# On the Existence and Uniqueness for High Order Fuzzy Fractional Differential Equations with Uncertainty

Authors: Abdourazek Souahi; Assia Guezane-Lakoud; Amara Hitta
Journal: Advances in Fuzzy Systems
 (2016)

---

## Abstract

A class fuzzy fractional differential equation (FFDE) involving Riemann-LiouvilleH-differentiability of arbitrary order q
>
1 is considered. Using Krasnoselskii-Krein type conditions, Kooi type conditions, and Rogers conditions we establish the uniqueness and existence of the solution after determining the equivalent integral form of the solution.


---

## Body

## 1. Introduction

In the last several years, fractional differential equations attracted more and more researchers and have proven to be very useful tools for modeling phenomena in physics, finance, and many other areas. The Riemann-Liouville formulation arises in a natural way for problems such as transport problems from the continuum random walk scheme or generalizes Chapman-Kolmogorov models [1, 2]. It was also applied for modeling the behavior of viscoelastic and viscoplastic materials under external influences [3, 4] and the continuum and statistical mechanics for viscoelasticity problems [5]. Some of the several research papers were published to consider the uniqueness of the solution for fractional differential equations under Nagumo like conditions (see, e.g., [6–13] and references therein). In [11, 12], Lakshmikantham and Leela established the uniqueness of the solution for the problem D
q
x
(
t
)
=
f
(
t
,
x
(
t
)
), where 0
<
q
<
1. Then, in [13], Yoruk et al. proved the uniqueness of the solution via Krasnoselskii-Krein, Rogers, and Kooi conditions, for 1
<
q
<
2.On the other hand, in order to obtain more realistic modeling of phenomena, one has to take uncertainty; see [14–16] and the references cited therein. Many researchers have worked in the theoretical and numerical aspect of fractional and fuzzy differential equations; the reader is kindly referred to [2, 17–31] and the references therein.In [32], Allahviranloo and Ahmadi introduced the fuzzy Laplace transform, which they used under the strongly generalized differentiability. Recently, ElJaoui et al. [33] developed it further. The newly defined fuzzy Laplace transform [32] for high order fuzzy derivatives is one of the most useful methods as mentioned by Jafarian et al. in [34]: “…, one of the important and interesting transforms in the problems of fuzzy equations is Laplace transforms. The fuzzy Laplace transform method solves fuzzy fractional differential equations and fuzzy boundary and initial value problems [35–38] ….”Motivated by the above works, we adopted the fuzzy Laplace transform to prove the uniqueness and existence for the following initial value problems (FFDE) for arbitrary orderq
>
1:(1)
D
q
x
t
=
f
t
,
x
t
,
D
q
-
1
x
t
,
x
0
=
y
0
,
D
q
-
i
x
0
=
0
~
,
i
=
1
,
…
,
q
, where y
0
∈
E and f
:
E
0
→
E is a continuous fuzzy-valued function with (2)
E
0
=
t
,
x
:
0
≤
t
≤
1
,
d
x
t
,
0
~
≤
b
, where d is the Hausdorff distance.Our aim is to both generalize and extend the previous uniqueness results of [13, 39].The organization of this paper is as follows. Section2 contains some basic definitions concerning fuzzy set theory and Riemann-Liouville generalized H-differentiability. In Section 3, using the fuzzy Laplace transform we determine the equivalent integral problem. Section 4 is devoted to the main results: a Krasnoselskii-Krein type of uniqueness theorem, a Kooi type uniqueness theorem, and a Rogers type uniqueness theorem; then we prove that the successive approximations converge to the unique solution.


