# The General Solution of Impulsive Systems with Caputo-Hadamard Fractional Derivative of Orderq

Authors: Xianmin Zhang; Xianzhen Zhang; Zuohua Liu; Hui Peng; Tong Shu; Shiyong Yang
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Motivated by some preliminary works about general solution of impulsive system with fractional derivative, the generalized impulsive differential equations with Caputo-Hadamard fractional derivative ofq
∈
C  (R
(
q
)
∈
(
1,2
)) are further studied by analyzing the limit case (as impulses approach zero) in this paper. The formulas of general solution are found for the impulsive systems.


---

## Body

## 1. Introduction

Hadamard fractional calculus is a key part of the theory of fractional calculus. The authors in [1–6] made an important development of the fractional calculus within the frame of Hadamard fractional derivative. For the general theory of Hadamard fractional calculus, one can see the monograph of Kilbas et al. [7].Recently, Jarad et al. made a progress on Hadamard fractional derivative to present the definition of Caputo-Hadamard fractional derivative in [8] and developed the fundamental theorem of this fractional derivative in [8, 9].Furthermore, impulsive differential equations are utilized as a valuable tool to describe the dynamics of processes in which sudden, discontinuous jumps occur, and impulsive differential equations with Caputo fractional derivative were widely researched in [10–26]. Next, the general solutions of several kinds of impulsive fractional differential equations have been found in [27–30], respectively.Motivated by the above-mentioned works, we will further seek the general solution of generalized impulsive system with Caputo-Hadamard fractional derivative:(1)
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
t
≠
t
k
k
=
1,2
,
…
,
m
,
t
≠
t
-
l
l
=
1,2
,
…
,
p
,
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
Δ
x
′
t
=
t
-
l
=
x
′
t
-
l
+
-
x
′
t
-
l
-
=
I
-
l
x
t
-
l
-
∈
C
,
l
=
1,2
,
…
,
p
,
x
a
=
x
a
∈
C
,
x
′
a
=
x
-
a
∈
C
.Here q
∈
C and R
(
q
)
∈
(
1,2
), D
a
+
q
C-H denotes left-sided Caputo-Hadamard fractional derivative of order q and a
>
0, f
:
[
a
,
T
]
×
C
→
C is an appropriate continuous function, a
=
t
0
<
t
1
<
⋯
<
t
m
<
t
m
+
1
=
T, and a
=
t
-
0
<
t
-
1
<
⋯
<
t
-
p
<
t
-
p
+
1
=
T. Here x
(
t
k
+
)
=
l
i
m
ε
→
0
+
x
(
t
k
+
ε
) and x
(
t
k
-
)
=
l
i
m
ε
→
0
-
x
(
t
k
+
ε
) represent the right and left limits of u
(
t
) at t
=
t
k, respectively, and x
′
(
t
-
l
+
) and x
′
(
t
-
l
-
) have similar meaning. Let us queue a
,
t
1
,
…
,
t
m
,
t
-
1
,
…
,
t
-
p
,
T to a
=
t
0
′
<
t
1
′
<
t
2
′
<
⋯
<
t
Π
′
<
t
Π
+
1
′
=
T such that(2)
s
e
t
t
1
,
…
,
t
m
,
t
-
1
,
…
,
t
-
p
=
s
e
t
t
Π
′
,
t
2
′
,
…
,
t
Π
′
.For each [
a
,
t
k
′
]
(
k
=
0,1
,
…
,
Π
), suppose [
a
,
t
k
1
]
⊆
[
a
,
t
k
′
]
⊂
[
a
,
t
k
1
+
1
] (here k
1
∈
{
1,2
,
…
,
m
}) and [
a
,
t
-
k
2
]
⊆
[
a
,
t
k
′
]
⊂
[
a
,
t
-
k
2
+
1
] (here k
2
∈
{
1,2
,
…
,
p
}), respectively.In order to get the solution of (1), we will first consider the following system:(3a)
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
t
≠
t
k
k
=
1
,
…
,
m
,
t
≠
t
-
l
l
=
1
,
…
,
p
,
(3b)
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
(3c)
Δ
δ
x
t
=
t
-
l
=
δ
x
t
-
l
+
-
δ
x
t
-
l
-
=
J
l
x
t
-
l
-
∈
C
,
l
=
1,2
,
…
,
p
,
(3d)
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
,where differential operator δ
=
t
(
d
/
d
t
), δ
0
x
(
t
)
=
x
(
t
).Next, some definitions and conclusions are introduced in Section2, and the formulas of general solution will be given for some impulsive differential equations with Caputo-Hadamard fractional derivative in Section 3.


## 2. Preliminaries

Definition 1 (see [7, p. 110]).
Let0
≤
a
≤
b
≤
∞ be finite or infinite interval of the half-axis R
+. The left-sided Hadamard fractional integral of order α
∈
C of function φ
(
x
) is defined by(4)
J
a
+
α
H
φ
x
=
1
Γ
α
∫
a
x
ln
⁡
x
s
α
-
1
φ
s
d
s
s
,
a
<
x
<
b
,where Γ
(
·
) is the Gamma function.Definition 2 (see [7, p. 110]).
