# Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

Authors: Peiguang Wang; Xiaowei Liu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.


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## Body

## 1. Introduction

It is now well recognized that impulsive differential equations are suitable mathematical models for many processes and phenomena in biology, physics, technology, and so forth. That is why in recent years the mathematical theory of such systems has gained increasing significance. We notice that most of the studies about initial value problems of impulsive differential equations are investigated only for perturbation or change of dependent variable keeping the initial time unchanged. However, in dealing with real world phenomena, it is impossible not to make errors in the starting time. When we consider such a change of initial time for each solution, we need to deal with the problem of comparing between any two solutions which start at different times.At present, the investigation of differential systems with initial time difference has attracted a lot of attention. There are two methods of comparing the differences of the two solutions. One is the differential inequalities technique and comparison principle; the other is variation of parameters. For the pioneering works in this area we can refer to the papers [1, 2]. Ever since then, many results for various differential and difference systems have been obtained. The results obtained by the former method can be seen in [3–10]; and those done by the latter can be found in [11–15]. However, up till now, to the best of our knowledge, there are few results for impulsive differential equations with initial time difference. To be specific, there are no results on variation of parameters formula for impulsive differential equations relative to initial time difference. The method of variation of parameters is an important and fruitful technique since it is a practical tool in the investigation of the properties of solutions. It has been applied to the study of the relations of unperturbed and perturbed systems with different initial conditions.In this paper, we will develop variation of parameters formula for impulsive differential equations with initial time changed and investigate Lipschitz stability by using one of the results obtained. The remainder of this paper is organized in the following manner. Some preliminaries are presented in Section2, and various types of nonlinear variation of parameters formulae are established in Section 3. Finally, as an application, one of the results is applied to impulsive differential equations and the stability properties are obtained.