## 2. Preliminaries

First, let us recall some basic definitions about fuzzy numbers and fuzzy sets. Here and in the rest of the paper, we denote byΓ the Gamma function and [
α
] the integer part of α.As defined in [40], E
=
{
u
:
R
→
[
0,1
]
;
u
s
a
t
i
s
f
i
e
s
A
1
–
(
A
4
)
} is the space of fuzzy numbers:(A1)
u is normal; that is, there exists y
0
∈
R such that u
(
y
0
)
=
1.
(A2)
u is fuzzy convex; that is, u
(
λ
y
+
(
1
-
λ
)
z
)
≥
m
i
n
{
u
(
y
)
,
u
(
z
)
} whenever y
,
z
∈
R and λ
∈
[
0,1
].
(A3)
u is upper semicontinuous; that is, for any y
0
∈
R and ε
>
0 there exists δ
(
y
0
,
ε
)
>
0 such that u
(
y
)
<
u
(
y
0
)
+
ε whenever |
y
-
y
0
|
<
δ, y
∈
R
n.
(A4)
The closure of the set{
y
∈
R
;
u
(
y
)
>
0
} is compact.The set[
u
]
α
=
{
u
∈
R
;
u
(
y
)
≥
α
} is called α-level set of u.It follows fromA
1
–
(
A
4
) that the α-level sets [
u
]
α are convex compact subsets of R for all α
∈
(
0,1
]. The fuzzy zero is defined by (3)
0
~
=
0
if
y
≠
0
,
1
if
y
=
0
.Definition 1 (see [40]).
A fuzzy numberu in the parametric form is a pair (
u
_
(
r
)
,
u
¯
(
r
)
) of functions u
_
(
r
)
,
u
¯
(
r
),  0
≤
r
≤
1, which satisfy the following conditions:(1)
u
_
(
r
) is a bounded nondecreasing left continuous function in (
0,1
] and right continuous at 0;
(2)
u
¯
(
r
) is a bounded nonincreasing left continuous function in (
0,1
] and right continuous at 0;
(3)
u
_
(
r
)
≤
u
¯
(
r
),  0
≤
r
≤
1.
Moreover, we also can present ther-cut representation of fuzzy numbers as [
u
]
r
=
[
u
_
(
r
)
,
u
¯
(
r
)
] for all 0
≤
r
≤
1.According to Zadeh’s extension principle, we have the following properties of fuzzy addition and multiplication by scalar onE: (4)
u
⊕
v
x
=
s
u
p
y
∈
R
min
⁡
u
y
,
v
x
-
y
,
x
∈
R
,
k
⊙
u
x
=
u
x
k
if
k
≥
0
,
0
~
if
k
=
0
. Seeking simplicity, we note ⊕
,
⊙ by the usual +
,
·. The Hausdorff distance between the fuzzy numbers is denoted by d
:
E
×
E
→
[
0
,
+
∞
[, such that d
(
u
,
v
)
=
s
u
p
r
∈
[
0,1
]
m
a
x
{
|
u
_
(
r
)
-
v
_
(
r
)
|
,
|
u
¯
(
r
)
-
v
¯
(
r
)
|
}. And (
d
,
E
) is a complete metric space.Definition 2.
Letx
,
y
∈
E. If there exists z
∈
E such that x
=
y
+
z, then z is called the H-difference of x and y, and it is denoted by x
⊖
y.Remark 3.
Note that the sign⊖ stands for H-difference and x
⊖
y
≠
x
+
(
-
1
)
y.We denote byC
F
[
0
,
a
] the space of all fuzzy-valued functions which are continuous on [
0
,
a
] and L
F
[
0
,
a
] the space of all Lebesgue integrable fuzzy-valued functions on [
0
,
a
], where a
>
0. We also denote by A
C
(
n
-
1
)
F
[
0
,
a
] the space of fuzzy-valued functions f which have continuous H-derivatives up to order n
-
1 on [
0
,
a
] such that f
(
n
-
1
) in A
C
F
[
0
,
a
].Definition 4 (see [41]).
Letf
∈
C
F
[
0,1
]
∩
L
F
[
0,1
]. The fuzzy fractional integral of the fuzzy-valued function f is defined by (5)
