# Integral Stability in Terms of Two Measures for Nonlinear Differential Systems with “Maxima”

Authors: Peiguang Wang; Qing Xu; Xiaojing Liu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper investigates relatively integral stability in terms of two measures for two differential systems with “maxima” by employing Lyapunov functions, Razumikhin method, and comparison principle. An example is given to illustrate our result.


---

## Body

## 1. Introduction

Recently, the interest in differential equations with “maxima” has increased exponentially. Such equations adequately model real world problems whose present state depends significantly on its maximum value on a past time interval. For example, many problems in the control theory correspond to the maximal deviation of the regulated quantity. Some qualitative properties of the solutions of ordinary differential equations with “maxima” can be found in [1–4] and references therein.Integral stability for ordinary differential equations was introduced by Vrkoc [5]. The concept of integral stability occurs in connection with the stability under persistent perturbations when the perturbations are small enough everywhere except on a small interval. Recent developments in this field have been focused on various types of differential equations. In [6, 7], the integral stability and integral ϕ
0-stability properties of ordinary differential equations were discussed, respectively. Later, Hristova [8] discussed the integral stability in terms of two measures for impulsive differential equations with “supremum.” Moreover the same author in [9] discussed the integral stability in terms of two measures for impulsive functional differential equations. However, the integral stability in terms of two measures for two differential systems has not been obtained until now.In this paper, we discuss the relatively integral stability in terms of two measures for two differential systems with “maxima.” Using Lyapunov functions, Razumikhin method, and comparison principle, sufficient conditions for uniform-relatively integral stability in terms of two measures are obtained.