## 2. Preliminaries

LetR
+
=
[
0
,
+
∞
) and let R
n denote the n-dimensional Euclidean space with appropriate norm ∥
·
∥.Consider the following unperturbed impulsive differential equations(1)
x
′
=
f
(
t
,
x
)
,
t
≠
t
k
,
x
(
t
0
+
)
=
x
0
,
x
(
t
k
+
)
=
x
(
t
k
)
+
I
t
k
(
x
(
t
k
)
)
,
(2)
x
′
=
f
(
t
,
x
)
,
t
≠
t
k
,
x
(
τ
0
+
)
=
y
0
,
x
(
t
k
+
)
=
x
(
t
k
)
+
I
t
k
(
x
(
t
k
)
)
,
whenever
t
k
≥
τ
0
,
together with the perturbed ones of (2)
(3)
y
′
=
F
(
t
,
y
)
,
t
≠
t
k
,
y
(
τ
0
+
)
=
y
0
,
y
(
t
k
+
)
=
y
(
t
k
)
+
I
t
k
(
y
(
t
k
)
)
,
whenever
t
k
≥
τ
0
,
where(1)
0
≤
t
0
<
t
1
<
t
2
<
⋯
<
t
k
<
⋯, and lim
⁡
k
→
∞
t
k
=
∞
,
k
=
1,2
,
…;
(2)
τ
0
>
t
0,  η
=
τ
0
-
t
0;
(3)
t
¯
k
=
t
k
-
η
≥
0;
(4)
S
1
=
{
t
k
},  S
2
=
{
t
¯
k
},  S
=
S
1
∪
S
2;
(5)
t
∈
R
+
,
x
∈
Ω
⊂
R
n
,
Ω
-
open;
(6)
f
,
F
:
R
+
×
Ω
→
R
n;
(7)
I
t
k
:
Ω
→
R
n;
(8)
f
(
t
,
0
)
=
0,  I
t
k
(
0
)
=
0, for all t
k.We are concerned in this paper with the variation of parameters formula for impulsive differential equations relative to initial time difference. Before we can proceed, we will introduce the following lemmas [12], which are necessary for completing our main results.Lemma 1.
Let the following conditions be fulfilled:(
A
1)
the functionf
:
R
+
×
Ω
→
R
n is continuous in (
t
k
-
1
,
t
k
]
×
Ω, k
=
1,2
,
… and for every k and x
0
∈
R
n, there exists a finite limit of f
(
t
,
x
) as (
t
,
x
)
→
(
t
k
,
x
0
), t
>
t
k;
(A
2)
the functionf is locally Lipschitzian in x on R
+
×
Ω;
(A
3)
fork
=
1,2
,
… the mapping ψ
k
:
Ω
→
Ω, x
→
z, z
=
ψ
k
(
x
)
≡
x
+
I
k
(
x
) is a homeomorphism;
(A
4)
the system (1) had a solution ϕ
(
t
) defined in [
α
,
β
], (
α
,
β
≠
t
k
,
k
=
1,2
,
…
).Then there exist a number ϵ
>
0 and a set
(4)
V
=
{
(
t
,
x
)
∈
R
+
×
Ω
,
α
≤
t
≤
β
,
|
x
-
ϕ
(
t
+
)
|
<
ϵ
}
,
such that,(i)
for every(
t
0
,
x
0
)
∈
V, there exists a unique solution x
(
t
,
t
0
,
x
0
) of the system (1) which is defined on [
α
,
β
];
(ii)
the functionx
(
t
,
t
0
,
x
0
) is continuous for
(5)
t
∈
[
α
,
β
]
,
(
t
0
,
x
0
)
∈
V
,
t
,
t
0
∉
S
1
;
(iii)
fork
=
1,2
,
…
,
x
0
∈
Ω, t
,
t
0 belonging to the interval of existence of solution x
(
t
,
t
0
,
x
0
) of (1), t
∉
S
1,
(6)
lim
⁡
ξ
→
t
0
ρ
→
x
0
⁡
x
(
t
,
ξ
,
ρ
)
=
x
(
t
,
t
0
,
x
0
)
.Lemma 2.
Let the following conditions be fulfilled:(A
5)
the functionf
:
R
+
×
Ω
→
R
n is continuous in (
t
k
-
1
,
t
k
]
×
Ω, k
=
1,2
,
…, and f
x
(
t
,
x
) is continuous in (
t
k
-
1
,
t
k
)
×
Ω, k
=
1,2
,
…;
(A
6)
for everyx
0
∈
Ω, k
=
1,2
,
…, there exist finite limits of functions f and f
x as (
t
,
x
)
→
(
t
k
,
x
0
), t
>
t
k;
(A
7)
fork
=
1,2
,
… the mapping ψ
k
:
Ω
→
Ω
,
x
→
z
,
z
=
ψ
k
(
x
)
≡
x
+
I
k
(
x
) is a diffeomorphism and for x
∈
Ω
(7)
det
⁡
(
I
+
∂
I
k
∂
x
(
x
)
)
≠
0
,
k
=
1,2
,
…
.Then,(i)
there existsδ
>
0 such that the solution x
(
t
,
t
0
,
x
0
) of (1) has continuous derivatives ∂
x
/
∂
t,  ∂
x
/
∂
t
0,  ∂
x
/
∂
x
0, in the domain
(8)
V
:
α
<
t
<
β
,
α
<
t
0
<
β
,
t
,
t
0
≠
t
k
,
k
=
1,2
,
…
|
x
0
-
ϕ
(
t
0
+
)
|
<
δ
;
(ii)
the derivativeΦ
(
t
,
t
0
,
x
0
)
=
(
∂
x
/
∂
x
0
)
(
t
,
t
0
,
x
0
) is a solution of the initial value problem
(9)
u
′
=
f
x
(
t
,
ϕ
(
t
)
)
u
,
t
≠
t
k
,
Δ
u
=
∂
I
k
∂
x
(
ϕ
(
t
k
)
)
u
,
t
=
t
k
,
u
(
t
0
+
)
=
I
,
where ϕ
(
t
) is the solution of (1) in [
α
,
β
]
α
,
β
≠
t
k
,
k
=
1,2
,
…;
(iii)
the derivative∂
x
/
∂
t
0 satisfies the relation
(10)
∂
x
∂
t
0
(
t
,
t
0
,
x
0
)
=
-
∂
x
∂
x
0
(
t
,
t
0
,
x
0
)
f
(
t
0
,
x
0
)
=
-
Φ
(
t
,
t
0
,
x
0
)
f
(
t
0
,
x
0
)
.