I
β
f
x
;
r
=
I
β
f
_
x
;
r
,
I
β
f
¯
x
;
r
,
0
≤
r
≤
1
, where (6)
I
β
f
_
x
;
r
=
1
Γ
β
∫
0
x
x
-
s
β
-
1
f
_
s
;
r
d
s
,
I
β
f
¯
x
;
r
=
1
Γ
β
∫
0
x
x
-
s
β
-
1
f
¯
s
;
r
d
s
.Definition 5 (see [41, Definition 6]).
Letf
∈
C
(
n
)
F
[
0,1
]
∩
L
F
[
0,1
],  x
0
∈
(
0,1
), and Φ
(
x
)
=
(
1
/
Γ
(
n
-
β
)
)
∫
0
x
(
f
(
t
)
d
t
/
(
x
-
t
)
β
-
n
+
1
), where n
=
[
β
]
+
1. One says that f is fuzzy Riemann-Liouville fractional differentiable of order β at x
0, if there exists an element D
0
β
f
(
x
0
)
∈
E, such that, for all h
>
0 sufficiently small, one has(i)
(7)
D
0
β
f
x
0
=
lim
h
→
0
⁡
Φ
n
-
1
x
0
+
h
⊖
Φ
n
-
1
x
0
h
=
lim
h
→
0
⁡
Φ
n
-
1
x
0
⊖
Φ
n
-
1
x
0
-
h
h
,
or
(ii)
(8)
D
0
β
f
x
0
=
lim
h
→
0
⁡
Φ
n
-
1
x
0
⊖
Φ
n
-
1
x
0
+
h
h
=
lim
h
→
0
⁡
Φ
n
-
1
x
0
-
h
⊖
Φ
n
-
1
x
0
h
.Denote byC
(
n
-
1
)
F
(
[
0
,
a
]
) the space of fuzzy-valued functions f on the bounded interval [
0
,
a
] which have continuous H-derivative up to order n
-
2 such that f
(
n
-
1
)
∈
C
F
[
0
,
a
]. C
(
n
-
1
)
F
(
[
0
,
a
]
) is a complete metric space endowed by the metric D such that for every g
,
h
∈
C
(
n
-
1
)
F
(
[
0
,
a
]
)
(9)
D
g
,
h
=
∑
i
=
0
n
-
1
sup
t
∈
0
,
a
⁡
d
g
i
t
,
h
i
t
.In the rest of the paper, we say that a fuzzy-valued functionf is [
(
i
)
-
β
]
R
L-differentiable if it is differentiable as in Definition 5 case (i) and is [
(
i
i
)
-
β
]
R
L-differentiable if it is differentiable as in Definition 5 case (ii).Definition 6 (see [41, Theorem 7]).
Letf
∈
C
(
n
)
F
[
0,1
]
∩
L
F
[
0,1
],  x
0
∈
(
0,1
), and Φ
(
x
)
=
(
1
/
Γ
(
n
-
β
)
)
∫
0
x
(
f
(
t
)
d
t
/
(
x
-
t
)
β
-
n
+
1
), where n
=
[
β
]
+
1 such that 0
≤
r
≤
1; then one has the following:(i)
iff is [
(
i
)
-
β
]
R
L-differentiable fuzzy-valued function, then (10)
D
0
β
f
x
0
;
r
=
D
0
β
f
_
x
0
;
r
,
D
0
β
f
¯
x
0
;
r
,
or
(ii)
iff is [
(
i
)
-
β
]
R
L-differentiable fuzzy-valued function, then (11)
D
0
β
f
x
0
;
r
=
D
0
β
f
¯
x
0
;
r
,
D
0
β
f
_
x
0
;
r
,
where(12)
D
0
β
f
_
x
0
;
r
=
1
Γ
n
-
β
d
d
x
n
∫
0
x
x
-
t
n
-
β
-
1
f
_
t
;
r
d
t
x
=
x
0
,
D
0
β
f
¯
x
0
;
r
=
1
Γ
n
-
β
d
d
x
n
∫
0
x
x
-
t
n
-
β
-
1
f
¯
t
;
r
d
t
x
=
x
0
.The following theorem is an important one about the fuzzy Laplace transformL of the Riemann-Liouville H-derivative for fuzzy-valued functions.Theorem 7.
Suppose thatf
∈
C
(
n
)
F
[
0
,
∞
)
∩
L
F
[
0
,
∞
); one has the following:(i)
iff is [
(
i
)
-
β
]
R
L-differentiable fuzzy-valued function, then(13)
L
D
0
β
f
x
=
p
β
L
f
t
⊖
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
f
0
or
(ii)
iff is [
(
i
)
-
β
]
R
L-differentiable fuzzy-valued function, then(14)
L
D
0
β
f
x
=
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
f
0
⊖
-
p
β
L
f
t
.The proof runs along similar lines as that of [41, Theorem 16], and we omit it here.