The left-sided Hadamard fractional derivative of orderα
∈
C (R
(
α
)
≥
0) on (a, b) is defined by(5)
D
a
+
α
H
φ
x
=
δ
n
J
a
+
n
-
α
H
φ
x
=
x
d
d
x
n
1
Γ
n
-
α
∫
a
x
ln
⁡
x
s
n
-
α
-
1
φ
s
d
s
s
,
a
<
x
<
b
,where n
=
[
R
(
α
)
]
+
1 and differential operator δ
=
x
(
d
/
d
x
) and δ
0
y
(
x
)
=
y
(
x
).Lemma 3 (see [7, p. 114–116]).
Letα
,
β
∈
C such that R
(
α
)
>
R
(
β
)
>
0. For 0
<
a
<
b
<
∞, if φ
∈
L
p
(
a
,
b
)
(
1
≤
p
<
∞
), then D
a
+
β
H
J
a
+
α
H
φ
=
J
a
+
α
-
β
H
φ and J
a
+
α
H
J
a
+
β
H
φ
=
J
a
+
α
+
β
H
φ.The left-sided Caputo-Hadamard fractional derivative was defined in [8] by (6)
D
a
+
α
C-H
φ
x
=
D
a
+
α
H
φ
x
-
∑
k
=
0
n
-
1
δ
k
φ
a
k
!
ln
⁡
t
a
k
x
;here R
(
α
)
≥
0, n
=
[
R
(
α
)
]
+
1, 0
<
a
<
b
<
∞, differential operator δ
=
x
(
d
/
d
x
), δ
0
y
(
x
)
=
y
(
x
), and(7)
φ
x
∈
A
C
δ
n
a
,
b
=
φ
:
a
,
b
⟶
C
:
δ
n
-
1
φ
x
∈
A
C
a
,
b
,
δ
=
x
d
d
x
.Theorem 4 (see [8, p. 4]).
LetR
(
α
)
≥
0, n
=
[
R
(
α
)
]
+
1, and φ
∈
A
C
δ
n
[
a
,
b
], 0
<
a
<
b
<
∞. Then, D
a
+
α
C
-
H
φ
(
x
) exist everywhere on [
a
,
b
] and(a)
ifα
∉
N
0,(8)
D
a
+
α
C
-
H
φ
x
=
1
Γ
n
-
α
∫
a
x
ln
⁡
x
s
n
-
α
-
1
δ
n
φ
s
d
s
s
=
J
a
+
n
-
α
H
δ
n
φ
x
,
(b)
ifα
=
n
∈
N
0,(9)
D
a
+
α
C
-
H
φ
x
=
δ
n
φ
x
.In particular,(10)
D
a
+
0
C
-
H
φ
x
=
φ
x
.Lemma 5 (see [8, p. 5]).
LetR
α
>
0, n
=
[
R
(
α
)
]
+
1 and φ
∈
C
[
a
,
b
]. If R
(
α
)
≠
0 or α
∈
N, then (11)
D
a
+
α
C
-
H
J
a
+
α
H
φ
x
=
φ
x
.Lemma 6 (see [8, p. 6]).
Letφ
∈
A
C
δ
n
[
a
,
b
] or let C
δ
n
[
a
,
b
] and α
∈
C; then,(12)
J
a
+
α
H
D
a
+
α
C
-
H
φ
x
=
φ
x
-
∑
k
=
0
n
-
1
δ
k
φ
a
k
!
ln
⁡
x
a
k
.Lemma 7 (see [29, p. 4]).
Letw
∈
C and R
(
w
)
∈
(
0,1
), and let ξ be a constant. A function u
(
t
)
:
[
a
,
T
]
→
C is general solution of system (13)
D
a
+
w
C
-
H
u
t
=
g
t
,
u
t
,
t
∈
a
,
T
,
t
≠
t
k
k
=
1,2
,
…
,
m
,
Δ
u
t
=
t
k
=
u
t
k
+
-
u
t
k
-
=
Δ
k
u
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
u
a
=
u
a
,
u
a
∈
C
,if and only if u
(
t
) satisfies the integral equation(14)
u
t
=
u
a
+
1
Γ
w
∫
a
t
ln
⁡
t
s
w
-
1
g
s
,
u
s
d
s
s
f
o
r
t
∈
a
,
t
1
,
u
a
+
∑
i
=
1
k
Δ
i
u
t
i
-
+
1
Γ
w
∫
a
t
l
n
t
s
w
-
1
g
s
,
u
s
d
s
s
+
ξ
∑
i
=
1
k
Δ
i
u
t
i
-
Γ
w
∫
a
t
i
l
n
t
i
s
w
-
1
g
s
,
u
s
d
s
s
+
∫
t
i
t
l
n
t
s
w
-
1
g
s
,
u
s
d
s
s
-
∫
a
t
l
n
t
s
w
-
1
g
s
,
u
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
k
=
1,2
,
…
,
mprovided that the integral in (14) exists.