## 3. Nonlinear Variation of Parameters Formula

We will present, in this section, the nonlinear variation of parameters formula for impulsive differential equations relative to initial time difference. It is very useful for investigating the stability properties of solutions.Theorem 3.
Let the system (1) satisfy the conditions of Lemma 2 and let x
(
t
,
t
0
,
x
0
) be a solution of (1). Then for any solution y
(
t
)
=
y
(
t
,
τ
0
,
y
0
) of the system (3). The following formula is valid:
(11)
y
(
t
+
η
,
τ
0
,
y
0
)
=
x
(
t
,
t
0
,
x
0
)
+
∫
0
1
‍
Φ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
d
s
+
∫
t
0
t
‍
f
~
(
s
,
y
~
(
s
)
,
η
)
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.Proof.
Setp
(
s
)
=
x
(
t
,
s
,
y
~
(
s
)
), where y
~
(
s
)
=
y
(
s
+
η
,
τ
0
,
y
0
), t
0
<
s
<
t. Then for s
∉
S, we have
(12)
p
′
(
s
)
=
∂
x
∂
s
(
t
,
s
,
y
~
(
s
)
)
+
∂
x
∂
y
~
(
t
,
s
,
y
~
(
s
)
)
F
(
s
+
η
,
y
~
(
s
)
)
≡
f
~
(
s
,
y
~
(
s
)
,
η
)
.
When s
∈
S, we have two cases.
Case
1. Consider
(13)
Δ
p
(
s
)
|
s
=
t
¯
k
=
x
(
t
,
t
¯
k
+
,
y
~
(
t
¯
k
+
)
)
-
x
(
t
,
t
¯
k
-
,
y
~
(
t
¯
k
-
)
)
=
x
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
-
x
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
)
=
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.
Case
2. Consider
(14)
Δ
p
(
s
)
|
s
=
t
k
∈
S
1
-
S
2
=
x
(
t
,
t
k
+
,
y
~
(
t
k
+
)
)
-
x
(
t
,
t
k
-
,
y
~
(
t
k
-
)
)
=
0
.
Integrating (12) from t
0 to t and using (13) and (14), we have
(15)
y
~
(
t
)
=
x
(
t
,
t
0
,
y
0
)
+
∫
t
0
t
‍
f
~
(
s
,
y
~
(
s
)
,
η
)
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.
Now let q
(
s
)
=
x
(
t
,
t
0
,
σ
(
s
)
), where σ
(
s
)
=
y
0
s
+
(
1
-
s
)
x
0, 0
≤
s
≤
1. Then we have
(16)
d
q
(
s
)
d
s
=
∂
x
∂
σ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
.
Integrating (16) from 0 to 1, we arrive at
(17)
x
(
t
,
t
0
,
y
0
)
=
x
(
t
,
t
0
,
x
0
)
+
∫
0
1
‍
∂
x
∂
σ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
d
s
.
Combining (15) and (17) yields
(18)
y
(
t
+
η
,
τ
0
,
y
0
)
=
x
(
t
,
t
0
,
x
0
)
+
∫
0
1
‍
Φ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
d
s
+
∫
t
0
t
‍
f
~
(
s
,
y
~
(
s
)
,
η
)
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.
The proof is complete.Corollary 4.
Suppose that the assumptions of Theorem3 hold except that F
(
t
+
η
,
y
) being replaced with f
(
t
,
y
)
+
R
(
t
+
η
,
y
); then the following formula is valid:
(19)
y
(
t
+
η
,
τ
0
,
y
0
)
=
x
(
t
,
t
0
,
x
0
)
+
∫
0
1
‍
Φ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
d
s
+
∫
t
0
t
‍
Φ
(
t
,
s
,
y
~
(
s
)
)
R
(
s
+
η
,
y
~
(
s
)
)
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.Theorem 5.
Suppose that the assumptions of Theorem3 hold; then the following formula is valid:
(20)
y
(
t
+
η
,
τ
0
,
y
0
)
-
x
(
t
,
t
0
,
x
0
)
=
x
(
t
,
t
0
,
η
)
+
∫
t
0
t
‍
H
~
(
s
,
w
(
s
)
,
η
)
d
s
+
∑
t
0
<
t
¯
k
<
t
t
¯
k
∈
S
2
-
S
1
∫
0
1
‍
Φ
(
+
s
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
t
,
t
¯
k
,
y
(
t
¯
k
+
η
)
-
x
(
t
¯
k
)
K
K
K
K
K
i
K
K
K
K
K
K
+
s
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
-
∑
t
0
<
t
k
<
t
t
k
∈
S
1
-
S
2
∫
0
1
‍
Φ
(
-
s
I
t
k
(
x
(
t
k
)
)
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
K
K
K
K
K
K
K
K
K
i
K
K
-
s
I
t
k
(
x
(
t
k
)
)
)
d
s
·
I
t
k
(
x
(
t
k
)
)
+