## 3. Fuzzy Fractional Integral Equation

In this section, we study the relation between problem (1) and the fuzzy integral form using the well-known fuzzy Laplace transform.In fact, by taking Laplace transform on both sides of(15)
D
q
x
t
=
f
t
,
x
t
,
D
q
-
1
x
t
≜
r
t
,
x we get (16)
L
D
q
x
t
=
L
f
t
,
x
t
,
D
q
-
1
x
t
. Based on the type of Riemann-Liouville H-differentiability, we obtain two cases.Case 1.
IfD
q
x is [
(
i
)
-
q
]
R
L-differentiable fuzzy-valued function, then(17)
L
r
t
,
x
=
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
0
⊖
p
q
L
x
t
, and based on the lower and upper functions of D
q
x the above equation becomes(18)
L
r
_
t
,
x
;
r
=
p
q
L
x
_
t
;
r
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
_
0
;
r
,
L
r
¯
t
,
x
;
r
=
p
q
L
x
¯
t
;
r
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
¯
0
;
r
, where (19)
L
r
_
t
,
x
;
r
=
min
⁡
r
t
,
u
∣
u
∈
x
_
t
;
r
,
x
¯
t
;
r
,
0
≤
r
≤
1
,
L
r
¯
t
,
x
;
r
=
max
⁡
r
t
,
u
∣
u
∈
x
_
t
;
r
,
x
¯
t
;
r
,
0
≤
r
≤
1
. In order to solve system (18), and for the sake of simplicity, we assume that(20)
L
x
_
t
;
r
=
H
1
p
;
r
,
L
x
¯
t
;
r
=
K
1
p
;
r
, where H
1
(
p
;
r
) and K
1
(
p
;
r
) are solutions of the previous system (18); it yields (21)
x
_
t
;
r
=
L
-
1
H
1
p
;
r
,
x
¯
t
;
r
=
L
-
1
K
1
p
;
r
.Case 2.
IfD
q
x is [
(
i
i
)
-
q
]
R
L-differentiable fuzzy-valued function, then (22)
L
r
t
,
x
=
p
q
L
x
t
⊖
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
0
, and based on the lower and upper functions of D
q
x the above equation becomes(23)
L
r
_
t
,
x
;
r
=
p
q
L
x
_
t
;
r
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
_
0
;
r
,
L
r
¯
t
,
x
;
r
=
p
q
L
x
¯
t
;
r
-
∑
k
=
0
n
-
1
p
k
D
β
-
k
-
1
x
¯
0
;
r
, where (24)
L
r
_
t
,
x
;
r
=
min
⁡
r
t
,
u
∣
u
∈
x
_
t
;
r
,
x
¯
t
;
r
,
0
≤
r
≤
1
,
L
r
¯
t
,
x
;
r
=
max
⁡
r
t
,
u
∣
u
∈
x
_
t
;
r
,
x
¯
t
;
r
,
0
≤
r
≤
1
. In order to solve system (23), and for the sake of simplicity, we assume that (25)
L
x
_
t
;
r
=
H
2
p
;
r
,
L
x
¯
t
;
r
=
K
2
p
;
r
, where H
2
(
p
;
r
) and K
2
(
p
;
r
) are solutions of the previous system (23). Then we obtain(26)
x
_
t
;
r
=
L
-
1
H
2
p
;
r
,
x
¯
t
;
r
=
L
-
1
K
2
p
;
r
.
Taking into account the initial conditions of problem (1) and using the linearity of the inverse Laplace transform on systems (20) and (26), we obtain the following for both cases.
x is a solution for problem (1) if and only if x is a solution for the following integral equation:(27)
x
t
=
y
0
+
1
Γ
q
∫
0
t
t
-
s
q
-
1
r
s
,
x
d
sin the sense of [
(
i
)
-
q
]
R
L-differentiability, and(28)
x
^
t
=
y
0
⊖
-
1
Γ
q
∫
0
t
t
-
s
q
-
1
r
s
,
x
d
s in the sense of [
(
i
i
)
-
q
]
R
L-differentiability.


