,1 are uniformly Hukuhara conformable fractional derivative of order q with derivative TqFα, i.e., for each t∈I and h>0 there exists a δ>0 such that(15)dHFαt+εt1−q−Fαtε,TqFαt<h,(16)dHFαt−Fαt−εt1−qε,TqFαt<h,for all 0≤ε<δ and α∈0,1. Then, F is q-differentiable, and the derivative is given by (14).Proof.
Consider the familyTqFαα∈0,1 By definition TqFαt is a compact, convex, and nonempty subset of ℝ.
Ifα1≤α2, then by assumption i,(17)Fα1t+εt1−q−Fα1t⊃Fα2t+εt1−q−Fα2t,for0≤ε<β,and consequently(18)TqFα1⊃TqFα2.
Letα>0 and αk be a nondecreasing sequence converging to α. For h>0 choose ε>0 such that equation(15) holds true. Then, the triangle inequality yields(19)dHTqFαt,TqFαkt≤dHTqFαt,Fαt+εt1−q−Fαtε+dHFαt+εt1−q−Fαtε,Fαkt+εt1−q−Fαktε+dHFαkt+εt1−q−Fαktε,TqFαkt≤2h+dHFαt+εt1−q−Fαtε,Fαkt+εt1−q−Fαktε.
By assumptioni, the rightmost term goes to zero as k⟶∞ and hence(20)limk⟶∞dHTqFαt,TqFαkt=0.
Now by Theorem3 and (18) we have(21)TqFαt=∪k≥1TqFαkt.
Ifα=0, then using equation 3.1 in [11], we deduce as before that(22)limk⟶∞dHTqF0t,TqFαkt=0,where αk is a nonincreasing sequence converging to zero and consequently(23)TqF0t=∪k≥1TqFαk¯.
Then, from Theorem1 it follows that there is an element u∈ℝℱ such that(24)uα=TqFαt,forα∈0,1.
Furthermore, lett∈I, h>0, and δ>0 be as in (ii). Then,(25)dFt+εt1−q⊖Ftε,u=sup0<α≤1dHFαt+εt1−q−Fαtε,TqFαt<h,for all 0≤ε<δ and similarly for dFt+εt1−q⊖Ft/ε,u. Hence, Fqt=u has the theorem.Theorem 5.
LetF:I⟶ℝℱ be q-differentiable. Denote Fαt=f1αt,f2αt, α∈0,1. Then, f1αt and f2αt are q-differentiable and(26)Fqtα=f1αqt,f2αqt.Proof.
Ifε>0 and α∈0,1, we have(27)Ft+εt1−q⊖Ftα=f1αt+εt1−q−f1αt,f2αt+εt1−q−f2αt.
Dividing byε, we have(28)Ft+εt1−q⊖Ftαε=f1αt+εt1−q−f1αtε,f2αt+εt1−q−f2αtε.
Similarly, we obtain(29)Ft⊖Ft−εt1−qαε=f1αt−f1αt−εt1−qε,f2αt−f2αt−εt1−qε,and passing to the limit gives the theorem.
Note that this definition and theorem of conformable fractional derivative are very restrictive; for instance, ifFt=c⋅gt, where c is a fuzzy number and g:a,b⟶ℝ+ is a function and is q-differentiable for some q∈0,1 with gqt<0, then F is not q-differentiable. To avoid this difficulty, we introduce a more general definition of the conformable fractional derivative for fuzzy-number-valued function.