∑
t
0
<
t
k
<
t
t
k
∈
S
1
∩
S
2
∫
0
1
‍
Φ
(
-
s
I
t
k
(
x
(
t
k
)
)
t
,
t
k
,
y
(
t
k

+

η
)

-

x
(
t
k
)

+

s
I
t
k
+
η
(
y
(
t
k

+

η
)
)
K
K
K
K
K
K
K
K
K
K
K
K
-
s
I
t
k
(
x
(
t
k
)
)
)
d
s
·
(
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
I
t
k
(
x
(
t
k
)
)
)
.Proof.
Setp
(
s
)
=
x
(
t
,
s
,
w
(
s
)
), where w
(
s
)
=
y
(
s
+
η
,
τ
0
,
y
0
)
-
x
(
s
,
t
0
,
x
0
), t
0
<
s
<
t. Then for s
∉
S, we have
(21)
p
′
(
s
)
=
∂
x
∂
s
(
t
,
s
,
w
(
s
)
)
+
∂
x
∂
w
(
t
,
s
,
w
(
s
)
)
H
(
s
,
w
(
s
)
,
η
)
≡
H
~
(
s
,
w
(
s
)
,
η
)
,
where H
(
s
,
w
(
s
)
,
η
)
=
F
(
s
+
η
,
w
(
s
)
+
x
(
s
)
)
-
f
(
s
,
x
(
s
)
).
Ifs
∈
S, we have three cases.
Case
1. Consider
(22)
Δ
p
(
s
)
|
s
=
t
¯
k
∈
S
2
-
S
1
=
x
(
t
,
t
¯
k
+
,
w
(
t
¯
k
+
)
)
-
x
(
t
,
t
¯
k
-
,
w
(
t
¯
k
-
)
)
=
x
(
t
,
t
¯
k
,
y
(
t
¯
k
+
+
η
)
-
x
(
t
¯
k
+
)
)
-
x
(
t
,
t
¯
k
-
,
y
(
t
¯
k
-
+
η
)
000000000000000000
00
000000
00
0000
-
x
(
t
¯
k
-
)
)
=
x
(
t
,
t
¯
k
,
y
(
t
¯
k
+
η
)
+
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
-
x
(
t
¯
k
)
)
-
x
(
t
,
t
¯
k
,
y
(
t
¯
k
+
η
)
-
x
(
t
¯
k
)
)
=
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
(
t
¯
k
+
η
)
-
x
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
(
t
¯
k
+
η
)
)
.
Case
2. Consider
(23)
Δ
p
(
s
)
|
s
=
t
k
∈
S
1
-
S
2
=
x
(
t
,
t
k
+
,
w
(
t
k
+
)
)
-
x
(
t
,
t
k
-
,
w
(
t
k
-
)
)
=
x
(
t
,
t
k
,
y
(
t
k
+
+
η
)

-

x
(
t
k
+
)
)

-

x
(
t
,
t
k
,
y
(
t
k
-
+
η
)

-

x
(
t
k
-
)
)
=
x
(
t
,
t
k
,
y
(
t
k
+
η
)
-
I
t
k
(
x
(
t
k
)
)
-
x
(
t
k
)
)
-
x
(
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
)
=
-
∫
0
1
‍
Φ
(
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
-
s
I
t
k
(
x
(
t
k
)
)
)
d
s
·
I
t
k
(
x
(
t
k
)
)
.
Case
3. Consider
(24)
Δ
p
(
s
)
|
s
=
t
k
∈
S
1
∩
S
2
=
x
(
t
,
t
k
+
,
w
(
t
k
+
)
)
-
x
(
t
,
t
k
-
,
w
(
t
k
-
)
)
=
x
(
t
,
t
k
,
y
(
t
k
+
+
η
)
-
x
(
t
k
+
)
)
-
x
(
K
-
x
(
t
k
-
)
t
,
t
k
,
y
(
t
k
-
+
η
)
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
-
x
(
t
k
-
)
)
=
x
(
t
,
t
k
,
y
(
t
k
+
η
)
+
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
I
t
k
(
x
(
t
k
)
)
K
K
K
K
-
x
(
t
k
)
)
-
x
(
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
)
=
∫
0
1
‍
Φ
(
K
(
y
(
t
k
+
η
)
)
-
s
I
t
k
(
x
(
t
k
)
)
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
K
K
K
K
K
K
+
s
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
s
I
t
k
(
x
(
t
k
)
)
)
d
s
·
(
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
I
t
k
(
x
(
t
k
)
)
)
.
Integrating (21) from t
0 to t and using (22) and (24), we have
(25)
y
(
t
+
η
,
τ
0
,
y
0
)
-
x
(
t
,
t
0
,
x
0
)
=
x
(
t
,
t
0
,
η
)
+
∫
t
0
t
‍
H
~
(
s
,
w
(
s
)
,
η
)
d
s
+
∑
t
0
<
t
¯
k
<
t
t
¯
k
∈
S
2
-
S
1
∫
0
1
‍
Φ
(
K
K
(
y
(
t
¯
k

