# The Uniqueness of Solution for a Class of Fractional Order Nonlinear Systems withp-Laplacian Operator

Authors: Jun-qi He; Xue-li Song
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We are concerned with the uniqueness of solutions for a class ofp-Laplacian fractional order nonlinear systems
with nonlocal boundary conditions. Based on some properties of the p-Laplacian operator, the criterion of uniqueness for solutions is established.


---

## Body

## 1. Introduction

Fractional order differential systems arise from many branches of applied mathematics and physics, such as gas dynamics, Newtonian fluid mechanics, nuclear physics, and biological process [1–12]. In the recent years, there has a significant development in fractional calculus. For example, by using the contraction mapping principle, ur Rehman and Khan [13] established the existence and uniqueness of positive solutions for the fractional order differential equation with multipoint boundary conditions:
(1)
D
t
α
y
(
t
)
=
f
(
t
,
y
(
t
)
,
D
t
β
y
(
t
)
)
,
t
∈
(
0,1
)
,
y
(
0
)
=
0
,
D
t
β
y
(
1
)
-
∑
i
=
1
m
-
2
ζ
i
D
t
β
y
(
ξ
i
)
=
y
0
,
where 1
<
α
≤
2, 0
<
β
<
1, ζ
i
∈
[
0
,
+
∞
), and 0
<
ξ
i
<
1, with ∑
i
=
1
m
-
2
ζ
i
ξ
i
<
1. In [14], by using the fixed point theorem of mixed monotone operator, Zhang et al. studied the existence and uniqueness of positive solution for the following fractional order differential systems with multipoint boundary conditions:
(2)
-
D
t
α
x
(
t
)
=
f
(
t
,
x
(
t
)
,
D
t
β
x
(
t
)
,
y
(
t
)
)
,
-
D
t
γ
y
(
t
)
=
g
(
t
,
x
(
t
)
)
,
t
∈
(
0,1
)
,
D
t
β
x
(
0
)
=
0
,
D
t
μ
x
(
1
)
=
∑
j
=
1
p
-
2
a
j
D
t
μ
x
(
ξ
j
)
,
y
(
0
)
=
0
,
D
t
ν
y
(
1
)
=
∑
j
=
1
p
-
2
‍
b
j
D
t
ν
y
(
ξ
j
)
,
where 1
<
γ
<
α
≤
2, 1
<
α
-
β
<
γ, 0
<
β
≤
μ
<
1, 0
<
ν
<
1, and 0
<
ξ
1
<
ξ
2
<
⋯
<
ξ
p
-
2
<
1, a
j
,
b
j
∈
[
0
,
+
∞
) with ∑
j
=
1
p
-
2
a
j
ξ
j
α
-
μ
-
1
<
1 and ∑
j
=
1
p
-
2
b
j
ξ
j
γ
-
1
<
1; D
t is the standard Riemann-Liouville derivative. Some interesting results were also obtained by Zhang et al. [1, 2, 5, 7, 9], Goodrich [15–17], and Ahmad and Nieto [18].On the other hand, thep-Laplacian equation
(3)
(
φ
p
(
x
′
(
t
)
)
)
′
=
f
(
t
,
x
(
t
)
,
x
′
(
t
)
)
,
where φ
p
(
s
)
=
|
s
|
p
-
2
s, p
>
1, can describe the turbulent flow in a porous medium; see [19]. Recently, by using Krasnoselskii’ s fixed point theorem and the Leggett-Williams theorem, Wang et al. [20] investigated the existence of positive solutions for the nonlocal fractional order differential equation with a p-Laplacian operator:
(4)
D
t
α
(
φ
p
(
D
t
β
x
)
)
(
t
)
+
f
(
t
,
x
(
t
)
)
=
0
,
x
(
0
)
=
0
,