## 3. Main Results

Firstly, let us consider some limit cases in system (3a), (3b), (3c), and (3d):(15)
lim
I
k
x
t
k
-
→
0
∀
k
∈
1,2
,
…
,
m
,
J
l
x
t
-
l
-
→
0
∀
l
∈
1,2
,
…
,
p
⁡
s
y
s
t
e
m
3
a
,
3
b
,
3
c
,
and
3
d
⟶
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
,
(16)
lim
J
l
x
t
-
l
-
→
0
,
∀
l
∈
1,2
,
…
,
p
⁡
s
y
s
t
e
m
3
a
,
3
b
,
3
c
,
and
3
d
⟶
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
t
≠
t
k
k
=
1
,
…
,
m
,
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
,
(17)
lim
I
k
x
t
k
-
→
0
,
∀
k
∈
1,2
,
…
,
m
⁡
s
y
s
t
e
m
3
a
,
3
b
,
3
c
,
and
3
d
⟶
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
t
≠
t
-
l
l
=
1
,
…
,
p
,
Δ
δ
x
t
=
t
-
l
=
δ
x
t
-
l
+
-
δ
x
t
-
l
-
=
J
l
x
t
-
l
-
∈
C
,
l
=
1,2
,
…
,
p
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
.Thus, (18)
(i)
lim
I
k
x
t
k
-
→
0
∀
k
∈
1,2
,
…
,
m
,
J
l
x
t
-
l
-
→
0
∀
l
∈
1,2
,
…
,
p
⁡
the  solution  of  system
3
a
,
3
b
,
3
c
,
a
n
d
3
d
=
the  solution  of  system
15
,
(ii)
lim
J
l
x
t
-
l
-
→
0
,
∀
l
∈
1,2
,
…
,
p
⁡
the  solution  of  system
3
a
,
3
b
,
3
c
,
a
n
d
3
d
=
the  solution  of  system
16
,
(iii)
lim
I
k
x
t
k
-
→
0
,
∀
k
∈
1,2
,
…
,
m
⁡
the  solution  of  system
3
a
,
3
b
,
3
c
,
a
n
d
3
d
=
the  solution  of  system
17
.Thus, the definition of solution of system (3a), (3b), (3c), and (3d) is presented as follows.Definition 8.
A functionz
(
t
)
:
[
a
,
T
]
→
C is said to be a solution of (3a), (3b), (3c), and (3d) if z
(
a
)
=
x
a and δ
z
(
a
)
=
x
^
a, the equation condition D
a
+
q
C-H
z
(
t
)
=
f
t
,
z
t for each t
∈
[
a
,
T
]
/
{
t
1
′
,
t
2
′
,
…
,
t
Π
′
} is verified, the impulsive conditions Δ
z
t
=
t
k
=
I
k
z
(
t
k
-
) (here k
=
1
,
2
,
…
,
m) and Δ
δ
z
t
=
t
-
l
=
J
l
z
(
t
-
l
-
) (here l
=
1
,
2
,
…
,
p) are satisfied, the restriction of z
(
·
) to the interval (
t
k
′
,
t
k
+
1
′
] (here k
=
0,1
,
2
,
…
,
Π) is continuous, and the conditions (i)–(iii) hold.Next, define a function by(19)
x
~
t
=
x
t
k
+
+
δ
x
t
k
+
ln
⁡
t
t
k
+
1
Γ
q
∫
t
k
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
,
for
t
∈
t
k
,
t
k
+
1
here
k
=
0,1
,
2
,
…
,
m
.By Theorem 4, we have(20)
D
a
+
q
C-H
x
~
t
t
∈
t
k
,
t
k
+
1
=
1
Γ
2
-
q
∫
a
t
ln
⁡
t
s
2
-
q
-
1
δ
2
x
t
k
+
+
δ
x
t
k
+
ln
⁡
s
t
k
+
1
Γ
q
∫
t
k
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
d
s
s
t
∈
t
k
,
t
k
+
1
=
1
Γ
2
-
q
Γ
q
∫
t
k
t
ln
⁡
t
s
2
-
q
-
1
δ
2
∫
t
k
s
l
n
s
η
q
-
1
f
η
,
x
η
d
η
η
d
s
s
t
∈
t
k
,
t
k
+
1
=
f
t
,
x
t
t
∈
t
k
,
t
k
+
1
.This means that x
~
(
t
) satisfies (3a), and x
~
(
t
) satisfies (3b)–(3d). However, x
~
(
t
) does not satisfy the conditions (i)–(iii), and it is not a solution of system (3a), (3b), (3c), and (3d). Therefore, x
~
(
t
) is considered as an approximate solution to seek the exact solutions of (3a), (3b), (3c), and (3d). Next, let us prove some useful conclusions.Lemma 9.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ξ is a constant. System (16) is equivalent to the integral equation(21)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
1
,
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
1
≤
k
≤
mprovided that the integral in (21) exists.Proof.
Necessity. Letting I
k
x
(
t
k
-
)
→
0
(
k
=
1,2
,
…
,
m
) in (16), we have(22)
l
i
m
I
k
x
t
k
-
→
0
,
∀
k
∈
1,2
,
…
,
m
s
y
s
t
e
m
16
⟶
D
a
+
q
C-H
x
t
=
f
t
,
x
t
,
t
∈
a
,
T
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
.That is, (23)
l
i
m
I
k
x
t
k
-
→
0
,
∀
k
∈
1,2
,
…
,
m
t
h
e
s
o
l
u
t
i
o
n
o
f
s
y
s
t
e
m
16
=
t
h
e
s
o
l
u
t
i
o
n
o
f
s
y
s
t
e
m
15
.In fact, we can verify that (21) satisfies the condition (23).