+

η
)
)
t
,
t
¯
k
,
y
(
t
¯
k
+
η
)
-
x
(
t
¯
k
)
K
K
K
K
K
K
K
K
K
K
K
+
s
I
t
¯
k
+
η
(
y
(
t
¯
k

+

η
)
)
K
K
e
f
22
e
f

)
d
s
·
I
t
¯
k
+
η
(
y
(
t
¯
k

+

η
)
)
-
∑
t
0
<
t
k
<
t
t
k
∈
S
1
-
S
2
∫
0
1
‍
Φ
(
t
,
t
k
,
y
(
t
k
+
η
)

-

x
(
t
k
)

-

s
I
t
k
(
x
(
t
k
)
)
)
d
s
·
I
t
k
(
x
(
t
k
)
)
+
∑
t
0
<
t
k
<
t
t
k
∈
S
1
∩
S
2
∫
0
1
‍
Φ
(
K
K
(
x
(
t
k
)
)
t
,
t
k
,
y
(
t
k
+
η
)
-
x
(
t
k
)
K
K
K
K
K
K
K
K
K
K
K
K
+
s
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
s
I
t
k
(
x
(
t
k
)
)

K
K
)
d
s
·
(
I
t
k
+
η
(
y
(
t
k
+
η
)
)
-
I
t
k
(
x
(
t
k
)
)
)
.
The proof is complete.


