Fuzzy Systems Theory and Application

244 Pages·2022·12.595 MB·English

by  Constantin Volosencu

Chapter 1
Existence, Uniqueness and
Approximate Solutions of Fuzzy
Fractional Differential Equations
Atimad Harir, Said Melliani and Lalla Saadia Chadli
Abstract
In this paper, the Cauchy problem of fuzzy fractional differential equations
Tγuð t Þ ¼ F ð t, uð t ÞÞ, u ðt 0Þ ¼ u0, with fuzzy conformable fractional derivative
( γ -differentiability, where γ ∈ ð 0, 1Þ are introduced. We study the existence and
uniqueness of solutions and approximate solutions for the fuzzy-valued mappings
of a real variable, we prove some results by applying the embedding theorem, and
the properties of the fuzzy solution are investigated and developed. Also, we show
the relation between a solution and its approximate solutions to the fuzzy fractional
differential equations of order γ.
Keywords: fuzzy conformable fractional derivative, fuzzy fractional differential
equations, existence and uniqueness of solution, approximate solutions, Cauchy
problem of fuzzy fractional differential equations
1. Introduction
In this paper, we will study Fuzzy solutions to
T uð t Þ ¼ F ð t, uð t ÞÞ, u ðt Þ ¼ u , γ ∈ ð 0, 1, (1)
γ 0 0
where subject to initial condition u for fuzzy numbers, by the use of the
0
concept of conformable fractional H-differentiability, we study the Cauchy
problem of fuzzy fractional differential equations for the fuzzy valued mappings
of a real variable. Several import-extant results are obtained by applying the
embedding theorem in [1] which is a generalization of the classical embedding
results [2, 3].
In Section 2 we recall some basic results on fuzzy number. In Section 3 we
introduce some basic results on the conformable fractional differentiability [4, 5]
and conformable integrability [5, 6] for the fuzzy set-valued mapping in [7]. In
Section 4 we show the relation between a solution and its approximate solution to
the Cauchy problem of the fuzzy fractional differential equation, and furthermore,
and we prove the existence and uniqueness theorem for a solution to the Cauchy
problem of the fuzzy fractional differential equation.
1
Fuzzy Systems - Theory and Applications
2. Preliminaries
We now recall some definitions needed in throughout the paper. Let us denote
by R the class of fuzzy subsets of the real axis fu : R ! ½0, 1gsatisfying the
F
following properties:
i. u is normal: there exists x ∈ R with uðx Þ¼1,
0 0
ii. u is convex fuzzy set: for all x, t ∈ R and 0 < λ ≤ 1, it holds that
uð λx þ ð1  λÞtÞ≥ min fuðxÞ, uðtÞg, (2)
iii. u is upper semicontinuous: for any x ∈ R, it holds that
0
uðx Þ≥ limuðxÞ, (3)
0
x!x0
iv. ½u0 ¼ clf x ∈ RjuðxÞ> 0gis compact.
Then R is called the space of fuzzy numbers see [8]. Obviously, R ⊂ R .Ifu is a
F F
fuzzy set, we define ½uα¼ fx ∈ RjuðxÞ≥ αgthe α-level (cut) sets of u, with
0 < α ≤ 1. Also, if u ∈ R then α-cut of u denoted by ½uα ¼ uα, uα :
F 1 2
Lemma 1 see [9] Let u, v : R ! ½0, 1be the fuzzy sets. Then u ¼ v if and only if
F
½uα ¼ ½vαfor all α ∈ ½0, 1:
For u, v ∈ R and λ ∈ R the sum u þ v and the product λu are defined by
F
½u þ vα ¼ uα þ vα, uα þ vα , (4)
1 1 2 2
½λuα¼ λ½ uα¼ λuα, λuα , λ ≥ 0; λuα, λuα , λ < 0, (5)
1 2 2 1
   