## 2. Preliminaries

Firstly, we give the following sets for convenience:(1)
K
=
{
a
(
t
)
∈
C
[
R
+
,
R
+
]
∣
a
(
t
)
is
strictly
increasing
,
a
(
0
)
=
0
[
R
+
,
R
+
]
}
;
C
K
=
{
a
(
t
)
∈
C
[
R
+
2
,
R
+
]
∣
∀
t
∈
R
+
,
a
(
t
,
u
)
∈
K
}
;
Γ
=
{
h
∈
C
[
R
+
×
R
n
,
R
+
]
∣
(
t
,
x
)
∈
R
+
×
R
n
,
inf
⁡
h
(
t
,
x
)
=
0
[
R
+
×
R
n
,
R
+
]
}
.Letρ, t, and r
>
0 be constants, h
0
∈
Γ. Define the following sets:
(2)
S
(
h
,
ρ
)
=
{
(
t
,
x
,
y
)
∈
R
+
×
R
n
×
R
n
∣
h
(
t
,
x
-
y
)
<
ρ
}
;
S
c
(
h
,
ρ
)
=
{
(
t
,
x
,
y
)
∈
R
+
×
R
n
×
R
n
∣
h
(
t
,
x
-
y
)
≥
ρ
}
;
Ω
(
t
,
h
,
ρ
)
=
{
(
x
,
y
)
∈
R
n
×
R
n
∣
h
(
t
,
x
)
≤
ρ
,
h
(
s
,
y
)
≤
ρ
,
s
∈
[
t
-
r
,
t
]
}
.We consider the following two differential systems with “maxima”:(3)
x
′
=
F
1
(
t
,
x
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
)
,
t
≥
t
0
,
x
(
t
+
t
0
)
=
ϕ
1
(
t
)
,
t
∈
[
-
r
,
0
]
,
y
′
=
F
2
(
t
,
y
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
)
,
t
≥
t
0
,
y
(
t
+
t
0
)
=
ϕ
2
(
t
)
,
t
∈
[
-
r
,
0
]
,
and the perturbed systems
(4)
x
′
=
F
1
(
t
,
x
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
)
+
G
1
(
t
,
x
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
)
,
t
≥
t
0
,
x
(
t
+
t
0
)
=
ϕ
1
(
t
)
,
t
∈
[
-
r
,
0
]
,
y
′
=
F
2
(
t
,
y
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
)
+
G
2
(
t
,
y
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
)
,
t
≥
t
0
,
y
(
t
+
t
0
)
=
ϕ
2
(
t
)
,
t
∈
[
-
r
,
0
]
,
where x
,
y
∈
R
n, F
1
,
F
2
,
G
1
,
G
2
∈
C
[
R
+
×
R
n
×
R
n
,
R
n
], F
1
(
t
,
0,0
)
=
F
2
(
t
,
0,0
)
=
G
1
(
t
,
0,0
)
=
G
2
(
t
,
0,0
)
≡
0 with R
+
=
[
0
,
∞
), r
>
0 is a given fixed number, t
0
∈
R
+, and ϕ
1
,
ϕ
2
∈
C
(
[
-
r
,
0
]
,
R
n
); R
n denote the n-dimensional Euclidean space with any convenient norm ∥
·
∥.We denote byx
(
t
;
t
0
,
ϕ
1
), y
(
t
;
t
0
,
ϕ
2
) the solutions of systems (3) satisfying the initial conditions x
(
t
+
t
0
)
=
ϕ
1
(
t
), y
(
t
+
t
0
)
=
ϕ
2
(
t
). Assume that solutions x
(
t
;
t
0
,
ϕ
1
), y
(
t
;
t
0
,
ϕ
2
) are defined on [
t
0
-
r
,
∞
) for any initial functions ϕ
1
,
ϕ
2
∈
C
(
[
-
r
,
0
]
,
R
n
).In further investigations, we need the following comparison scalar ordinary differential equations:(5)
u
′
=
f
(
t
,
u
)
,
(6)
u
′
=
g
(
t
,
u
)
,
and its perturbed scalar ordinary differential equation
(7)
ω
′
=
g
(
t
,
ω
)
+
q
(
t
)
,
where u
,
ω
∈
R, f, g
∈
R
+
×
R
→
R, and q
∈
R
+
→
R.The following definitions will be needed in the sequel.Definition 1.
Lettingh
0
,
h
∈
Γ, then (i)
h
0 is finer than h if there exits a δ
>
0 and a function a
∈
C
K such that
(8)
h
0
(
t
,
x
)
<
δ
implies
h
(
t
,
x
)
≤
a
(
t
,
h
0
(
t
,
x
)
)
;
(ii)
h
0 is uniformly finer than h if there exists a δ
>
0 and a function a
∈
K such that
(9)
h
0
(
t
,
x
)
<
δ
implies
h
(
t
,
x
)
≤
a
(
h
0
(
t
,
x
)
)
.Definition 2.
The functionV
(
t
,
x
,
y
) belongs to class V
0, if V
(
t
,
x
,
y
)
∈
C
[
Ω
×