## 4. Main Results

Now, we state the Krasnoselskii-Krein type conditions for FFDE (1).Theorem 8.
Letf
∈
C
(
E
0
,
E
) satisfy the following Krein type conditions: (H1)
d
(
f
(
t
,
x
,
y
)
,
f
(
t
,
x
¯
,
y
¯
)
)
≤
m
i
n
{
Γ
(
q
)
,
1
}
k
+
α
(
q
-
[
q
]
)
/
2
t
1
-
α
(
q
-
[
q
]
)
d
(
x
,
x
¯
)
+
d
(
y
,
y
¯
),  t
≠
0 and 0
<
α
<
1,
(H2)
d
(
f
(
t
,
x
,
y
)
,
f
(
t
,
x
¯
,
y
¯
)
)
≤
δ
d
(
x
,
x
¯
)
α
+
t
α
(
q
-
[
q
]
)
d
(
y
,
y
¯
)
α, where δ and k are positive constants and k
(
1
-
α
)
<
1
+
α
(
q
-
[
q
]
); then in the sense of [
(
i
)
-
q
]
R
L-differentiability, the solution x is unique and in the sense of [
(
i
)
-
q
]
R
L-differentiability, the solution x
^ is unique on [
0
,
η
], where η
=
m
i
n
1
,
b
Γ
(
1
+
q
)
/
M
1
/
q
,
d
/
M and M is the bound for f on E
0: that is, d
(
f
,
0
~
)
≤
M.Proof.
First we establish the uniqueness; supposex and y are any two solutions of (1) in [
(
i
)
-
q
]
R
L-differentiability and let ϕ
(
t
)
=
d
(
x
(
t
)
,
y
(
t
)
) and θ
(
t
)
=
d
(
D
q
-
1
x
(
t
)
,
D
q
-
1
y
(
t
)
). Note that ϕ
(
0
)
=
θ
(
0
)
=
0.
We defineR
(
t
)
=
∫
0
t
ϕ
α
(
s
)
+
s
α
(
q
-
[
q
]
)
θ
α
(
s
)
d
s; clearly R
(
0
)
=
0.
Using (27) and condition (H2), we get (29)
ϕ
t
≤
δ
Γ
q
∫
0
t
t
-
s
q
-
1
ϕ
α
s
+
s
α
q
-
q
θ
α
s
d
s
≤
δ
Γ
q
t
q
-
1
R
t
,
θ
t
≤
∫
0
t
δ
ϕ
s
α
+
t
α
q
-
q
θ
s
α
d
s
≤
δ
R
t
. For the sake of simplicity we use the same symbol C to denote all different constants arising in the rest of the proof.
We have(30)
R
′
t
=
ϕ
α
t
+
t
α
q
-
q
θ
α
t
≤
C
t
α
q
-
1
+
t
α
q
-
q
R
α
t
. Since R
(
t
)
>
0 for t
>
0, multiplying both sides of (30) by (
1
-
α
)
R
-
α
(
t
) and then integrating the resulting inequality, we get(31)
R
t
≤
C
t
α
/
1
-
α
q
+
1
+
t
α
/
1
-
α
q
+
1
-
α
q
/
1
-
α
. Using the fact that(32)
a
+
b
1
-
α
≤
1
2
1
-
α
-
1
a
1
-
α
+
b
1
-
α for every a
,
b
∈
(
0,1
), (31) becomes(33)
R
t
≤
C
t
α
/
1
-
α
q
+
1
+
t
α
/
1
-
α
q
+
1
-
α
q
/
1
-
α
. This leads to the following estimates on ϕ and θ, for t
∈
[
0
,
η
]:(34)
ϕ
t
≤
C
t
q
/
1
-
α
+
t
q
/
1
-
α
+
α
1
-
q
/
1
-
α
,
θ
t
≤
C
t
α
/
1
-
α
q
+
1
+
t
α
/
1
-
α
q
+
1
-
α
q
/
1
-
α
. Define the function ψ
(
t
)
=
t
-
k
m
a
x
{
ϕ
(
t
)
,
θ
(
t
)
} for t
∈
(
0,1
]. When either t
-
k
ϕ
(
t
) or t
-
k
θ
(
t
) is the maximum, we get(35)
0
≤
ψ
t
≤
C
t
q
/
1
-
α
-
k
+
t
q
/
1
-
α
+
α
1
-
q
/
1
-
α
-
k
, or(36)
0
≤
ψ
t
≤
C
t
α
/
1
-
α
q
+
1
-
k
+
t
α
q
/
1
-
α
+
1
-
α
q
/
1
-
α
-
k
. Since k
(
1
-
α
)
<
1
+
α
(
q
-
[
q
]
) (by assumption), we have(37)
k
1
-
α
<
1
+
α
q
-
q
⟹
k
1
-
α
<
q
k
-
1
1
-
α
<
α
q
k
1
-
α
<
q
+
α
-
α
q
k
1
-
α
<
α
q
+
1
-
α
q
. So all of the exponents of t in the above inequalities are positive. Hence, l
i
m
t
→
0
+
ψ
(
t
)
=
0. Therefore, if we define ψ
(
0
)
=
0, the function ψ is continuous in [
0
,
η
].
We want to prove thatψ
≡
0. In fact, since the function ψ is continuous, if ψ does not vanish at some points t, that is, ψ
(
t
)
>
0 on ]
0
,
η
], then there exists a maximum m
>
0 reached when t is equal to some t
1: 0
<
t
1
≤
η
≤
1 such that ψ
(
s
)
<
m
=
ψ
(
t
1
), for s
∈
]
0
,
t
1