^ ^
∀α ∈ ½0, 1. Additionally if we denote0 ¼ χ , then 0 ∈ R is a neutral element
f0g F
with respert to þ:
Let d : R  R ! R ∪ f0gby the following equation:
F F þ
dðu, vÞ¼ sup d ð½uα, ½vαÞ, forall u, v ∈ R , (6)
H F
α ∈ ½0, 1
where d is the Hausdorff metric defined as:
H
d ð½uα, ½vαÞ¼ max juα vαj, uα vαjj (7)
H 1 1 2 2
 
The following properties are well-known see [10]:
dðuþ w, v þ wÞ¼dðu, vÞ and d ðu, vÞ¼dðv, uÞ, ∀ u, v, w ∈ R , (8)
F
dðku, kvÞ¼∣k∣dðu, vÞ, ∀k ∈ R, u, v ∈ R (9)
F
dðuþ v, w þ eÞ≤ dðu, wÞþdðv, eÞ, ∀ u, v, w, e ∈ R , (10)
F
and ðR , dÞis a complete metric space.
F
Definition 1 The mapping u : ½0, a! R for some interval ½0, ais called a
F
fuzzy process. Therefore, its α-level set can be written as follows:
½uðtÞα¼ uαðtÞ, uαðtÞ, t ∈ ½0, a, α ∈ ½0, 1: (11)
1 2
 