Next, taking fractional derivative to (21) for t
∈
(
t
k
,
t
k
+
1
] (here k
=
0,1
,
2
,
…
,
m), we get(24)
D
a
+
q
C-H
x
t
t
∈
t
k
,
t
k
+
1
=
1
Γ
2
-
q
∫
a
t
ln
⁡
t
s
2
-
q
-
1
δ
2
x
a
+
x
^
a
ln
⁡
s
a
+
∑
i
=
1
k
I
i
x
t
i
-
+
1
Γ
q
∫
a
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
η
q
-
1
f
η
,
x
η
d
η
η
+
∫
t
i
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
-
∫
a
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
+
ln
⁡
s
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
η
q
-
2
f
η
,
x
η
d
η
η
d
s
s
t
∈
t
k
,
t
k
+
1
=
1
Γ
2
-
q
Γ
q
∫
a
t
ln
⁡
t
s
2
-
q
-
1
δ
2
∫
a
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
∫
t
i
t
ln
⁡
t
s
2
-
q
-
1
δ
2
∫
t
i
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
d
s
s
-
∫
a
t
ln
⁡
t
s
2
-
q
-
1
δ
2
∫
a
s
ln
⁡
s
η
q
-
1
f
η
,
x
η
d
η
η
d
s
s
t
∈
t
k
,
t
k
+
1
=
f
t
,
x
t
t
≥
a
+
∑
i
=
1
k
ξ
I
i
x
t
i
-
f
t
,
x
t
t
≥
t
i
-
f
t
,
x
t
t
≥
a
t
∈
t
k
,
t
k
+
1
=
f
t
,
x
t
t
∈
t
k
,
t
k
+
1
.So, (21) satisfies the condition of fractional derivative in system (16).
Finally, using (21) for each t
k (here k
∈
{
1,2
,
…
,
m
}), we have(25)
x
t
k
+
-
x
t
k
-
=
lim
t
→
t
k
+
⁡
x
t
-
x
t
k
=
x
a
+
x
^
a
ln
⁡
t
k
a
+
∑
i
=
1
k
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
k
ln
⁡
t
k
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
ln
⁡
t
k
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
k
l
n
t
k
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
k
/
t
i
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
-
x
a
-
x
^
a
ln
⁡
t
k
a
-
∑
i
=
1
k
-
1
I
i
x
t
i
-
-
1
Γ
q
∫
a
t
k
l
n
t
k
s
q
-
1
f
s
,
x
s
d
s
s
-
ξ
∑
i
=
1
k
-
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
l
n
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
l
n
t
k
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
k
l
n
t
k
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
k
/
t
i
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
=
I
k
x
t
k
-
.It means that (21) satisfies the impulsive condition of (16). Hence, (21) satisfies all conditions of system (16).
Sufficiency (by mathematical induction). By Lemma 6, the solution of (16) satisfies(26)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
for
t
∈
a
,
t
1
.Using (26), we obtain(27)
x
t
1
+
=
x
t
1
-
+
I
1
x
t
1
-
=
x
a
+
x
^
a
ln
⁡
t
1
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
,
δ
x
t
1
+
=
δ
x
t
1
-
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
.Thus, the approximate solution x
~
(
t
) is given by(28)
x
~
t
=
x
t
1
+
+
δ
x
t
1
+
l
n
t
t
1
+
1
Γ
q
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
1
l
n
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
1
,
t
2
.Let e
1
t
=
x
t
-
x
~
t, for t
∈
(
t
1
,
t
2
]. By (26), the exact solution x
(
t
) of system (16) satisfies(29)
lim
I
1
x
t
1
-
→
0
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
,
for
t
∈
t
1
,
t
2
.Then,(30)
lim
I
1
x
t
1
-
→
0
⁡
e
1
t
=
lim
I
1
x
t
1
-
→
0
⁡
x
t
-
x
~
t
=
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
.This shows that e
1
(
t
) is connected with I
1
x
(
t
1
-
) and l
i
m
I
1
x
(
t
1
-
)
e
1
(
t
). Thus, we assume(31)
e
1
t
=
χ
I
1
x
t
1
-
l
i
m
I
1
x
t
1
-
→
0
e
1
t
=
-
χ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
l
n
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
l
n
t
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
1
,
t
2
,where function χ is an undetermined function with χ
(
0
)
=
1. So,(32)
x
t
=
x
~
t
+
e
1
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
1
-
χ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
l
n
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
l
n
t
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
1
,
t
2
.Letting γ
I
1
x
t
1
-
=
1
-
χ
I
1
(
x
(
t
1
-
)
), we get(33)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
1
,
t
2
.