D
t
β
x
(
0
)
=
0
,
x
(
1
)
=
a
x
(
ξ
)
,
where 0
<
β
≤
2, 0
<
α
≤
1, 0
≤
a
≤
1, and 0
<
ξ
<
1. And then, by looking for a more suitable upper and lower solution, Ren and Chen [21] established the existence of positive solutions for four points fractional order boundary value problem:
(5)
D
t
β
(
φ
p
(
D
t
α
x
)
)
(
t
)
=
f
(
t
,
x
(
t
)
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
0
,
x
(
1
)
=
a
x
(
ξ
)
,
D
t
α
x
(
0
)
=
0
,
D
t
α
x
(
1
)
=
b
D
t
α
x
(
η
)
,
where D
t
α and D
t
β are the standard Riemann-Liouville derivatives, p-Laplacian operator is defined as φ
p
(
s
)
=
|
s
|
p
-
2
s, p
>
1, and the nonlinearity f may be singular at both t
=
0,1 and x
=
0.Inspired by the above work, in this paper, we study the uniqueness of positive solutions for the following fractional order differential system withp-Laplacian operator:
(6)
D
t
β
(
φ
p
1
(
D
t
α
x
)
)
(
t
)
=
λ
f
(
t
,
y
(
t
)
)
,
D
t
γ
(
φ
p
2
(
D
t
δ
y
)
)
(
t
)
=
ρ
g
(
t
,
x
(
t
)
)
,
x
(
0
)
=
0
,
x
(
1
)
=
a
x
(
ξ
)
,
D
t
α
x
(
0
)
=
0
,
D
t
α
x
(
1
)
=
b
D
t
α
x
(
η
)
,
y
(
0
)
=
0
,
y
(
1
)
=
c
y
(
ζ
)
,
D
t
δ
y
(
0
)
=
0
,
D
t
δ
y
(
1
)
=
d
D
t
δ
y
(
μ
)
,
where D
t
α, D
t
β, D
t
γ, and D
t
δ are the standard Riemann-Liouville derivatives with α
,
β
,
γ
,
δ
∈
(
1,2
], a
,
b
,
c
,
d
∈
[
0,1
], and ξ
,
η
,
ζ
,
μ
∈
(
0,1
), λ and ρ are positive parameters, p-Laplacian operator is defined as φ
p
1
(
s
)
=
|
s
|
p
1
-
2
s, p
1
>
1, (
φ
p
1
)
-
1
=
φ
q
1
,
1
/
p
1
+
1
/
q
1
=
1, and φ
p
2
(
s
)
=
|
s
|
p
2
-
2
s, p
2
>
1, (
φ
p
2
)
-
1
=
φ
q
2, 1
/
p
2
+
1
/
q
2
=
1. In the rest of paper, we assume that f
,
g
:
[
0,1
]
×
ℝ
→
ℝ are continuous.Normally, we cannot apply the contraction mapping principle for solving the BVP (1) like ur Rehman and Khan [13] since p-Laplacian operator is nonlinear. In this paper, by using a property of the p-Laplacian operator, we overcome this difficulty and establish the uniqueness of solution for the eigenvalue problem of the fractional differential system (6).


## 2. Preliminaries and Lemmas

We firstly list the necessary definitions from fractional calculus theory here, which can be found in [10–12].Definition 1.
Letβ
>
0. The fractional integral operator of a function f
:
(
0
,
+
∞
)
→
ℝ is given by
(7)
I
β
f
(
t
)
=
1
Γ
(
β
)
∫
0
t
(
t
-
s
)
β
-
1
f
(
s
)
d
s
.Definition 2.
Letβ
>
0. The Riemann-Liouville fractional derivative of a function f
:
(
0
,
+
∞
)
→
ℝ is given by
(8)
D
t
β
f
(
t
)
=
1
Γ
(
n
-
β
)
(
d
d
t
)
n
∫
0
t
(
t
-
s
)
n
-
β
-
1
f
(
s
)
d
s
,
where n
=
[
β
]
+
1, [
β
] denotes the integer part of the number β, and Γ denotes the gamma function.Property 1.