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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations
DOI: http://dx.doi.org/10.5772/ITexLi.94000
Theorem 1.1 [11] Let u : ½0, a!R be Seikkala differentiable and denote
F
½uðtÞα ¼ uαðtÞ, uαðtÞ . Then, the boundary function uαðtÞand uαðtÞare differentiable
1 2 1 2
and  
½u0ðtÞα ¼ uα 0ðtÞ, uα 0ðtÞ , α ∈ ½0, 1: (12)
1 2
h    i
Definition 2 [12] Let u : ½0, a!R . The fuzzy integral, denoted by
F
cuðtÞdt, b, c ∈ ½0, a, is defined levelwise by the following equation:
b
Ð
c α c c
uðtÞdt ¼ uαðtÞdt, uαðtÞdt , (13)
ð  ð 1 ð 2 
b b b
for all 0 ≤ α ≤ 1. In [12], if u : ½0, a!R is continuous, it is fuzzy integrable.
F
Theorem 1.2 [13] If u ∈ R , then the following properties hold:
F
i. ½uα2 ⊂ ½uα1, if 0 ≤ α ≤ α ≤ 1; (14)
1 2
ii. fα g⊂ ½0, 1is a nondecreasing sequence which converges to α then
k
½uα ¼ ⋂½uαk: (15)
k ≥ 1
Conversely if A ¼f uα, uα ; α ∈ ð0, 1gis a family of closed real intervals
α 1 2
verifying ðiÞand ðiiÞ, then fAαgdefined a fuzzy number u ∈ RF such that ½uα¼ Aα:
From [1], we have the following theorems:
Theorem 1.3 There exists a real Banach space X such that  can be the embedding
F
as a convex cone C with vertex 0 into X. Furthermore, the following conditions hold:
i. the embedding j is isometric,
ii. addition in X induces addition in  , i.e, for any u, v ∈  ,
F F
iii. multiplication by a nonnegative real number in X induces the
corresponding operation in  , i.e., for any u ∈  ,
F F
iv. C-C is dense in X,
v. C is closed.
3. Fuzzy conformable fractional differentiability and integral
Definition 3 [4] Let F : ð0, aÞ!F be a fuzzy function. γth order “fuzzy
conformable fractional derivative” of F is defined by
Fðtþ εt1γÞ⊖ FðtÞ FðtÞ⊖ Fðt εt1γÞ
TγðFÞðtÞ ¼ε l!im0þ ε ¼ ε!lim0þ ε : (16)
for all t > 0, γ ∈ ð0, 1Þ. Let FðγÞðtÞstands for T ðFÞðtÞ. Hence
γ
3
Fuzzy Systems - Theory and Applications
Fðtþ εt1γÞ⊖ FðtÞ FðtÞ⊖ Fðt εt1γÞ
FðγÞðtÞ ¼ lim ¼ lim : (17)
ε!0þ ε ε!0þ ε
If F is γ- differentiable in some ð0, aÞ, and limt!0þFðγÞðtÞexists, then
FðγÞð0Þ¼ limFðγÞðtÞ (18)
t!0þ
and the limits ( in the metric d).
Remark 1 From the definition, it directly follows that if F is γ-differentiable then the
multivalued mapping F is γ-differentiable for all α ∈ ½0, 1and
α
α
TγFα¼ FðγÞðtÞ , (19)
h i
where T F is denoted from the conformable fractional derivative of F of order
γ α α
γ. The converse result does not hold, since the existence of Hukuhara difference
½uα⊖ ½vα, α ∈ ½0, 1does not imply the existence of H-difference u ⊖ v:
Theorem 1.4 [4] Let γ ∈ ð0, 1.
If F is differentiable and F is γ-differentiable then
TγFðtÞ ¼t1γF0ðtÞ (20)
Theorem 1.5 [5, 14] If F : ð0, aÞ! is γ-differentiable then it is continuous.
F
Remark 2 If F : ð0, aÞ! is γ-differentiable and FðγÞfor all γ ∈ ð0, 1is
F
continuous, then we denote F ∈ C1ðð0, aÞ,  Þ.
F
Theorem 1.6 [5, 14] Let γ ∈ ð0, 1and if F, G : ð0, aÞ! are γ-differentiable
F
and λ ∈  then
TγðF þ GÞðtÞ ¼TγðFÞðtÞþTγðGÞðtÞ and TγðλFÞðtÞ¼λTγðFÞðtÞ: (21)
Definition 4 [5] Let F ∈ Cð ð0, aÞ,  Þ∩L1ðð0, aÞ,  Þ, Define the fuzzy
F F
fractional.
integral for a ≥ 0 and γ ∈ ð0, 1:
t F
IaðFÞðtÞ ¼Ia tγ1F ðtÞ ¼ ðsÞds, (22)
γ 1 ð s1γ
  a
where the integral is the usual Riemann improper integral.
Theorem 1.7 [5] TγIaγðFÞðtÞ, for t ≥ a, where F is any continuous function in the
domain of Ia.
γ
Theorem 1.8 [5] Let γ ∈ ð0, 1and F be γ-differentiable in ð0, aÞand assume