Using (33), we obtain(34)
x
t
2
+
=
x
t
2
-
+
I
2
x
t
2
-
=
x
a
+
x
^
a
ln
⁡
t
2
a
+
I
1
x
t
1
-
+
I
2
x
t
2
-
+
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
2
/
t
1
Γ
q
-
1
∫
a
t
1
l
n
t
1
s
q
-
2
f
s
,
x
s
d
s
s
,
δ
x
t
2
+
=
δ
x
t
2
-
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
Γ
q
-
1
∫
a
t
1
l
n
t
1
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
1
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
.Therefore, the approximate solution x
~
t is provided by(35)
x
~
t
=
x
t
2
+
+
δ
x
t
2
+
ln
⁡
t
t
2
+
1
Γ
q
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1,2
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
2
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
Γ
q
-
1
ln
⁡
t
t
2
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
1
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.Let e
2
(
t
)
=
x
(
t
)
-
x
~
(
t
) for t
∈
(
t
2
,
t
3
]. Moreover, by (33), the exact solution x
(
t
) of system (16) satisfies(36)
lim
I
1
x
t
1
-
→
0
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
2
x
t
2
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
γ
I
2
x
t
2
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
,
lim
I
2
x
t
2
-
→
0
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
,
lim
I
1
x
t
1
-
→
0
,
I
2
x
t
2
-
→
0
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.Then,(37)
lim
I
1
x
t
1
-
→
0
⁡
e
2
t
=
lim
I
1
x
t
1
-
→
0
⁡
x
t
-
x
~
t
=
-
1
+
γ
I
2
x
t
2
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
,
lim
I
2
x
t
2
-
→
0
⁡
e
2
t
=
lim
I
2
x
t
2
-
→
0
⁡
x
t
-
x
~
t
=
-
1
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
-
γ
I
1
x
t
1
-
1
Γ
q
∫
t
1
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
t
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
t
1
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
,
lim
I
1
x
t
1
-
→
0
,
I
2
x
t
2
-
→
0
⁡
e
2
t
=
lim
I
1
x
t
1
-
→
0
,
I
2
x
t
2
-
→
0
⁡
x
t
-
x
~
t
=
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.By (37), we get(38)
e
2
t
=
γ
I
1
x
t
1
-
+
γ
I
2
x
t
2
-
-
1
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
-
γ
I
1
x
t
1
-
1
Γ
q
∫
t
1
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
t
1
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.Then,(39)
x
t
=
x
~
t
+
e
2
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1,2
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
γ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
+
γ
I
2
x
t
2
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
l
n
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.Consider the following limit case(40)
lim
t
2
→
t
1
⁡
D
C-H
a
+
q
x
t
=
f
t
,
x
t
,
t
∈
a
,
t
3
,
t
≠
t
1
,
t
≠
t
2
,
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
.
(41)
⟶
D
C-H
a
+
q
x
t
=
f
t
,
x
t
,
t
∈
a
,
t
3
,
t
≠
t
1
Δ
x
t
=
t
1
=
I
1
x
t
1
-
+
I
2
x
t
2
-
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
.Using (33) and (39) for (41) and (40), respectively, we have(42)
χ
I
1
x
t
1
-
+
I
2
x
t
2
-
=
χ
I
1
x
t
1
-
+
χ
I
2
x
t
2
-
f
o
r
∀
I
1
x
t
1
-
,
I
2
x
t
2
-
∈
C
.Therefore, χ
(
z
)
=
ξ
z  ∀
z
∈
C; here ξ is a constant. Thus, (43)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
1
,
t
2
,
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
I
1
x
t
1
-
+
I
2
x
t
2
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
I
1
x
t
1
-
1
Γ
q
∫
a
t
1
ln
⁡
t
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
1
Γ
q
-
1
∫
a
t
1
ln
⁡
t
1
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
I
2
x
t
2
-
1
Γ
q
∫
a
t
2
ln
⁡
t
2
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
2
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
2
Γ
q
-
1
∫
a
t
2
ln
⁡
t
2
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
2
,
t
3
.Next, suppose(44)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
.Using (44), we obtain(45)
x
t
k
+
1
+
=
x
t
k
+
1
-
+
I
k
+
1
x
t
k
+
1
-
=
x
a
+
x
^
a
ln
⁡
t
k
+
1
a
+
∑
i
=
1
k
+
1
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
k
+
1
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
,
δ
x
t
k
+
1
+
=
δ
x
t
k
+
1
-
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
.Therefore, the approximate solution x
~
(
t
) is presented by(46)
x
~
t
=
x
t
k
+
1
+
+
δ
x
t
k
+
1
+
ln
⁡
t
t
k
+
1
+
1
Γ
q
∫
t
k
+
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
+
1
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
k
+
1
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
+
1
,
t
k
+
2
.Let e
k
+
1
(
t
)
=
x
(
t
)
-
x
~
(
t
) for t
∈
(
t
k
+
1
,
t
k
+
2
]. In addition, by (44), the exact solution x
(
t
) of system (16) satisfies(47)
lim
I
i
x
t
i
-