## 4. Application

In this section, we turn to the Lipschitz stability of system (3):
(26)
y
′
=
F
(
t
,
y
)
,
t
≠
t
k
,
y
(
τ
0
+
)
=
y
0
,
y
(
t
k
+
)
=
y
(
t
k
)
+
I
t
k
(
y
(
t
k
)
)
,
whenever
t
k
≥
τ
0
,
where F
(
t
+
η
,
y
)
=
f
(
t
,
y
)
+
R
(
t
+
η
,
y
).Definition 6.
The solutiony
(
t
+
η
,
τ
0
,
y
0
) of the system (3) is said to be initial time difference Lipschitz stable (
ITDLS
) with respect to the solution x
(
t
,
t
0
,
x
0
) for t
≥
t
0, where x
(
t
,
t
0
,
x
0
) is any solution of the system (1), if and only if there exists an M
=
M
(
τ
0
) such that
(27)
∥
y
(
t
+
η
,
τ
0
,
y
0
)
-
x
(
t
,
t
0
,
x
0
)
∥
≤
M
(
∥
y
0
-
x
0
∥
+
τ
0
-
t
0
)
.Theorem 7.
Let the following conditions be fulfilled:(
B
1
)
the assumptions of Corollary4 hold;
(
B
2
)
the zero solution of (1) is Lipschitz stable;
(
B
3
)
∥
Φ
(
t
,
s
,
y
~
(
s
)
)
R
(
s
+
η
,
y
~
(
s
)
)
∥
≤
γ
(
s
)
∥
y
~
(
s
)
∥  for t
0
<
s
≤
t;
(
B
4
)
∥
Φ
(
t
,
t
0
,
σ
(
s
)
)
∥
≤
M
1
(
∥
y
0
-
x
0
∥
+
η
)
/
∥
y
0
-
x
0
∥ and M
1 is a constant;
(
B
5
)
∥
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
∥
≤
β
k
∥
y
~
(
t
¯
k
)
∥ and β
k
≥
0 are constants;
(
B
6
)
∥
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
∥
≤
α
k and α
k
≥
0 are constants;
(
B
7
)
∫
t
0
∞
‍
γ
(
s
)
d
s
<
∞, γ
(
s
)
∈
C
[
R
+
,
R
+
] and ∏
t
0
<
t
¯
k
<
t
‍
(
1
+
α
k
β
k
)
<
∞.Then the solution y
(
t
+
η
,
τ
0
,
y
0
) of the system (3) is ITDLS with respect to the solution x
(
t
,
t
0
,
x
0
).Proof.
From Corollary4, it follows that
(28)
y
~
(
t
)
-
x
(
t
,
t
0
,
x
0
)
=
∫
0
1
‍
Φ
(
t
,
t
0
,
σ
(
s
)
)
(
y
0
-
x
0
)
d
s
+
∫
t
0
t
‍
Φ
(
t
,
s
,
y
~
(
s
)
)
R
(
s
+
η
,
y
~
(
s
)
)
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
‍
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
d
s
·
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
.
Taking the norm and using the triangle inequality on both sides, we have
(29)
∥
y
~
(
t
)
-
x
(
t
,
t
0
,
x
0
)
∥
≤
∫
0
1
∥
Φ
(
t
,
t
0
,
σ
(
s
)
)
∥
‍
∥
(
y
0
-
x
0
)
∥
d
s
+
∫
t
0
t
∥
Φ
(
t
,
s
,
y
~
(
s
)
)
R
(
s
+
η
,
y
~
(
s
)
)
∥
‍
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
∫
0
1
∥
Φ
(
t
,
t
¯
k
,
y
~
(
t
¯
k
)
+
s
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
)
∥
‍
d
s
·
∥
I
t
¯
k
+
η
(
y
~
(
t
¯
k
)
)
∥
.
From conditions (
B
2
)
–
(
B
5
), we obtain
(30)
∥
y
~
(
t
)
-
x
(
t
,
t
0
,
x
0
)
∥
≤
M
1
(
∥
y
0
-
x
0
∥
+
η
)
+
∫
t
0
t
‍
γ
(
s
)
∥
y
~
(
s
)
∥
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
∥
y
~
(
t
¯
k
)
∥
.
Setting M
*
(
t
)
=
∥
y
~
(
t
)
-
x
(
t
,
t
0
,
x
0
)
∥, we have
(31)
M
*
(
t
)
≤
M
1
(
∥
y
0
-
x
0
∥
+
η
)
+
∫
t
0
t
‍
γ
(
s
)
M
*
(
s
)
d
s
+
∫
t
0
t
‍
γ
(
s
)
∥
x
(
s
,
t
0
,
x
0
)
∥
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
M
*
(
t
¯
k
)
+
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
∥
x
(
t
¯
k
)
∥
.
Since ∥
x
(
t
,
t
0
,
x
0
)
∥
≤
M
2
∥
x
0
∥, as long as ∥
x
0
∥
<
ε, then we have
(32)
∥
x
(
t
,
t
0
,
x
0
)
∥
≤
M
2
ε
,
M
*
(
t
)
≤
M
1
(
∥
y
0
-
x
0
∥
+
η
)
+
∫
t
0
t
‍
γ
(
s
)
M
*
(
s
)
d
s
+
M
2
ε
∫
t
0
t
γ
(
s
)
‍
d
s
+
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
M
*
(
t
¯
k
)
+
M
2
ε
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
.
Applying Gronwall’s inequality to (32), we get
(33)
M
*
(
t
)
≤
{
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
M
1
(
∥
y
0
-
x
0
∥
+
η
)
K
K
K
K
+
M
2
ε
∫
t
0
t
‍
γ
(
s
)
d
s
+
M
2
ε
∑
t
0
<
t
¯
k
<
t
‍
α
k
β
k
}
·
∏
t
0
<
t
¯
k
<
t
‍
(
1
+
α
k
β
k
)
exp
⁡
{
∫
t
0
t
‍
γ
(
s
)
d
s
}
.
Setting  M
3
=
{
M
1
+
(
M
2
ε
∫
t
0
t
γ
(
s
)
d
s
/
∥
y
0
-
x
0
∥
+
η
)
+
(
M
2
ε
∑
t
0
<
t
¯
k
<
t
α
k
β
k
/
∥
y
0
-
x
0
∥
+
η
)
}
∏
t
0
<
t
¯
k
<
t
(
1
+
α
k
β
k
)
exp
⁡
{
∫
t
0
t
γ
(
s
)
d
s
}, we have
(34)
M
*
(
t
)
≤
M
3
(
∥
y
0
-
x
0
∥
+
η
)
.
From condition (
B
7
), it follows that the solution y
(
t
+
η
,
τ
0
,
y
0
) of the system (3) is ITDLS with respect to the solution x
(
t
,
t
0
,
x
0
).
The proof is complete.



---

*Source: 725832-2014-06-05.xml*