that the conformable derivative FðγÞ is integrable over ð0, aÞ. Then for each
s ∈ ð0, aÞwe have
FðsÞ¼FðaÞþIaFðγÞðtÞ (23)
γ
4. Existence and uniqueness solution to fuzzy fractional differential
equations
In this section we state the main results of the paper, i.e. we will concern
ourselves with the question of the existence theorem of approximate solutions by
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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations
DOI: http://dx.doi.org/10.5772/ITexLi.94000
using the embedding results on fuzzy number space ð , dÞand we prove the
F
uniqueness theorem of solution for the Cauchy problem of fuzzy fractional
differential equations of order γ ∈ ð0, 1.
4.1 Solution and its approximate solutions
Assume that F : ð0, aÞ !  is continuous Cð ðð0, aÞ Þ,  Þ. Consider
F F F F
the fractional initial value problem
TγðuÞðtÞ ¼Fðt, uðtÞÞ, uðt0Þ¼u0, (24)
where u ∈  and γ ∈ ð0, 1:
0 F
From Theorems (1.5), (1.7) and (1.8), it immediately follows:
Theorem 1.9 A mapping u : ð0, aÞ!  is a solution to the problem (24) if and
F
only if it is continuous and satisfies the integral equation
t
uðtÞ¼ u þ sγ1Fðs, uðsÞÞds (25)
0
ð
t0
for all t ∈ ð0, aÞand γ ∈ ð0, 1:
In the following we give the relation between a solution and its approximate
solutions.
We denote Δ ¼ ½t , t þ θBðu , μÞwhere θ, μ be two positive real numbers
0 0 0 0
u ∈  , Bðu , μÞ¼fx ∈  jdðu, u Þ≤ μg:
0 F 0 F 0
Theorem 1.10 Let γ ∈ ð0, 1and F ∈ C ðΔ ,  Þ, η ∈ ð0, θÞ,
0 F
u ∈ C1ð ½t , t þ η, Bðu , μÞÞsuch that
n 0 0 0
juðγÞðtÞ ¼jF ðt, u ðtÞÞþB ðtÞ, u ðt Þ¼u , ∥B ðtÞ∥≤ε (26)
n n n n 0 0 n n
∀t ∈ ½t , t þ η, n ¼ 1, 2, …:
0 0
where ε > 0, ε ! 0, B ðtÞ ∈ Cð½t , t þ η, XÞ, and j s the isometric embedding
n n n 0 0
from ð , dÞonto its range in the Banach space X. For each t ∈ ½t , t þ ηthere
F 0 0
exists an β > 0 such that the H-differences u ðt þ εt1γÞ⊖ u ðtÞand
n n
u ðtÞ⊖ u ðt  εt1γÞexist for all 0 ≤ ε < β and n ¼ 1, 2, …: If we have
n n
dðuðtÞ, uðtÞÞ!0 (27)
n
uniform convergence (u.c) for all t ∈ ½t , t þ η, n ! ∞, then
0 0
u ∈ C1ð½ t , t þ η, Bðu , μÞÞand
0 0 0
TγðuðtÞÞ¼ Fðt, uðtÞÞ, uðt0Þ¼u0, t ∈ ½t0, t0 þ η: (28)
Proof: By (27) we know that uðtÞ ∈ Cð½ t , t þ η, Bðu , μÞÞ. For fixed
0 0 0
t ∈ ½t , t þ ηand any t ∈ ½t , t þ η, t > t , denote ε ¼ htγ1 and ∀ γ ∈ ð0, 1
1 0 0 0 0 1 1
ju t þ εt1γ  ju ðt Þ
n 1 1 n 1
Gðt, nÞ¼    jF ðt , u ðt ÞÞB ðt Þ: (29)
ε 1 n 1 n 1
ju ðt þ hÞju ðt Þ
¼ n 1 n 1  jF ðt, u ðt ÞÞB ðt Þ: (30)
htγ1 1 n 1 n 1
1
ju ðtÞju ðt Þ
¼ t1γ n n 1  jF ðt , u ðtÞÞB ðt Þ: (31)
1 t  t 1 n 1 n 1
1
5
Fuzzy Systems - Theory and Applications
It is well know that
limGðt, nÞ¼jðu ÞðγÞðt ÞjF ðt, u ðt ÞÞB ðt Þ (32)
n 1 1 n 1 n 1
t!t1
¼ jðu ÞðγÞðt ÞjF ðt , u ðtÞÞB ðt Þ¼ Θ ∈ X (33)
n 1 1 n 1 n 1
juð tÞju ðt Þ
limGðt, nÞ¼t1γ 1  jF ðt , uðtÞÞ (34)
n!∞ 1 t  t 1 1
1
From F ∈ C1ðΔ ,  Þ, is know that for any ε > 0, there exists β > 0 such that
0 F 1
ε
dðFðt, vÞ, Fðt , uðt ÞÞÞ< (35)
1 1 4
whenever t < t < t þ β and dðv, uðt ÞÞ< β with v ∈ Bðu , μÞTake natural
1 1 1 1 1 0
number N > 0 such hat
ε β
ε < , dðu ðtÞ, uðtÞÞ< 1 for any n > N, t ∈ ½t , t þ η (36)
n n 0 0
4 2
Take β > 0 such that β < β and
1
β
dðuðtÞ, uðtÞÞ< 1 (37)
1 2
whenever t < t < t þ β:
1 1
By the definition of Gðt, nÞand (26), we have ∀γ ∈ ð0, 1
ju t þ εt1γ  ju ðt ÞðεÞjðu ÞðγÞðt Þ¼ðεÞjF ðt, u ðt ÞÞ (38)
n 1 1 n 1 n 1 1 n 1
 