Lettingβ
>
α
>
0 and f
∈
L
1
(
0,1
), then(1)
(9)
I
β
I
α
f
(
t
)
=
I
β
+
α
f
(
t
)
,
D
t
α
I
β
f
(
t
)
=
I
β
-
α
f
(
t
)
,
D
t
α
I
α
f
(
t
)
=
f
(
t
)
,
(2)
(10)
I
β
D
t
β
f
(
x
)
=
f
(
x
)
+
c
1
x
β
-
1
+
c
2
x
β
-
2
+
⋯
+
c
n
x
β
-
n
,where c
i
∈
ℝ
(
i
=
1,2
,
…
,
n
) and n is the smallest integer greater than or equal to β.
The main results of this paper are based on the following property ofp-Laplacian operator, which is easy to be proved.Lemma 3.
(1) Ifq
≥
2 and |
x
|
,
|
y
|
≤
M, then
(11)
|
φ
q
(
x
)
-
φ
q
(
y
)
|
≤
(
q
-
1
)
M
q
-
2
|
x
-
y
|
.
(2) If1
<
q
<
2, x
y
>
0, and |
x
|
,
|
y
|
≥
m
>
0, then
(12)
|
φ
q
(
x
)
-
φ
q
(
y
)
|
≤
(
q
-
1
)
m
q
-
2
|
x
-
y
|
.
Applying Definitions1 and 2 and Property 1, we have the following lemma.Lemma 4.
Lety
∈
L
1
[
0,1
], 1
<
α
,
β
≤
2, 0
<
ξ
,
η
<
1, and 0
≤
a
,
b
≤
1. The fractional order boundary value problem,
(13)
D
t
β
(
φ
p
1
(
D
t
α
x
)
)
(
t
)
=
h
(
t
)
,
t
∈
(
0,1
)
,
x
(
0
)
=
0
,
x
(
1
)
=
a
x
(
ξ
)
,
D
t
α
x
(
0
)
=
0
,
D
t
α
x
(
1
)
=
b
D
t
α
x
(
η
)
,
has the unique solution
(14)
x
(
t
)
=
∫
0
1
K
1
(
t
,
s
)
φ
q
1
(
∫
0
1
‍
K
2
(
s
,
τ
)
h
(
τ
)
d
τ
)
d
s
,
where
(15)
K
1
(
t
,
s
)
=
k
1
(
t
,
s
)
+
a
k
1
(
ξ
,
s
)
t
α
-
1
1
-
a
ξ
α
-
1
,
K
2
(
t
,
s
)
=
k
2
(
t
,
s
)
+
b
1
k
2
(
η
,
s
)
t
β
-
1
1
-
b
1
η
β
-
1
,
k
1
(
t
,
s
)
=
{
(
t
(
1
-
s
)
)
α
-
1
-
(
t
-
s
)
α
-
1
Γ
(
α
)
,
0
≤
s
≤
t
≤
1
,
(
t
(
1
-
s
)
)
α
-
1
Γ
(
α
)
,
0
≤
t
≤
s
≤
1
,
k
2
(
t
,
s
)
=
{
(
t
(
1
-
s
)
)
β
-
1
-
(
t
-
s
)
β
-
1
Γ
(
β
)
,
0
≤
s
≤
t
≤
1
,
(
t
(
1
-
s
)
)
β
-
1
Γ
(
β
)
,
0
≤
t
≤
s
≤
1
,
and b
1
=
b
p
1
-
1.Similar to (14), the fractional order boundary value problem,
(16)
D
t
γ
(
φ
p
2
(
D
t
δ
y
)
)
(
t
)
=
h
(
t
)
,
t
∈
(
0,1
)
,
y
(
0
)
=
0
,
y
(
1
)
=
c
y
(
ζ
)
,
D
t
γ
y
(
0
)
=
0
,
D
t
δ
y
(
1
)
=
d
D
t
δ
y
(
μ
)
,
has unique solution
(17)
y
(
t
)
=
∫
0
1
K
3
(
t
,
s
)
φ
q
2
(
∫
0
1
‍
K
4
(
s
,
τ
)
h
(
τ
)
d
τ
)
d
s
,
where
(18)
K
3
(
t
,
s
)
=
k
3
(
t
,
s
)
+
c
k
3
(
ζ
,
s
)
t
δ
-
1
1
-
c
ζ
δ
-
1
,
K
4
(
t
,
s
)
=
k
4
(
t
,
s
)
+
d
1
k
4
(
μ
,
s
)
t
γ
-
1
1
-
d
1
μ
γ
-
1
,
k
3
(
t
,
s
)
=
{
(
t
(
1
-
s
)
)
δ
-
1
-
(
t
-
s
)
δ
-
1
Γ
(
δ
)
,
0
≤
s
≤
t
≤
1
,
(
t
(
1
-
s
)
)
δ
-
1
Γ
(
δ
)
,
0
≤
t
≤
s
≤
1
,
k
4
(
t
,
s
)
=
{
(
t
(
1
-
s
)
)
γ
-
1
-
(
t
-
s
)
γ
-
1
Γ
(
γ
)
,
0
≤
s
≤
t
≤
1
,
(
t
(
1
-
s
)
)
γ
-
1
Γ
(
γ
)
,
0
≤
t
≤
s
≤
1
,
and d
1
=
d
p
2
-
1.Lemma 5.