). But, from condition (H1) we get for either cases(38)
m
=
ψ
t
1
=
t
1
-
k
ϕ
t
1
≤
min
⁡
Γ
q
,
1
m
t
1
q
-
1
+
α
q
-
q
<
m or(39)
m
=
ψ
t
1
=
t
1
-
k
θ
t
1
≤
min
⁡
Γ
q
,
1
m
t
1
α
q
-
q
<
m which is a contradiction. Thus, the uniqueness of the solution is established in the sense of [
(
i
)
-
q
]
R
L-differentiability. The second part of the proof is almost completely similar to the [
(
i
)
-
q
]
R
L-differentiability; thus, we omit it.Remark 9.
For the case1
<
q
<
2, of the deterministic case, Theorem 8 is reduced to [13, Theorem 3.1].Theorem 10 (Kooi’s type uniqueness theorem).
Letf satisfy the following conditions: (J1)
d
(
f
(
t
,
x
,
y
)
,
f
(
t
,
x
¯
,
y
¯
)
)
≤
m
i
n
{
Γ
(
q
)
,
1
}
k
+
α
(
q
-
[
q
]
)
/
2
t
1
-
α
(
q
-
[
q
]
)
d
(
x
,
x
¯
)
+
d
(
y
,
y
¯
),  t
≠
0 and 0
<
α
<
1;
(J2)
t
β
d
(
f
(
t
,
x
,
y
)
,
f
(
t
,
x
¯
,
y
¯
)
)
≤
c
d
(
x
,
x
¯
)
α
+
t
α
(
q
-
[
q
]
)
d
(
y
,
y
¯
)
α,where c and k are positive constants and k
(
1
-
α
)
<
1
+
α
(
q
-
[
q
]
)
-
β, for (
t
,
x
,
y
)
,
(
t
,
x
¯
,
y
¯
)
∈
R
0; then in the sense of [
(
i
)
-
q
]
R
L-differentiability, the solution x is unique and in the sense of [
(
i
)
-
q
]
R
L-differentiability, the solution x
^ is unique.Proof.
It is similar to that of Theorem8; thus, we omit it.Lemma 11.
Letϕ and θ be two nonnegative continuous functions in the interval [
0
,
η
] for a real number a
>
0. Let ψ
(
t
)
=
∫
0
t
ϕ
(
s
)
+
s
q
-
[
q
]
θ
(
s
)
/
2
s
q
-
[
q
]
+
2
d
s. Assume the following: (i)
ϕ
(
t
)
≤
t
q
-
[
q
]
ψ
(
t
),
(ii)
θ
(
t
)
≤
ψ
(
t
),
(iii)
ϕ
(
t
)
=
o
(
t
q
-
[
q
]
e
-
1
/
t
),
(iv)
θ
(
t
)
=
o
(
e
-
1
/
t
). Then ϕ
≡
θ
≡
0.Proof.
Letψ
(
t
)
=
∫
0
t
ϕ
(
s
)
+
s
q
-
[
q
]
θ
(
s
)
/
2
s
q
-
[
q
]
+
2
d
s. After differentiating ψ and using (ii), we obtain, for t
>
0,  ψ
′
(
t
)
≤
1
/
t
2
ψ
(
t
), so that e
1
/
t
ψ
(
t
) is decreasing. Now, from (iii) and (iv), if ϵ
>
0 then, for a small t, we have(40)
e
1
/
t
ψ
t
≤
e
1
/
t
∫
0
t
1
2
s
2
2
ϵ
e
-
1
/
s
d
s
=
ϵ
. Hence, l
i
m
t
→
0
e
1
/
t
ψ
(
t
)
=
0 which implies that ψ
(
t
)
≤
0. Finally, ψ is nonnegative due to (i), and thus ψ
≡
0.Theorem 12 (Rogers’ type uniqueness theorem).
Let the functionf verify the following conditions: (K
1)
d
f
(
t
,
x
,
y
)
,
0
~
≤
m
i
n
{
Γ
(
q
)
,
1
}
o
e
-
1
/
t
/
t
2, uniformly for positive and bounded x and y on E,
(K
2)
d
(
f
(
t
,
x
,
y
)
,
f
(
t
,
x
¯
,
y
¯
)
)
≤
m
i
n
{
Γ
(
q
)
,
1
}
1
/
2
t
q
-
[
q
]
+
2
d
(
x
,
x
¯
)
+
t
(
q
-
[
q
]
)
d
(
y
,
y
¯
). Then the problem has at most one solution.The proof of this theorem is essentially based on Lemma11.Proof.
Supposex and y are any two solutions of (1) in [
(
i
)
-
q
]
R
L-differentiability, and let ϕ
(
t
)
=
d
(
x
(
t
)
,
y
(
t
)
) and θ
(
t
)
=
d
(
D
q
-
1
x
(
t
)
,
D
q
-
1
y
(
t
)
); we get for t
∈
[
0
,
η
]
⊂
[
0,1
]
(41)
ϕ
t
≤
1
Γ
q
∫
0
t
t
-
s
q
-
1
d
r
s
,
x
,
r
s
,
y
≤
∫
0
t
t
-
s
q
-
1
2
s
q
-
q
+
2
ϕ
s
+
s
q
-
q
θ
s
d
s
≤
t
q
-
1
∫
0
t
1
2
s
q
-
q
+
2
ϕ
s
+
s
q
-
1
θ
s
d
s
≤
t
q
-
q
∫
0
t
1
2
s
q
-
q
+
2
ϕ
s
+
s
q
-
1
θ
s
d
s
≤
t
q
-
q
ψ
t
,
θ
s
≤
∫
0
t
d
r
s
,
x
,
r
s
,
y
≤
∫
0
t
min
⁡
Γ
q
,
1
2
s
q