t1γ ju ðtÞju ðt Þ  ðt  t Þt1γjðuÞ0ðt Þ¼ðt  t ÞjFð t , u ðt ÞÞ (39)
1 n n 1 1 1 n 1 1 1 n 1
 
We choose ψ ∈ X∗ such that ∥ψ ∥ ¼ 1 and for all γ ∈ ð0, 1
ψ t1γ ju ðtÞju ðt Þ  ðt  t Þt1γjðu Þ0ðt Þ (40)
1 n n 1 1 1 n 1
 
 
¼ ∥t1γ ju ðtÞju ðt Þ  ðt  t Þt1γjðu Þ0ðt Þ∥ (41)
1 n n 1 1 1 n 1
 
Let t1γφð tÞ¼t1γψ ju ðtÞ  ðt  t Þt1γjðu Þ0ðt Þ, consequently
1 1 n 1 1 n 1
 
t1γφ0ðtÞ¼t1γψ ju0ðtÞ  t1γjðu Þ0ðt Þ (42)
1 1 n 1 n 1
 
hence
∥t1γ ju ðtÞju ðt Þ  ðt  tÞt1γjðu Þ0ðt Þ∥ (43)
1 n n 1 1 1 n 1
 
¼ t1γðφð tÞφ ðt ÞÞ¼t1γφ0ð^tÞðt  t Þ (44)
1 1 1 1
¼ ψ t1γ ju0ð^tÞju0ðt Þ ðt  t Þ (45)
1 n n 1 1
  
≤∥ψ ∥∥t1γ ju0ð^tÞju0ðtÞ ∥ð t  t Þ (46)
1 n n 1 1
 
¼ ∥t1γ ju0ð^tÞju0ðt Þ ∥ ðt  t Þ, (47)
1 n n 1 1
^
where t ≤t ≤ t: In view of (39), we have
1
 
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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations
DOI: http://dx.doi.org/10.5772/ITexLi.94000
∥Gðt, nÞ∥≤∥t1γ ju0ð^tÞju0 ðt Þ ∥, t ≤^t ≤ t: (48)
1 n n 1 1
 