Let1
<
α
,
β
,
γ
,
δ
≤
2, 0
<
ξ
,
ζ
,
η
,
μ
<
1, and 0
≤
a
,
b
,
c
,
d
≤
1. The functions K
i
(
t
,
s
)
,
i
=
1,2
,
3,4, are continuous on [
0,1
]
×
[
0,1
] and satisfy(i)
K
i
(
t
,
s
)
≥
0
,
i
=
1,2
,
3,4 for t
,
s
∈
[
0,1
];
(ii)
fort
,
s
∈
[
0,1
],
(19)
σ
1
(
s
)
t
β
-
1
≤
K
2
(
t
,
s
)
≤
σ
3
(
s
)
t
β
-
1
,
σ
2
(
s
)
t
γ
-
1
≤
K
4
(
t
,
s
)
≤
σ
4
(
s
)
t
γ
-
1
,
where(20)
σ
1
(
s
)
=
b
1
k
2
(
η
,
s
)
1
-
b
1
η
β
-
1
,
σ
3
(
s
)
=
(
1
-
s
)
β
-
1
Γ
(
β
)
+
b
1
k
2
(
η
,
s
)
1
-
b
1
η
β
-
1
,
σ
2
(
s
)
=
d
1
k
4
(
μ
,
s
)
1
-
d
1
μ
γ
-
1
,
σ
4
(
s
)
=
(
1
-
s
)
γ
-
1
Γ
(
γ
)
+
d
1
k
4
(
μ
,
s
)
1
-
d
1
μ
γ
-
1
.
(iii)
Fort
,
s
∈
[
0,1
],
(21)
K
1
(
t
,
s
)
≤
r
1
(
1
-
s
)
α
-
1
,
K
3
(
t
,
s
)
≤
r
2
(
1
-
s
)
δ
-
1
,
where(22)
r
1
=
1
Γ
(
α
)
[
1
+
a
1
-
a
ξ
α
-
1
]
,
r
2
=
1
Γ
(
δ
)
[
1
+
c
1
-
c
ξ
δ
-
1
]
.Proof.