→
0
,
∀
i
∈
1,2
,
…
,
k
+
1
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
lim
I
j
x
t
j
-
→
0
⁡
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
1
≤
i
≤
k
+
1
,
i
≠
j
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
1
≤
i
≤
k
+
1
,
i
≠
j
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
j
∈
1,2
,
…
,
k
+
1
.Then,(48)
lim
I
i
x
t
i
-
→
0
,
∀
i
∈
1,2
,
…
,
k
+
1
⁡
e
k
+
1
t
=
lim
I
i
x
t
i
-
→
0
,
∀
i
∈
1,2
,
…
,
k
+
1
⁡
x
t
-
x
~
t
=
-
1
Γ
q
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
lim
I
j
x
t
j
-
→
0
⁡
e
k
+
1
t
=
lim
I
j
x
t
j
-
→
0
⁡
x
t
-
x
~
t
=
-
1
+
ξ
∑
1
≤
i
≤
k
+
1
,
i
≠
j
I
i
x
t
i
-
1
Γ
q
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
a
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
-
ξ
∑
1
≤
i
≤
k
+
1
,
i
≠
j
I
i
x
t
i
-
1
Γ
q
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
t
i
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
t
i
t
k
+
1
ln
⁡
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
k
,
t
k
+
1
,
j
∈
1,2
,
…
,
k
+
1
.By (48), we obtain(49)
e
k
+
1
t
=
-
1
+
ξ
∑
i
=
1
k
+
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
k
+
1
l
n
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
a
t
k
+
1
l
n
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
-
ξ
∑
i
=
1
k
+
1
I
i
x
t
i
-
1
Γ
q
∫
t
i
t
k
+
1
l
n
t
k
+
1
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
k
+
1
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
t
i
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
/
t
k
+
1
Γ
q
-
1
∫
t
i
t
k
+
1
l
n
t
k
+
1
s
q
-
2
f
s
,
x
s
d
s
s
for
t
∈
t
k
,
t
k
+
1
,
j
∈
1,2
,
…
,
k
+
1
.Thus, we have(50)
x
t
=
x
~
t
+
e
k
+
1
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
+
1
I
i
x
t
i
-
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
+
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
+
1
,
t
k
+
2
.Then, the solution of system (16) satisfies (21).
By the proof of Sufficiency and Necessity, system (16) is equivalent to (21). The proof is completed.Lemma 10.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ζ is a constant. System (17) is equivalent to the integral equation(51)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
a
,
t
-
1
,
x
a
+
x
^
a
ln
⁡
t
a
+
∑
j
=
1
l
J
j
x
t
-
j
-
ln
⁡
t
t
-
j
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
l
J
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
-
j
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
t
-
l
,
t
-
l
+
1
,
1
≤
l
≤
pprovided that the integral in (51) exists.Remark 11.
For (17), we have(52)
lim
J
1
x
t
-
1
-
→
0
,
…
,
J
p
x
t
-
p
-
→
0
⁡
s
y
s
t
e
m
17
⟶
s
y
s
t
e
m
15
.Then,(53)
lim
J
1
x
t
-
1
-
→
0
,
…
,
J
p
x
t
-
p
-
→
0
⁡
the  solution  of  system
17
=
the  solution  of  system
15
.In fact, we can verify that (51) satisfies the condition (53). Moreover, the approximate solution x
~
(
t
) of system (17) is defined by(54)
x
~
t
=
x
t
-
l
+
+
δ
x
t
-
l
+
l
n
t
t
-
l
+
1
Γ
q
∫
t
-
l
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
-
l
,
t
-
l
+
1
;here x
(
t
-
l
+
)
=
x
(
t
-
l
-
) and δ
x
(
t
-
l
+
)
=
δ
x
(
t
-
l
-
)
+
J
l
(
x
(
t
-
l
-
)
), l
=
1,2
,
…
,
p.
Due to similarity with Lemma9, the proof is omitted.Corollary 12.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ξ is a constant. A function x
(
t
)
:
[
a
,
T
]
→
C is general solution of the system (16); then,(55)
δ
x
t
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
1
,
x
^
a
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
∫
a
t
i
l
n
t
i
s
q
-
2
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
,
t
k
+
1
,
1
≤
k
≤
mprovided that the integral in (55) exists.Corollary 13.
Letq
∈
C, R
(
q
)
∈
1,2, and ζ is a constant. A function x
(
t
)
:
[
a
,
T
]
→
C is general solution of the system (17); then,(56)
δ
x
t
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
-
1
,
x
^
a
+
∑
j
=
1
l
J
j
x
t
-
j
-
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
l
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
-
l
,
t
-
l
+
1
,
1
≤
l
≤
pprovided that the integral in (56) exists.Remark 14.
By Corollaries12 and 13, it is shown that two kinds of impulses Δ
x
t
=
t
k
(
k
=
1,2
,
…
,
m
) and Δ
δ
x
t
=
t
-
l
(
l
=
1
,
2
,
…
,
p
) have similar effect on δ
x
(
t
) of system (15).Lemma 15.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ξ and ζ are two constants. A function x
(
t
)
:
[
a
,
T
]
→
C is general solution of the system (3a), (3b), (3c), and (3d); then,(57)
δ
x
t
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
1
′
,
x
^
a
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Πprovided that the integral in (57) exists.Proof.