R
n
×
R
n
,
R
+
], Ω
⊂
R
+, and V
(
t
,
x
,
y
) is Lipschitz with respect to x and y.
LettingV
(
t
,
x
,
y
)
∈
V
0, we define a derivative of the function V
(
t
,
x
,
y
) along the trajectory of systems (3) as follows:
(10)
D
(
3
)
V
(
t
,
ϕ
1
(
t
)
,
ϕ
2
(
t
)
)
=
limsup
⁡
ϵ
→
0
1
ϵ
{
V
(
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
2
(
t
+
s
)
)
t
+
ϵ
,
ϕ
1
(
t
)
+
ϵ
F
1
(
t
,
ϕ
1
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
1
(
t
+
s
)
)
,
ϕ
2
(
t
)
+
ϵ
F
2
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
2
(
t
+
s
)
)
)
-
V
(
t
,
ϕ
1
(
t
)
,
ϕ
2
(
t
)
)
(
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
2
(
t
+
s
)
)
t
+
ϵ
,
ϕ
1
(
t
)
}
and a derivative of the function V
(
t
,
x
,
y
) along the trajectory of systems (4) as follows:
(11)
D
(
4
)
V
(
t
,
ϕ
1
(
t
)
,
ϕ
2
(
t
)
)
=
limsup
⁡
ϵ
→
0
1
ϵ
{
V
(
G
2
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
ϕ
2
(
t
+
s
)
)
)
t
+
ϵ
,
ϕ
1
(
t
)
+
ϵ
(
F
1
(
t
,
ϕ
1
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
1
(
t
+
s
)
)
+
G
1
(
t
,
ϕ
1
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
ϕ
1
(
t
+
s
)
)
)
,
ϕ
2
(
t
)
+
ϵ
(
F
2
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
-
r
,
0
]
ϕ
2
(
t
+
s
)
)
+
G
2
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
ϕ
2
(
t
+
s
)
)
)
)
-
V
(
t
,
ϕ
1
(
t
)
,
ϕ
2
(
t
)
)
(
G
2
(
t
,
ϕ
2
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
ϕ
2
(
t
+
s
)
)
)
t
+
ϵ
,
ϕ
1
(
t
)
}
.Definition 3.
LettingV
∈
V
0 and h
0
,
h
∈
Γ, then V is said to be(i)
relativelyh-positive definite if there exists a γ
>
0 and a function a
∈
K such that
(12)
h
(
t
,
x
-
y
)
<
δ
implies
a
(
h
(
t
,
x
-
y
)
)
≤
V
(
t
,
x
,
y
)
;
(ii)
relativelyh
0-decrescent if there exists a λ
>
0 and a function b
∈
K such that
(13)
h
0
(
t
,
x
-
y
)
<
λ
implies
V
(
t
,
x
,
y
)
≤
b
(
h
0
(
t
,
x
-
y
)
)
;
(iii)
weak-relativelyh
0-decrescent if there exists a λ
>
0 and a function b
∈
C
K such that
(14)
h
0
(
t
,
x
-
y
)
<
λ
implies
V
(
t
,
x
,
y
)
≤
b
(
t
,
h
0
(
t
,
x
-
y
)
)
.
One will introduce relatively integral stability in terms of two measures for differential systems (3).Definition 4.
Lettingh
0
,
h
∈
Γ, differential systems (3) are said to be uniform-relatively integrally stable in terms of measures (
h
0
,
h
), if for α
>
0 and any t
0
≥
0, there exists β
=
β
(
α
)
∈
K such that, for any initial functions ϕ
1
,
ϕ
2
∈
C
(
[
-
r
,
0
]
,
R
n
) and any perturbations G
1
,
G
2
∈
C
(
R
+
×
R
n
×
R
n
,
R
n
), the inequality
(15)
h
(
t
,
x
1
-
y
2
)
<
β
,
t
≥
t
0
holds, provided that
(16)
max
⁡
s
∈
[
-
r
,
0
]
h
0
(
t
0
+
s
,
ϕ
1
-
ϕ
2
)
<
α
,
(17)
∫
t
0
t
0
+
T
‍
sup
⁡
(
x
-
y
,
x
1
-
y
2
)
∈
Ω
(
s
,
h
,
β
)
∥
G
1
(
s
,
x
,
x
1
)
-
G
2
(
s
,
y
,
y
2
)
∥
d
s
≤
α
,
T
>
0
,
where x
1
(
t
;
t
0
,
ϕ
1
), y
2
(
t
;
t
0
,
ϕ
2
) are the solutions of the initial value problem for perturbed differential systems with “maxima” (4).