The proof is obvious; we omit the proof.The basic space used in this paper isE
=
C
(
[
0,1
]
;
ℝ
)
×
C
(
[
0,1
]
;
ℝ
), where ℝ is a real number set. Obviously, the space E is a Banach space if it is endowed with the norm as follows:
(23)
∥
(
u
,
v
)
∥
:
=
∥
u
∥
+
∥
v
∥
,
∥
u
∥
=
max
⁡
t
∈
[
0,1
]
⁡
|
u
(
t
)
|
,
∥
v
∥
=
max
⁡
t
∈
[
0,1
]
⁡
|
v
(
t
)
|
,
for any (
u
,
v
)
∈
E. By Lemma 4, (
x
,
y
)
∈
E is a solution of the fractional order system (1) if and only if (
x
,
y
)
∈
E is a solution of the integral equation
(24)
x
(
t
)
=
λ
q
1
∫
0
1
K
1
(
t
,
s
)
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
y
(
τ
)
)
d
τ
)
d
s
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
t
∈
[
0,1
]
,
y
(
t
)
=
ρ
q
2
∫
0
1
K
3
(
t
,
s
)
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
x
(
τ
)
)
d
τ
)
d
s
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
t
∈
[
0,1
]
.We define an operatorT
:
E
→
E by
(25)
T
(
x
,
y
)
(
t
)
=
(
F
(
x
,
y
)
,
G
(
x
,
y
)
)
,
where
(26)
F
(
x
,
y
)
=
λ
q
1
∫
0
1
K
1
(
t
,
s
)
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
y
(
τ
)
)
d
τ
)
d
s
,
G
(
x
,
y
)
=
ρ
q
2
∫
0
1
K
3
(
t
,
s
)
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
x
(
τ
)
)
d
τ
)
d
s
.It is easy to see that(
x
,
y
) is the solution of the boundary value problem (6) if and only if (
x
,
y
) is the fixed point of T. As f
,
g
∈
C
(
[
0,1
]
×
ℝ
,
ℝ
), we know that T
:
E
→
E is a continuous and compact operator.


## 3. Main Results

Now we here introduce a new concept: the𝒟-contraction mapping.Definition 6.
A functionψ
:
(
-
∞
,
+
∞
)
→
[
0
,
+
∞
) is called a nonlinear 𝒟-contraction mapping if it is continuous and nondecreasing and satisfies ψ
(
r
)
≤
r
,
r
>
0.Theorem 7.
Suppose thatp
1
,
p
2
>
2, if there exist nonnegative functions a
i
(
t
)
,
i
=
1,2
,
3,4, such that
(27)
0
<
∫
0
1
δ
i
(
t
)
a
i
(
t
)
d
t
<
+
∞
,
i
=
1,2
,
3,4
,
and the following conditions are satisfied:(
H
1
)
for any(
t
,
w
)
∈
(
0,1
)
×
ℝ,
(28)
f
(
t
,
w
)
≥
a
1
(
t
)
,
g
(
t
,
w
)
≥
a
2
(
t
)
,
(
H
2
)
there exist𝒟-contraction mappings ψ
1, ψ
2 as
(29)
|
f
(
t
,
u
)
-
f
(
t
,
v
)
|
≤
a
3
(
t
)
ψ
1
(
|
u
-
v
|
)
,
k
k
k
k
k
l
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
,
|
g
(
t
,
u
)
-
g
(
t
,
v
)
|
≤
a
4
(
t
)
ψ
2
(
|
u
-
v
|
)
,
k
k
k
k
k
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
.Then the fractional order differential system (6) has a unique solution provided that
(30)
Λ
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
<
1
.Proof.
In the casep
1
,
p
2
>
2, we have 1
<
q
1, q
2
<
2. Now we prove that T is a contraction mapping. By (27)-(28) and Lemma 5, we have
(31)
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
y
(
τ
)
)
d
τ
≥
s
β
-
1
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
,
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
x
(
τ
)
)
d
τ
≥
s
γ
-
1
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
.
By (12), (28), and (31), for any (
u
1
,
v
1
)
,
(
u
2
,
v
2
)
∈
E and for t
>
0, we have
(32)
|
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
l
-
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
|
≤
(
q
1
-
1
)
(
s
β
-
1
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
K
2
(
s
,
τ
)
|
f
(
τ
,
v
1
(
τ
)
)
-
f
(
τ
,
v
2
(
τ
)
)
|
d
τ
≤
(
q
1
-
1
)
(
s
β
-
1
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
ψ
1
(
∥
v
1
-
v
2
∥
)
≤
(
q
1
-
1
)
s
(
β
-
1
)
(
q
1
-
2
)
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
∥
v
1
-
v
2
∥
.
Similarly, we also have
(33)
|
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
1
(
τ
)
)
d
τ
)
l
-
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
2
(
τ
)
)
d
τ
)
|
≤
(
q
2
-
1
)
s
(
γ
-
1
)
(
q
2
-
2
)
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
∥
u
1
-
u
2
∥
.