According to Corollaries12 and 13, the solutions of system (3a), (3b), (3c), and (3d) satisfy(58)
δ
x
t
=
x
^
a
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
a
,
t
1
′
.By the definition of Caputo-Hadamard fractional derivative, system (3a), (3b), (3c), and (3d) satisfies(59)
system
3
a
,
3
b
,
3
c
,
a
n
d
3
d
⟺
D
a
+
q
-
1
C-H
δ
x
t
=
f
t
,
x
t
,
t
∈
J
=
a
,
T
,
t
≠
t
k
k
=
1
,
…
,
m
,
t
≠
t
-
l
l
=
1
,
…
,
p
,
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
Δ
δ
x
t
=
t
-
l
=
δ
x
t
-
l
+
-
δ
x
t
-
l
-
=
J
l
x
t
-
l
-
∈
C
,
l
=
1,2
,
…
,
p
,
x
a
=
x
a
∈
C
,
δ
x
a
=
x
^
a
∈
C
.Moreover, it is reasonable that impulses Δ
x
t
=
t
k(k
=
1
,
2
,
…
,
m) are considered as special impulses Δ
δ
x
t
=
t
-
l (l
=
1,2
,
…
,
p) in system (59) by Remark 14. Therefore, using Lemma 7 for system (59) (as t
∈
(
t
k
′
,
t
k
+
1
′
], here k
=
1
,
2
,
…
,
Π), we have(60)
δ
x
t
=
x
^
a
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
+
1
Γ
q
-
1
∫
a
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
+
∑
i
=
1
k
1
ξ
i
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
+
∑
j
=
1
k
2
ζ
j
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
,where ξ
i (i
=
1
,
2
,
…
,
k
1) and ζ
j (j
=
1
,
2
,
…
,
k
2) are undetermined constants. Letting J
j
x
t
-
j
-
=
0 (for all j
∈
{
1
,
2
,
…
,
k
2
}) and I
i
x
(
t
i
-
)
=
0 (for all i
∈
{
1
,
2
,
…
,
k
1
}), respectively, we get ξ
i
=
ξ (for all i
∈
{
1
,
2
,
…
,
k
1
}) and ζ
j
=
ζ (for all j
∈
{
1
,
2
,
…
,
k
2
}) by Corollaries 12 and 13. Thus, (61)
δ
x
t
=
x
^
a
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
+
1
Γ
q
-
1
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Π
.This proof is completed.Theorem 16.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ξ and ζ are two constants. System (3a), (3b), (3c), and (3d) is equivalent to the integral equation(62)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
1
′
,
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
ln
⁡
t
t
-
j
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
-
j
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Πprovided that the integral in (62) exists.Proof.
Necessity. We can verify that (62) satisfies conditions (i)–(iii) by Lemmas 9, 10, and 6.
Next, taking the fractional derivative to (62) for t
∈
(
t
k
′
,
t
k
+
1
′
] (here k
=
0
,
1
,
2
,
…
,
Π), we get(63)
D
a
+
q
C-H
x
t
t
∈
t
k
′
,
t
k
+
1
′
=
1
Γ
2
-
q
∫
a
t
ln
⁡
t
η
2
-
q
-
1
δ
2
x
a
+
x
^
a
ln
⁡
η
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
ln
⁡
η
t
-
j
+
1
Γ
q
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
η
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
-
j
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
η
/
t
-
j
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
d
η
η
t
∈
t
k
′
,
t
k
+
1
′
=
1
Γ
2
-
q
Γ
q
∫
a
t
ln
⁡
t
η
2
-
q
-
1
δ
2
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
∫
t
i
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
∫
t
-
j
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
1
f
s
,
x
s
d
s
s
d
η
η
t
∈
t
k
′
,
t
k
+
1
′
=
f
t
,
x
t
t
≥
a
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
f
t
,
x
t
t
≥
t
i
-
f
t
,
x
t
t
≥
a
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
f
t
,
x
t
t
≥
t
-
j
-
f
t
,
x
t
t
≥
a
t
∈
t
k
′
,
t
k
+
1
′
=
f
t
,
x
t
t
∈
t
k
′
,
t
k
+
1
′
.So, (62) satisfies (3a).
Finally, it is straightforward to verify that (62) satisfies (3b) and (3c). So, (62) satisfies all conditions of system (3a), (3b), (3c), and (3d).