-
q
+
2
ϕ
s
+
s
q
-
q
θ
s
d
s
≤
∫
0
t
1
2
s
q
-
q
+
2
ϕ
s
+
s
q
-
q
θ
s
d
s
≤
ψ
t
, where ψ is defined as in Lemma 11.
Also, ifϵ
>
0, then from the condition (K1) for small t, we have (42)
ϕ
t
≤
t
q
-
1
Γ
q
∫
0
t
d
r
s
,
x
,
r
s
,
y
<
t
q
-
1
2
ϵ
∫
0
t
e
-
1
/
s
s
2
d
s
≤
t
q
-
1
e
-
1
/
t
2
ϵ
<
ϵ
t
q
-
q
e
-
1
/
t
2
,
θ
t
≤
∫
0
t
d
r
s
,
x
,
r
s
,
y
<
2
ϵ
min
⁡
1
,
Γ
q
∫
0
t
e
-
1
/
s
s
2
d
s
≤
2
ϵ
e
-
1
/
t
. By applying Lemma 11, we obtain d
(
x
(
t
)
,
y
(
t
)
)
=
0 for every t
∈
[
0,1
], and this proves the uniqueness of the solution of the FFDE (1) in [
(
i
)
-
q
]
R
L-differentiability. The second part of the proof is almost completely similar; thus, we omit it.Theorem 13.
Letf
∈
C
(
E
0
,
E
) satisfy the conditions of Theorem 8. Then the successive approximations (43)
x
n
t
=
y
0
+
1
Γ
q
∫
0
t
t
-
s
q
-
1
r
s
,
x
n
-
1
d
s in the sense of [
(
i
)
-
q
]
R
L-differentiability or(44)
x
^
n
t
=
y
0
⊖
-
1
Γ
q
∫
0
t
t
-
s
q
-
1
r
s
,
x
n
-
1
d
s in the sense of [
(
i
i
)
-
q
]
R
L-differentiability converge to the unique solution of the FFDE (1).Proof.
Without loss of generality, we prove Theorem13 for the sequence {
x
n
} in the sense of [
(
i
)
-
q
]
R
L-differentiability using Ascoli-Arzela Theorem. The convergence of the sequence {
x
^
n
} in the sense of [
(
i
i
)
-
q
]
R
L-differentiability is completely similar so we omit it.
Step 1. The sequences {
x
j
}
j
≥
0 and {
D
q
-
1
x
j
}
j
≥
0 are well defined and continuous and uniformly bounded on [
0
,
η
]; in fact(45)
d
x
j
+
1
t
,
y
0
≤
1
Γ
q
∫
0
t
t
-
s
q
-
1
d
r
s
,
x
j
,
0
~
d
s
,
d
D
q
-
1
x
j
+
1
t
,
y
0
≤
∫
0
t
d
r
s
,
x
j
,
0
~
d
s
. For j
=
0 and t
∈
[
0
,
η
], we have(46)
d
x
1
t
,
y
0
≤
M
t
q
Γ
q
+
1
≤
b
,
d
D
q
-
1
x
1
t
,
y
0
≤
M
t
≤
d
. Moreover, for every i
∈
{
0
,
…
,
n
-
1
} we have (47)
d
x
1
i
t
,
0
~
=
d
D
i
I
q
f
t
,
x
0
t
,
D
q
-
1
x
0
t
,
0
~
=
d
I
q
-
i
f
t
,
x
0
t
,
D
q
-
1
x
0
t
,
0
~
=
1
Γ
q
-
i
∫
0
t
t
-
s
q
-
i
-
1
d
f
s
,
x
0
s
,
D
q
-
1
x
0
s
,
0
~
d
s
≤
M
Γ
q
-
i
∫
0
t
t
-
s
q
-
i
-
1
d
s
≤
M
t
q
-
i
q
-
i
Γ
q
-
i
≤
M
t
q
-
i
Γ
q
-
i
+
1
. By induction, the sequences {
x
j
+
1
(
t
)
} and {
D
q
-
1
x
j
+
1
(
t
)
} are well defined and uniformly bounded on [
0
,
η
].
Step 2. We prove that the functions y and z are continuous in [
0
,
η
], where y and z are defined by (48)
y
t
=
lim
⁡
sup
j
→
∞
⁡
ζ
j
0
t
,
z
t
=
lim
⁡
sup
j
→
∞
⁡
ξ
j
t
, such that (49)
ζ
j
0
t
=
d
x
j
t
,
x
j
-
1
t
,
ξ
j
t
=
d
D
q
-
1
x
j
t
,
D
q
-
1
x
j
-
1
t
. Let us note(50)
m
t
=
∑
i
≤
n
-
1
lim
⁡
sup
j
→
∞
⁡
ζ
j
i
t
, where(51)
ζ
j
i
t
=
d
x
j
i
t
,
x
j
-
1
i
t
. For 0
≤
t
1
≤
t
2 and for every i
∈
{
0
,
…
,
n
-
1
}, we obtain (52)
ζ
j
i
t
1
-
ζ
j
i
t
2
=
d
x
j
+
1
i
t
1
,
x
j
i
t
1
-
d
x
j
+
1
i
t
2
,
x
j
i
t
2
≤
1
Γ
q
-
i
∫
0
t
1
t
1
-
s
q
-
1
-
i
d
r
s
,
x
j
,
r
s
,
x
j
-
1
d
s
-
∫
0
t
2
t
2
-
s
q
-
1
-
i
d
r
s
,
x
j
,
r
s
,
x
j
-
1
d
s
≤
2
M
Γ
q
-
i
∫
0
t
1
t
1
-
s
q
-
1
-
i
-
t
2
-
s
q
-
1
-
i
d
s
-
∫
t
1
t
2
t
2
-
s
q
-
1
-
i
d
s
≤
2
M
q
-
i
Γ
q
-
i
t
1
q
-
i
-
t
2
q
-
i
+
2
t
2
-
t
1
q
-
i
≤
4
M
Γ
q
-
i
+
1
t
2
-
t
1
q
-
i