So it follows from (14), (17), and (31)-(32) that
(34)
|
F
(
u
1
,
v
1
)
(
t
)
-
F
(
u
2
,
v
2
)
(
t
)
|
=
|
λ
q
1
∫
0
1
‍
K
1
(
t
,
s
)
k
k
k
k
k
k
k
k
k
×
[
φ
q
1
(
∫
0
1
‍
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
k
k
lklk
-
φ
q
1
(
∫
0
1
‍
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
]
d
s
|
≤
λ
q
1
r
1
∫
0
1
(
1
-
s
)
α
-
1
k
k
k
k
k
k
k
k
l
k
×
|
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
k
k
lklk
-
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
|
d
s
≤
λ
q
1
r
1
(
q
1
-
1
)
∫
0
1
(
1
-
s
)
α
-
1
s
(
β
-
1
)
(
q
1
-
2
)
d
s
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
∥
v
1
-
v
2
∥
≤
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
∥
v
1
-
v
2
∥
,
|
G
(
u
1
,
v
1
)
(
t
)
-
G
(
u
2
,
v
2
)
(
t
)
|
=
|
ρ
q
2
∫
0
1
K
3
(
t
,
s
)
k
k
k
k
k
k
k
k
k
×
[
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
k
k
k
k
k
-
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
2
(
τ
)
)
d
τ
)
]
d
s
|
≤
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
∥
u
1
-
u
2
∥
.
Hence
(35)
|
T
(
u
1
,
v
1
)
-
T
(
u
2
,
v
2
)
|
=
|
(
F
(
u
1
,
v
1
)
-
F
(
u
2
,
v
2
)
,
G
(
u
1
,
v
1
)
-
G
(
u
2
,
v
2
)
)
|
≤
∥
F
(
u
1
,
v
1
)
-
F
(
u
2
,
v
2
)
∥
+
∥
G
(
u
1
,
v
1
)
-
G
(
u
2
,
v
2
)
∥
≤
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
∥
v
1
-
v
2
∥
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
∥
u
1
-
u
2
∥
≤
Λ
(
∥
v
1
-
v
2
∥
+
∥
u
1
-
u
2
∥
)
=
Λ
∥
(
u
1
,
v
1
)
-
(
u
2
,
v
2
)
∥
,
where
(36)
Λ
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
.
Noticing that 0
<
Λ
<
1, we obtain that F
:
C
[
0,1
]
→
C
[
0,1
] is a contraction mapping. By means of the Banach contraction mapping principle, we get that T has a unique fixed point in E which implies that the fractional order differential system (6) has a unique solution.Theorem 8.
Suppose that1
<
p
1, p
2
≤
2, if there exist nonnegative functions b
i
(
t
), i
=
1,2
,
3,4, such that
(37)
0
<
∫
0
1
δ
i
(
t
)
b
i
(
t
)
d
t
<
+
∞
,
i
=
1,2
,
3,4
,
and the following conditions are satisfied:(
H
3
)
for any(
t
,
w
)
∈
(
0,1
)
×
ℝ,
(38)
|
f
(
t
,
w
)
|
≤
b
3
(
t
)
,
g
(
t
,
w
)
≤
b
4
(
t
)
,
(
H
4
)
there exist𝒟-contraction mappings ϕ
1, ϕ
2 as
(39)
|
f
(
t
,
u
)
-
f
(
t
,
v
)
|
≤
b
1
(
t
)
ϕ
1
(
|
u
-
v
|
)
,
k
k
k
k
k
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
,
|
g
(
t
,
u
)
-
g
(
t
,
v
)
|
≤
b
2
(
t
)
ϕ
2
(
|
u
-
v
|
)
,
k
k
k
k
k
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
.Then the fractional order differential system (6) has a unique solution provided that
(40)
Λ
~
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
)
q
2
-
2
∫
0
1
δ
4
(
τ
)
b
2
(
τ
)
d
τ
<
1
.Proof.