## 4. The Generalized Fuzzy Conformable Fractional Differentiability

We consider the following definition.Definition 3.
LetF:I⟶ℝℱ be a fuzzy function and q∈0,1. One says, F is q1-differentiable at point t>0 if there exists an element Fqt∈ℝℱ such that for all ε>0 sufficiently near to 0 there exist Ft+εt1−q⊖Ft, Ft⊖Ft−εt1−q, and the limits (in the metric d):(30)limε⟶0+Ft+εt1−q⊖Ftε=limε⟶0+Ft⊖Ft−εt1−qε=Fqt,where F is q2-differentiable at t>0 if for all ε<0 sufficiently near to 0, then there exist Ft+εt1−q⊖FtandFt⊖Ft−εt1−q:(31)limε⟶0−Ft+εt1−q⊖Ftε=limε⟶0−Ft⊖Ft−εt1−qε=Fqt.
IfF is qn-differentiable at t>0, we denote its q-derivatives q∈0,1 by Fnqt, for n=1,2.Example 1.
Letg:I⟶ℝ+ and define F:I⟶ℝℱ by Ft=c⋅gt for all t∈I, where c is the fuzzy number. If g is q-differentiable at t0∈I, then F is the generalized fuzzy conformable fractional derivative at t0∈I and we have Fqt0=c⋅gqt0. For instance, if gqt0>0, F is q1-differentiable. If gqt0<0, then F is q2-differentiable.Remark 2.
In the previous definition,q1-differentiable corresponds to Definition 3, so this differentiability concept is a generalization of Definition 2 and obviously more general. For instance, in the previous example, for Ft=c⋅gt with gqt0<0, we have. Fqt0=c⋅gqt0Theorem 6.
LetF:I⟶ℝℱ be fuzzy function, where Fαt=f1αt,f2αt, α∈0,1:(i)
IfF is q1-differentiable, then f1αt and f2αt are q-differentiable and(32)Fq1tα=f1αqt,f2αqt.(ii)
IfF is q2-differentiable, then f1αt and f2αt are q-differentiable and(33)Fq2tα=f2αqt,f1αqt.Proof.
(i)
See demonstration of Theorem5.(ii)
Ifε<0, q∈0,1, and α∈0,1, then we have(34)Ft+εt1−q⊖Ftα=f1αt+εt1−q−f1αt,f2αt+εt1−q−f2αt.and multiplying by1/ε, we have(35)Ft+εt1−q⊖Ftαε=f2αt+εt1−q−f2αtε,f1αt+εt1−q−f1αtε.
Similarly, we obtain(36)Ft⊖Ft−εt1−qαε=f2αt−f2αt−εt1−qε,f1αt−f1αt−εt1−qε,and passing to the limit we have Fq2tα=f2αqt,f1αqt.Theorem 7.
LetF:I⟶ℝℱ and g:I⟶ℝ be two q-differentiable functions at a point t>0 (F is the generalized fuzzy conformable fractional derivative Definition 3 and q∈0,1:(i)
Ifgt⋅Tqgt>0 and F is q1-differentiable, then g⋅F is q1-differentiable and(37)Tqg⋅Ft=gt⋅TqFt+Ft⋅Tqgt.(ii)
Ifgt⋅Tqgt<0 and F is q2-differentiable, then g⋅F is q2-differentiable and(38)Tqg⋅Ft=gt⋅TqFt+Ft⋅Tqgt.Proof.
We present the details only for the casei, since the other case is analogous. F is q1-differentiable at t>0 the H-difference Ft⊖Ft+εt1−q exists for ε>0 sufficiently small, i.e., there exists u1t,εt1−q∈ℝℱ, such that(39)Ft=Ft+εt1−q+u1t,εt1−q.
Also,(40)gt=gt+εt1−q+u2t,εt1−q,where u2t,εt1−q=gt−gt+εt1−q for ε>0 sufficiently small.
By Theorem2, we obtain(41)Ft⋅gt=Ft+εt1−q⋅gt+εt1−q+Ft+εt1−q⋅u2t,εt1−q+gt+εt1−q⋅u1t,εt1−q+u1t,εt1−q⋅u2t,εt1−q,that is, the H-difference Ft⋅gt⊖Ft+εt1−q⋅gt+εt1−q exists and we have(42)Ft⋅gt⊖Ft+εt1−q⋅gt+εt1−q=Ft+εt1−q⋅u2t,εt1−q+gt+εt1−q⋅u1t,εt1−q+u1t,εt1−q⋅u2t,εt1−q.