Sufficiency. According to Lemmas 9 and 10, the solutions of system (3a), (3b), (3c), and (3d) satisfy(64)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
a
,
t
1
′
.Next, by Lemma 15, the solutions of system (3a), (3b), (3c), and (3d) satisfy(65)
δ
x
t
=
x
^
a
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
+
1
Γ
q
-
1
∫
a
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
2
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Π
.Using (65), we have(66)
x
t
=
C
+
∫
x
^
a
η
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
η
+
1
Γ
q
-
1
∫
a
η
ln
⁡
η
s
q
-
2
1
η
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
1
η
f
s
,
x
s
d
s
s
+
∫
t
i
η
ln
⁡
t
s
q
-
2
1
η
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
2
1
η
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
1
η
f
s
,
x
s
d
s
s
+
∫
t
-
j
η
ln
⁡
η
s
q
-
2
1
η
f
s
,
x
s
d
s
s
-
∫
a
η
ln
⁡
η
s
q
-
2
1
η
f
s
,
x
s
d
s
s
d
η
=
C
+
x
^
a
ln
⁡
t
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
ln
⁡
t
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
∑
i
=
1
k
1
ξ
I
i
x
t
i
-
ln
⁡
t
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∑
j
=
1
k
2
ζ
J
j
x
t
-
j
-
ln
⁡
t
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
+
∑
i
=
1
k
1
ξ
I
i
x
t
i
-
Γ
q
∫
t
i
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
η
s
q
-
1
f
s
,
x
s
d
s
s
+
∑
j
=
1
k
2
ζ
J
j
x
t
-
j
-
Γ
q
∫
t
-
j
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
l
n
t
s
q
-
1
f
s
,
x
s
d
s
s
.Letting J
j
x
(
t
-
j
-
)
=
0 (for all j
∈
{
1
,
2
,
…
,
p
}) and I
i
x
(
t
i
-
)
=
0 (for all i
∈
{
1
,
2
,
…
,
m
}) in (66), respectively, by Lemmas 9 and 10, we obtain(67)
C
=
x
a
-
x
^
a
ln
⁡
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
-
∑
j
=
1
k
2
J
j
x
t
-
j
-
t
-
j
+
∑
i
=
1
k
1
ξ
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
l
n
t
i
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
i
Γ
q
-
1
∫
a
t
i
l
n
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
∑
j
=
1
k
2
ζ
J
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
l
n
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
-
ln
⁡
t
-
j
Γ
q
-
1
∫
a
t
-
j
l
n
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
.Thus,(68)
x
t
=
x
a
+
x
^
a
ln
⁡
t
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
+
∑
j
=
1
k
2
J
j
x
t
-
j
-
ln
⁡
t
t
-
j
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
x
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
J
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
-
j
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Π
.So, the solutions of system (3a), (3b), (3c), and (3d) satisfy (62). This proof is completed.Corollary 17.
Letq
∈
C, R
(
q
)
∈
(
1,2
), and ξ and ζ are two constants. System (1) is equivalent to the integral equation(69)
x
t
=
x
a
+
a
x
-
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
f
o
r
t
∈
a
,
t
1
′
,
x
a
+
a
x
-
a
ln
⁡
t
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
+
∑
j
=
1
k
2
t
-
j
I
-
j
x
t
-
j
-
ln
⁡
t
t
-
j
+
1
Γ
q
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ξ
∑
i
=
1
k
1
I
i
y
t
i
-
1
Γ
q
∫
a
t
i
ln
⁡
t
i
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
i
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
i
Γ
q
-
1
∫
a
t
i
ln
⁡
t
i
s
q
-
2
f
s
,
x
s
d
s
s
+
ζ
∑
j
=
1
k
2
t
-
j
I
-
j
x
t
-
j
-
1
Γ
q
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
f
s
,
x
s
d
s
s
+
∫
t
-
j
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
1
-
∫
a
t
ln
⁡
t
s
q
-
1
f
s
,
x
s
d
s
s
+
ln
⁡
t
/
t
-
j
Γ
q
-
1
∫
a
t
-
j
ln
⁡
t
-
j
s
q
-
2
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Πprovided that the integral in (69) exists.Remark 18.
Substitutingx
^
a
=
a
x
-
a and J
j
(
x
(
t
-
j
-
)
)
=
t
-
j
I
-
j
(
x
(
t
-
j
-
)
) into (62), (69) can be obtained. Next, let us analyze the limited case of system (1):(70)
l
i
m
q
→
2
-
s
y
s
t
e
m
1
⟶
δ
2
x
t
=
f
t
,
x
t
,
t
∈
J
=
a
,
T
,
t
≠
t
k
k
=
1
,
…
,
m
,
t
≠
t
-
l
l
=
1
,
…
,
p
,
Δ
x
t
=
t
k
=
x
t
k
+
-
x
t
k
-
=
I
k
x
t
k
-
∈
C
,
k
=
1,2
,
…
,
m
,
Δ
x
′
t
=
t
-
l
=
x
′
t
-
l
+
-
x
′
t
-
l
-
=
I
-
l
x
t
-
l
-
∈
C
,
l
=
1,2
,
…
,
p
,
x
a
=
x
a
∈
C
,
x
′
a
=
x
-
a
∈
C
.On the other hand, by (69), we have(71)
l
i
m
q
→
2
-
x
t
=
x
a
+
a
x
-
a
ln
⁡
t
a
+
1
Γ
q
∫
a
t
ln
⁡
t
s
f
s
,
x
s
d
s
s
,
f
o
r
t
∈
a
,
t
1
′
,
x
a
+
a
x
-
a
ln
⁡
t
a
+
∑
i
=
1
k
1
I
i
x
t
i
-
+
∑
j
=
1
k
2
I
-
j
x
t
-
j
-
t
-
j
ln
⁡
t
t
-
j
+
∫
a
t
ln
⁡
t
s
f
s
,
x
s
d
s
s
f
o
r
t
∈
t
k
′
,
t
k
+
1
′
,
1
≤
k
≤
Π
.
It can be verified that (71) is the solution of (70), which indirectly supports our conclusion.



---

*Source: 8101802-2016-03-31.xml*