## 3. Main Results

In further investigations, we need the following comparison result.Lemma 5.
Let the following conditions hold:(
H
1
)
F
1
,
F
2
∈
C
(
[
t
0
,
T
]
×
R
n
×
R
n
,
R
n
), where t
0
,
T
∈
R
+, t
0
<
T;
(
H
2
)
V
:
[
t
0
,
T
]
×
R
n
×
R
n
→
R
+, V
∈
V
0 and
(
i
)
for any numbert
∈
[
t
0
,
T
] and any function ψ
1
,
ψ
2
∈
C
(
[
t
-
r
,
t
]
,
R
n
) such that V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
>
V
(
t
+
s
,
ψ
1
(
t
+
s
)
,
ψ
2
(
t
+
s
)
) for s
∈
[
-
r
,
0
) the inequality
(18)
D
(
3
)
V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
g
(
t
,
V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
holds, whereg
∈
C
(
[
t
0
,
T
]
×
R
+
,
R
+
)
,
g
(
t
,
0
)
≡
0;
(
H
3
)
x
(
t
;
t
0
,
ϕ
1
), y
(
t
;
t
0
,
ϕ
2
) are the solutions of the initial value problem for differential systems with “maxima” (3);
(
H
4
)
u
*
(
t
)
=
u
*
(
t
;
t
0
,
u
0
) is the maximal solution of (6) with initial condition u
*
(
t
0
)
=
u
0, which is defined for t
∈
[
t
0
,
T
].
Then the inequalitymax
⁡
s
∈
[
-
r
,
0
]
⁡
V
(
t
0
+
s
,
ϕ
1
(
s
)
,
ϕ
2
(
s
)
)
≤
u
0 implies the validity of the inequality V
(
t
,
x
(
t
;
t
0
,
ϕ
1
)
,
y
(
t
;
t
0
,
ϕ
2
)
)
≤
u
*
(
t
), t
∈
[
t
0
,
T
].Proof.
Letu
0
∈
R
+ and ϕ
1
,
ϕ
2
∈
C
(
[
-
r
,
0
]
,
R
n
) be such that
(19)
max
⁡
s
∈
[
-
r
,
0
]
V
(
t
0
+
s
,
ϕ
1
(
s
)
,
ϕ
2
(
s
)
)
≤
u
0
.
Let υ
n
(
t
) be the maximal solution of the initial value problem
(20)
u
′
=
g
(
t
,
u
)
+
1
n
,
u
(
t
0
)
=
u
0
+
1
n
.
Let m
(
t
)
∈
C
(
[
t
0
-
r
,
T
]
,
R
+
)
:
m
(
t
)
=
V
(
t
,
x
(
t
;
t
0
,
ϕ
1
)
,
y
(
t
;
t
0
,
ϕ
2
)
).
Because of the fact thatu
*
(
t
;
t
0
,
u
0
)
=
lim
⁡
n
→
∞
⁡
υ
n
(
t
), it is enough to prove that for any n the inequality
(21)
m
(
t
)
≤
υ
n
(
t
)
,
t
∈
[
t
0
,
T
]
holds. Then the inequality m
(
t
0
)
<
υ
n
(
t
0
) holds.
Assume that inequality (21) is not true; then there exists a point η
∈
(
t
0
,
T
]
:
m
(
η
)
>
υ
n
(
η
). Let t
*
=
sup
⁡
{
t
∈
[
t
0
,
T
]
:
m
(
s
)
<
υ
n
(
s
)
,
s
∈
[
t
0
,
t
)
}. According to the assumption t
*
<
T, we have
(22)
m
(
t
*
)
=
υ
n
(
t
*
)
,
m
(
t
)
<
υ
n
(
t
)
,
t
∈
[
t
0
,
t
*
)
,
m
(
t
)
≥
υ
n
(
t
)
,
t
∈
[
t
*
,
t
*
+
δ
)
,
where δ
>
0 is a small enough number. From inequality (22) it follows that
(23)
m
′
(
t
)
≥
υ
n
′
(
t
)
=
g
(
t
*
,
υ
n
(
t
*
)
)
+
1
n
=
g
(
t
*
,
m
(
t
*
)
)
+
1
n
.
From g
(
t
,
u
)
+
1
/
n
>
0 on [
t
*
-
r
,
t
*
]
∩
[
t
0
,
T
], it follows that the function υ
n
(
t
) is nondecreasing on [
t
*
-
r
,
t
*
]
∩
[
t
0
,
T
].
Thereforem
(
t
*
)
>
m
(
s
) for s
∈
[
t
*
-
r
,
t
*
).
According to condition (i) of Lemma5 and definition of function m
(
t
), we get m
′
(
t
*
)
≤
g
(
t
*
,
m
(
t
*
)
)
<
g
(
t
*
,
m
(
t
*
)
)
+
1
/