From (36) and (37) we know that
β
dðuð^tÞ, uðt ÞÞ< 1 (49)
1
2
and
^ ^ ^ ^
dðu ðtÞ, uðt ÞÞ≤ dðu ðtÞ, uðtÞÞþdðuðtÞ, uðt ÞÞ (50)
n 1 n 1
β β
< 1 þ 1¼ β (51)
2 2 1
Hence by (35) and (48) we have for all γ ∈ ð0, 1:
∥Gðt, nÞ∥≤∥t1γ ju0ð^tÞju0ðt Þ ∥ (52)
1 n n 1
^ ^ ^  
¼ ∥jFðt, uðtÞÞþB ðtÞjF ðt , u ðt ÞÞB ðt Þ∥ (53)
n n 1 n 1 n 1
^ ^
≤∥jFðt, u ðtÞÞjF ðt , uðt ÞÞ∥ (54)
n 1 1
þ∥jFð t , uðt ÞÞjF ðt , u ðt ÞÞ∥ þ 2ε (55)
1 1 1 n 1 n
^ ^
≤ dðj Fðt, u ðtÞÞjF ðt , uðt ÞÞÞ (56)
n 1 1
þdðj Fðt , uðt ÞÞjF ðt , u ðtÞÞÞþ2ε (57)
1 1 1 n 1 n
ε ε
< þ þ 2ε < ε (58)
n
4 4
whenever n > N and t < t < t þ β:
1 1
Let n ! ∞, and applying (34), we have
juð tÞju ðt Þ
∥t1γ 1  jF ðt , uðt ÞÞ∥≤ε, t < t < t þ β: (59)
1 t  t 1 1 1 1
1
On the other hand, from the assumption of Theorem (1.9), there exists an
β ðt Þ ∈ ð0, βÞsuch that the H-differences u ðtÞ⊖ u ðt Þexist for all
1 n n 1
t ∈ ½t , t þ βð tÞand n ¼ 1, 2, …:
1 1 1
Now let v ðtÞ ¼u ðtÞ⊖ u ðt Þwe verify that the fuzzy number-valued sequence
n n n 1
fv ðtÞguniformly converges on ½t , t þ β ðt Þ. In fact, from the assumption
n 1 1 1
dðu ðtÞ, uðtÞÞ!0 u.c. for all t ∈ ½t , t þ η, we know
n 0 0
dðv ðtÞ, v ðtÞÞ¼dðv ðtÞþu ðt Þ, v ðtÞþu ðt ÞÞ (60)
n m n n 1 m n 1
≤ dðu ðtÞ, u ðtÞÞþdðu ðtÞ, v ðtÞþu ðt ÞÞ (61)
n m m m n 1
¼ dðu ðtÞ, u ðtÞÞþdðv ðtÞþu ðt Þ, v ðtÞþu ðt ÞÞ (62)
n m m m 1 m n 1
¼ dðu ðtÞ, u ðtÞÞþdðu ðt Þ, u ðt ÞÞ (63)
n m m 1 n 1
! u:c ∀t ∈ ½t, t þ β ðt Þ n, m ! ∞: (64)
1 1 1
Since ð , dÞis complete, there exists a fuzzy number-valued mapping vðtÞsuch
F
that fvðtÞgu.c to vðtÞon ½t , t þ β ðt Þas n ! ∞:
n 1 1 1
In addition, we have
dðuðt ÞþvðtÞ, uðtÞÞ≤ dð uðt ÞþvðtÞ, u ðt þ v ðt ÞÞþdðu ðt þ v ðtÞ, uðtÞÞ (65)
1 1 n 1 n 1 n 1 n
≤ dðuðt ÞþvðtÞ, uðt ÞþvðtÞÞ (66)
1 1 n
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Fuzzy Systems - Theory and Applications
þdðuðt Þþv ðtÞ, u ðt Þþv ðtÞÞþdðu ðtÞ, uðtÞÞ (67)
1 n n 1 n n
¼ dðv ðtÞ, uðtÞÞþdðu ðt Þ, uðt ÞÞþdðu ðtÞ, uðtÞÞ (68)
n n 1 1 n
∀t ∈ ½t , t þ β ðt Þ:
1 1 1
Let n ! ∞: It follows that
uðt ÞþvðtÞuðtÞ forall t ∈ ½t , t þ β ðt Þ: (69)
1 1 1 1
Hence the H-difference uðtÞ⊖ uðt Þexist for all t ∈ ½t , t þ β ðt Þ:
1 1 1 1
Thus from (59) we have for all γ ∈ ð0, 1:
u t þ t1γε ⊖ uðtÞ
1 1 1
d0  ε  , Fðt1, uðt1ÞÞ1≤ ε , t ∈ ½t1, t1þ β ðt1Þ: (70)
@ A
So, limε!0þu t1þ t11γε ⊖ uðt1Þ=ε ¼ Fðt1, uðt1ÞÞ: Similarty, we have
 