Multiplying with1/ε and passing to limit with ε>0, by Proof of Lemma 3.2 [12], we obtain(43)limε⟶0+Ft⋅gt⊖Ft+εt1−q⋅gt+εt1−qε=limε⟶0+Ft+εt1−q⋅u2t,εt1−qε+limε⟶0+gt+εt1−q⋅u1t,εt1−qε+limε⟶0+u1t,εt1−qε⋅u2t,εt1−q,limε⟶0+Ft⋅gt⊖Ft+εt1−q⋅gt+εt1−qε=Ft⋅Tqgt+gt⋅TqFt+Tqgt⋅limε⟶0+u2t,εt1−q=Ft⋅Tqgt+gt⋅TqFt.
Analogously, we obtain(44)limε⟶0+Ft⋅gt⊖Ft−εt1−q⋅gt−εt1−qε=Ft⋅Tqgt+gt⋅TqFt,and the conclusion in the case i is obtained.Theorem 8.
Letq∈0,1:(i)
IfF is 1-differentiable and F is q1-differentiable, then(45)Tq1Ft=t1−qD11Ft.(ii)
IfF is 2-differentiable and F is q2-differentiable, then(46)Tq2Ft=t1−qD21Ft.Note that the definition ofn-differentiable or Dn1 for n∈1,2, see [12–14].Proof.
We present the details only for Case (i), since the other case is analogous. Leth=εt1−q in Definition 7, and then ε=tq−1h. Therefore, if ε>0 and α∈0,1, we have(47)Ft+εt1−q⊖Ftα=f1αt+εt1−q−f1αt,f2αt+εt1−q−f2αt.
Dividing byε, we have(48)Ft+εt1−q⊖Ftαε=f1αt+εt1−q−f1αtε,f2αt+εt1−q−f2αtε,and passing to the limit,(49)limε⟶0+Ft+εt1−q⊖Ftαε=limε⟶0+f1αt+εt1−q−f1αtε,f2αt+εt1−q−f2αtε=limh⟶0+f1αt+h−f1αttq−1h,f2αt+h−f2αttq−1h=t1−qlimh⟶0+f1αt+h−f1αth,f2αt+h−f2αth=t1−qf1α′t,f2α′t.
Similarly, we obtain(50)Ft⊖Ft−εt1−qαε=f1αt−f1αt−εt1−qε,f2αt−f2αt−εt1−qε,and passing to the limit and ε=tq−1h gives Tq1Ft=t1−qf1α′t,f2α′t.Example 2.
Letc∈ℝℱ and g0,a⟶ℝ, if g is q-differentiable on 0,a, then the function F0,a⟶ℝ defined by Ft=c⋅gt,t∈0,a. Then, F is qi-differentiable, for i=1,2, and we have Fqit=c⋅gqitfori=1,2.
c
=
1,2,3 and. cα=1+α,3−α.(i)
Letgt=et, so Ft=1,2,3et and Tqet=t1−qet, then it is easy to see that Tq1Ft=1,2,3t1−qet, i.e.,(51)Fq1tα=1+αt1−qet,3−αt1−qet.(ii)
Letgt=e−t, so Ft=1,2,3e−t and Tqe−t=−t1−qe−t, then it is easy to see that Tq2Ft=−1,2,3t1−qe−t, i.e.,(52)Fq2tα=−3+αt1−qe−t,−1−αt1−qe−t.


## 5. Conclusion

We have investigated generalized fuzzy conformable fractional differentiability. The conformableq-differentiability introduced here is a very general differentiability concept, being also practically applicable, and we can calculate by the fuzzy conformable derivative of the product of two functions Tqf⋅g because all fractional derivatives do not satisfy the known formula Tqf⋅g=Tqfg+fTqg.The disadvantage of fuzzy generalized conformable differentiability of a function seems to be that a simple fuzzy differential equationyq+y=0,0<q≤1,y0=y0∈ℝℱ has not got a unique solution, so it may have several solutions. The advantage of the existence of these solutions is that we can choose the solution that reflects better the behaviour of the modelled real-world system.For further research we propose the study for fuzzy fractional differential equations, by using the generalized conformable differentiability concept. In addition, we propose to extend the results of the present paper and to combine them with the results in [15] for fuzzy fractional differential equations.



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