In the case1
<
p
1, p
2
≤
2, we get q
1
,
q
2
≥
2; here we still prove that T is a contraction mapping if the conditions of theorem are satisfied. By (37)-(38) and Lemma 5, for any (
x
,
y
)
∈
E, we have
(41)
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
y
(
τ
)
)
d
τ
≤
s
β
-
1
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
,
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
x
(
τ
)
)
d
τ
≤
s
γ
-
1
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
.
By (11), (39), and (41), for any (
u
1
,
v
1
)
,
(
u
2
,
v
2
)
∈
E and for t
>
0, we have
(42)
|
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
l
-
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
|
≤
(
q
1
-
1
)
(
s
β
-
1
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
K
2
(
s
,
τ
)
|
f
(
τ
,
v
1
(
τ
)
)
-
f
(
τ
,
v
2
(
τ
)
)
|
d
τ
≤
(
q
1
-
1
)
(
s
β
-
1
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
ϕ
1
(
∥
v
1
-
v
2
∥
)
≤
(
q
1
-
1
)
s
(
β
-
1
)
(
q
1
-
2
)
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
∥
v
1
-
v
2
∥
.
Similarly, we also have
(43)
|
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
1
(
τ
)
)
d
τ
)
l
-
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
2
(
τ
)
)
d
τ
)
|
≤
(
q
2
-
1
)
s
(
γ
-
1
)
(
q
2
-
2
)
×
(
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
b
2
(
τ
)
d
τ
∥
u
1
-
u
2
∥
.
So it follows from (14), (17), and (42)-(43) that
(44)
|
F
(
u
1
,
v
1
)
(
t
)
-
F
(
u
2
,
v
2
)
(
t
)
|
=
|
λ
q
1
∫
0
1
‍
K
1
(
t
,
s
)
k
k
k
k
k
×
[
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
l
k
-
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
]
d
s
|
≤
λ
q
1
r
1
∫
0
1
(
1
-
s
)
α
-
1
k
k
k
k
k
×
|
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
k
l
-
φ
q
1
(
∫
0
1
K
2
(
s
,
τ
)
f
(
s
,
v
2
(
τ
)
)
d
τ
)
|
d
s
≤
λ
q
1
r
1
(
q
1
-
1
)
∫
0
1
(
1
-
s
)
α
-
1
s
(
β
-
1
)
(
q
1
-
2
)
d
s
×
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
∥
v
1
-
v
2
∥
≤
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
∥
v
1
-
v
2
∥
,
|
G
(
u
1
,
v
1
)
(
t
)
-
G
(
u
2
,
v
2
)
(
t
)
|
=
|
ρ
q
2
∫
0
1
K
3
(
t
,
s
)
k
k
k
k
k
k
k
×
[
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
1
(
τ
)
)
d
τ
)
k
k
k
k
k
k
k
k
k
l
-
φ
q
2
(
∫
0
1
K
4
(
s
,
τ
)
g
(
s
,
u
2
(
τ
)
)
d
τ
)
]
d
s
|
≤
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
b
2
(
τ
)
d
τ
∥
u
1
-
u
2
∥
.
Hence
(45)
|
T
(
u
1
,
v
1
)
-
T
(
u
2
,
v
2
)
|
=
|
(
F
(
u
1
,
v
1
)
-
F
(
u
2
,
v
2
)
,
G
(
u
1
,
v
1
)
-
G
(
u
2
,
v
2
)
)
|
≤
∥
F
(
u
1
,
v
1
)
-
F
(
u
2
,
v
2
)
∥
+
∥
G
(
u
1
,
v
1
)
-
G
(
u
2
,
v
2
)
∥
≤
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
×
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
∥
v
1
-
v
2
∥
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
b
2
(
τ
)
d
τ
∥
u
1
-
u
2
∥
≤
Λ
~
(
∥
v
1
-
v
2
∥
+
∥
u
1
-
u
2
∥
)
=
Λ
~
∥
(
u
1
,
v
1
)
-
(
u
2
,
v
2
)
∥
,
where
(46)
Λ
~
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
3
(
τ
)
b
3
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
b
1
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
4
(
τ
)
b
4
(
τ
)
d
τ
)
q
2
-
2
∫
0
1
δ
4
(
τ
)
b
2
(
τ
)
d
τ
.