. The right-hand side in the above inequalities is at most 4
M
/
Γ
(
q
-
i
+
1
)
(
t
2
-
t
1
)
q
-
i
+
ϵ for large n if ϵ
>
0 provided that(53)
t
2
-
t
1
≤
η
≤
4
M
Γ
q
-
i
+
1
t
2
-
t
1
q
-
i
, for every i
≤
n
-
1. And since ϵ is arbitrary and t
1
,
t
2 can be interchangeable, we get(54)
m
t
1
-
m
t
2
≤
∑
i
≤
n
-
1
4
M
Γ
q
-
i
+
1
t
2
-
t
1
q
-
i
≤
4
M
n
-
1
Γ
q
+
1
t
2
-
t
1
q
. The same goes for z
(
t
), and we obtain(55)
z
t
1
-
z
t
2
≤
2
M
t
2
-
t
1
. These imply that y and z are continuous on [
0
,
η
].
Step 3. We verify that the family {
D
q
-
1
j
n
+
1
(
t
)
} is equicontinuous in C
F
(
[
0
,
η
]
,
E
) and that the family {
x
j
+
1
(
t
)
} is equicontinuous in C
(
n
-
1
)
F
(
[
0
,
η
]
,
E
).
We may prove that by using condition (H2) and the definition of successive approximations (43) we obtain (56)
ζ
j
+
1
0
t
≤
C
∫
0
t
t
-
s
q
-
1
ζ
j
0
s
α
+
s
α
q
-
q
ξ
j
s
α
d
s
,
ζ
j
+
1
i
t
≤
C
∫
0
t
t
-
s
q
-
i
-
1
ζ
j
0
s
α
+
s
α
q
-
q
ξ
j
s
α
d
s
. As a consequence, we obtain the following estimation: (57)
D
x
j
+
1
,
x
j
≤
∑
i
≤
n
-
1
C
∫
0
1
1
-
s
q
-
i
-
1
x
j
s
-
x
j
-
1
s
α
+
s
α
q
-
q
D
q
-
1
x
j
s
-
D
q
-
1
x
j
-
1
s
α
d
s
. By Arzela-Ascoli Theorem, there exists a subsequence of integers {
j
k
}, such that(58)
d
x
j
k
t
,
x
j
k
-
1
t
⟶
y
t
as
j
k
⟶
∞
,
d
D
q
-
1
x
j
k
t
,
D
q
-
1
x
j
k
-
1
t
⟶
y
t
as
j
k
⟶
∞
. Let us note (59)
m
∗
t
=
lim
⁡
sup
k
→
∞
⁡
d
x
j
k
t
-
x
j
k
-
1
t
,
z
∗
t
=
lim
⁡
sup
k
→
∞
⁡
d
D
q
-
1
x
j
k
t
-
D
q
-
1
x
j
k
-
1
t
. Further, if {
d
(
x
j
,
x
j
-
1
)
}
→
0 and {
d
(
D
q
-
1
x
j
,
D
q
-
1
x
j
-
1
)
}
→
0 as j
→
∞, then the limit of any successive approximation of x
n is the solution x of (1), which was proved to be unique in Theorem 8. It follows that a selection of subsequences is unnecessary and that the entire sequence {
x
j
} converges uniformly to x
(
t
). For that, it is sufficient to show that y
≡
0 and z
≡
0 which will lead to m
∗
(
t
) and z
∗
(
t
) being null.
Setting(60)
R
t
=
∫
0
t
y
s
α
+
s
α
q
-
q
z
s
α
d
s
, and by defining ψ
∗
(
t
)
=
t
-
k
m
a
x
{
y
(
t
)
,
z
(
t
)
}, we show that l
i
m
t
→
0
+
ψ
∗
(
t
)
=
0.
Now we shall prove thatψ
∗
(
t
)
≡
0. Suppose that ψ
∗
(
t
)
>
0 at any point in [
0
,
η
]; then there exists t
1 such that 0
<
m
¯
=
ψ
∗
(
t
1
)
=
m
a
x
0
≤
t
≤
η
ψ
∗
(
t
). Hence, from condition (H1), we obtain(61)
m
¯
=
ψ
t
1
=
t
1
-
k
y
t
1
≤
min
⁡
Γ
q
,
1
m
¯
t
1
q
-
1
+
α
q
-
q
<
m
¯ or(62)
m
¯
=
ψ
t
1
=
t
1
-
k
z
t
1
≤
min
⁡
Γ
q
,
1
m
¯
t
1
α
q
-
q
<
m
¯
. In both cases, we end up with a contradiction. So ψ
∗
(
t
)
≡
0. Therefore, iteration (43) converges uniformly to the unique solution x of (1) on [
0
,
η
].


## 5. Conclusion

In this paper we established the uniqueness and existence of the solution under a fuzzy version of the Krasnoselskii-Krein conditions. Our work generalizes and extends the work of Yoruk et al. to arbitrary order in the fuzzy version. We finally hope to study other classes of fuzzy fractional differential problems in future works.



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*Source: 5246430-2016-06-08.xml*