u t þ t1γε ⊖ uðt Þ
1 1 1
lim   ¼ Fðt , uðt ÞÞ:
ε!0 ε 1 1
Hence uðγÞðt Þexists and
1
uðγÞðt Þ¼ Fðt , uðt ÞÞ: (71)
1 1 1
from t ∈ ½t , t þ ηis arbitrary, we know that Eq. (28) holds true and
1 0 0
u ∈ C1ð½ t , t þ η, Bðu , μÞÞ:The proof is concluded.
0 0 0
Lemma 2 For all t ∈ ½t , t þ η, n ¼ 1, 2, … and γ ∈ ð0, 1:
0 0
If we replace Eq. (26) by
ju ðtÞ ¼jFð t , u ðtÞÞþB ðtÞ, u ðt Þ¼u , ∥B ðtÞ∥≤ε , (72)
nþ1 n n n 0 0 n n
retain other assumptions, then the conclusions also hold true.
Proof: This is completely similar to the proof of Theorem (1.10), hence itis
omitted here.
4.2 Uniqueness solution
In this section, by using existence theorom of approximate solutions, and the
embedding results on fuzzy number space ð , dÞ,wegivetheexistenceanduniqueness
F
theorem for the Cauchy problem of the fuzzy fractional differential equations of order γ:
Theorem 1.11
i. Let F ∈ C ðΔ ,  Þand d Fðt, uÞ,^0 ≤ σ for all ðt, uÞ∈ Δ :
0 F 0
 
ii. G ∈ Cð½ t , t þ θ½0, μ, Þ, Gðt,0Þ0, and 0 ≤ Gðt, yÞ≤ σ , for all
0 0 1
t ∈ ½t , t þ θ,0≤ y ≤ μ such that Gðt, yÞis noncreasing on y the fractional
0 0
initial value problem
TγyðtÞ ¼Gðt, yðtÞÞ, yðt0Þ¼0 (73)
has only the solution yðtÞ0on ½t , t þ θ.
0 0
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Existence, Uniqueness and Approximate Solutions of Fuzzy Fractional Differential Equations
DOI: http://dx.doi.org/10.5772/ITexLi.94000
iii. dðF ðt, uÞ, Fðt, vÞÞ≤ Gðt, dðu, vÞÞfor all ðt, uÞ, ðt, vÞ ∈ Δ , and dðu, vÞ≤ μ:
0
Then the Cauchy problem (28) has unique solution u ∈ C1ð½t , t þ η, Bðu , μÞÞ
0 0 0
on ½t , t þ η, where η ¼ min fθ, μ=σ, μ=σ g, and the successive iterations
0 0 1
t
u ðtÞ¼u þ sγ1Fðs, u ðsÞÞds (74)
nþ1 0 n
ð
t0
uniformly converge to uðtÞon ½t , t þ η:
0 0
Proof: In the proof of Theorem 4.1 in [15], taking the conformable derivative uðγÞ
for all γ ∈ ð0, 1, using theorem (1.4) and properties (9), then we obtain the proof of
Theorem (1.11).
Example 1 Let L > 0 is a constant, taking Gðt, yÞ¼Ly in the proof of Theorem (4.2),
then obtain the proof of Corollary 4.1 in [15] whereσ1¼ Lμ,henceη ¼
min fθ, μ=σ,1=Lg. Then the Cauchy problem (28) has unique solution
u ∈ C1ð½ t , t þ η, B ðΔ , μÞÞ, and the successive iterations (74) uniformly converge to
0 0 0
uðtÞon ½t , t þ η:
0 0
5. Conclusion
In this work, we introduce the concept of conformable differentiability for fuzzy
mappings, enlarging the class of γ-differentiable fuzzy mappings where γ ∈ ð0, 1.
Subsequently, by using the γ-differentiable and embedding theorem, we study the
Cauchy problem of fuzzy fractional differential equations for the fuzzy valued
mappings of a real variable. The advantage of the γ-differentiability being also
practically applicable, and we can calculate by this derivative the product of two
functions because all fractional derivatives do not satisfy see [4].
On the other hand, we show and prove the relation between a solution and its
approximate solutions to the Cauchy problem of the fuzzy fractional differential
equation, and the existence and uniqueness theorem for a solution to the problem
(1) are proved.
For further research, we propose to extend the results of the present paper and
to combine them the results in citeref for fuzzy conformable fractional differential
equations.
Conflict of interest
The authors declare no conflict of interest.
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