Noticing that 0
<
Λ
~
<
1, we obtain that F
:
C
[
0,1
]
→
C
[
0,1
] is a contraction mapping. By means of the Banach contraction mapping principle, we get that T has a unique fixed point in E which implies that the fractional order differential system (6) has a unique solution.It follows from Theorems7 and 8 that the following corollaries for mixed cases hold.Corollary 9.
Suppose thatp
1
>
2 and 1
<
p
2
≤
2 if there exist nonnegative functions a
i
(
t
)
,
i
=
1,2
,
3,4, such that
(47)
0
<
∫
0
1
δ
i
(
t
)
a
i
(
t
)
d
t
<
+
∞
,
i
=
1,2
,
3,4
,
and the following conditions are satisfied:(
H
1
)
for any(
t
,
w
)
∈
(
0,1
)
×
ℝ,
(48)
f
(
t
,
w
)
≥
a
1
(
t
)
,
|
g
(
t
,
w
)
|
≤
a
2
(
t
)
,
(
H
2
)
there exist𝒟-contraction mappings ψ
1, ψ
2 as
(49)
|
f
(
t
,
u
)
-
f
(
t
,
v
)
|
≤
a
3
(
t
)
ψ
1
(
|
u
-
v
|
)
,
k
k
k
k
k
l
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
,
|
g
(
t
,
u
)
-
g
(
t
,
v
)
|
≤
a
4
(
t
)
ψ
2
(
|
u
-
v
|
)
,
k
k
k
k
k
l
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
.Then the fractional order differential system (6) has a unique solution provided that
(50)
Λ
~
1
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
1
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
4
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
<
1
.Corollary 10.
Suppose thatp
2
>
2 and 1
<
p
1
≤
2 if there exist nonnegative functions a
i
(
t
), i
=
1,2
,
3,4, such that
(51)
0
<
∫
0
1
‍
δ
i
(
t
)
a
i
(
t
)
d
t
<
+
∞
,
i
=
1,2
,
3,4
,
and the following conditions are satisfied:(
H
1
)
for any(
t
,
w
)
∈
(
0,1
)
×
ℝ,
(52)
|
f
(
t
,
w
)
|
≤
a
1
(
t
)
,
g
(
t
,
w
)
≥
a
2
(
t
)
,
(
H
2
)
there exist𝒟-contraction mappings ψ
1, ψ
2 as
(53)
|
f
(
t
,
u
)
-
f
(
t
,
v
)
|
≤
a
3
(
t
)
ψ
1
(
|
u
-
v
|
)
,
k
k
k
k
k
l
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
,
|
g
(
t
,
u
)
-
g
(
t
,
v
)
|
≤
a
4
(
t
)
ψ
2
(
|
u
-
v
|
)
,
k
k
k
k
k
l
k
k
k
a
.
e
.
(
t
,
u
)
,
(
t
,
v
)
∈
[
0,1
]
×
ℝ
.Then the fractional order differential system (6) has a unique solution provided that
(54)
Λ
~
2
=
λ
q
1
(
q
1
-
1
)
r
1
B
(
α
,
(
β
-
1
)
(
q
1
-
2
)
+
1
)
×
(
∫
0
1
δ
3
(
τ
)
a
1
(
τ
)
d
τ
)
q
1
-
2
∫
0
1
δ
3
(
τ
)
a
3
(
τ
)
d
τ
+
ρ
q
2
(
q
2
-
1
)
r
2
B
(
δ
,
(
γ
-
1
)
(
q
2
-
2
)
+
1
)
×
(
∫
0
1
δ
2
(
τ
)
a
2
(
τ
)
d
τ
)
q
2
-
2
×
∫
0
1
δ
4
(
τ
)
a
4
(
τ
)
d
τ
<
1
.



---

*Source: 921209-2014-05-06.xml*