n that contradicts (23).
Therefore inequality (21) holds and the conclusion of Lemma 5 follows.In the following results, we will obtain sufficient conditions for uniform-relatively integral stability in terms of two measures.Theorem 6.
Let the following conditions hold:(
A
1
)
h
0
,
h
∈
Γ, h
0 is uniformly finer than h;
(
A
2
)
there existsV
1
∈
V
0, it is relatively h
0-decrescent and
(i)
for any numbert
≥
0 and functions ψ
1
,
ψ
2
∈
C
(
[
t
-
r
,
t
]
,
R
n
) such that V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
>
V
1
(
t
+
s
,
ψ
1
(
t
+
s
)
,
ψ
2
(
t
+
s
)
) for s
∈
[
-
r
,
0
) and (
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
∈
S
(
h
,
ρ
), the inequality
(24)
D
(
4
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
f
(
t
,
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
holds, wheref
∈
C
(
R
+
×
R
,
R
), ρ
>
0 is a constant;
(
A
3
)
for any numberμ
>
0, there exists V
2
(
μ
)
∈
V
0 such that
(ii)
b
(
h
(
t
,
x
-
y
)
)
≤
V
2
(
μ
)
(
t
,
x
,
y
)
≤
a
(
h
0
(
t
,
x
-
y
)
), where a
,
b
∈
K and lim
⁡
u
→
∞
⁡
b
(
u
)
=
∞;
(iii)
for any numbert
≥
0 and functions ψ
1
,
ψ
2
∈
C
(
[
-
r
,
0
]
,
R
n
) such that (
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
∈
S
(
h
,
ρ
)
∩
S
C
(
h
0
,
μ
) and
(25)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
>
V
1
(
t
+
s
,
ψ
1
(
t
+
s
)
,
ψ
2
(
t
+
s
)
)
+
V
2
(
μ
)
(
t
+
s
,
ψ
1
(
t
+
s
)
,
ψ
2
(
t
+
s
)
)
fors
∈
[
-
r
,
0
), the inequality
(26)
D
(
3
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
D
(
3
)
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
g
(
t
,
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
holds, whereg
∈
C
(
R
+
×
R
,
R
), g
(
t
,
0
)
≡
0;
(
A
4
)
the zero solution of differential equation (5) is equistable;
(
A
5
)
the zero solution of differential equation (6) is uniform-integrally stable.
Then differential systems with “maxima” (3) are uniform-relatively integrally stable in terms of measures (
h
0
,
h
).Proof.
SinceV
1
(
t
,
x
,
y
) is relatively h
0-decrescent, there exist ρ
1
∈
(
0
,
ρ
) and ψ
3
∈
K such that h
0
(
t
,
x
-
y
)
<
ρ
1, the inequality
(27)
V
1
(
t
,
x
,
y
)
≤
ψ
3
(
h
0
(
t
,
x
-
y
)
)
holds.
Sinceh
0 is uniformly finer than h, there exist ρ
0
∈
(
0
,
ρ
1
) and ψ
4
∈
K
:
ψ
4
(
ρ
0
)
<
ρ
1 such that h
0
(
t
,
x
-
y
)
<
ρ
0 implies
(28)
h
(
t
,
x
-
y
)
≤
ψ
4
(
h
0
(
t
,
x
-
y
)
)
.
Letα
>
0 be a number such that α
<
ρ
0. According to condition (
A
3
), there exist V
2
(
α
)
(
t
,
x
,
y
) with Lipschitz constant M
2. Let M
1 be Lipschitz constant of the function V
1
(
t
,
x
,
y
).
Denote(
M
1
+
M
2
)
α
=
α
1. Without loss of generality, we assume α
1
<
b
(
ρ
).
From condition(
A
4
), it follows that there exists a δ
1
=
δ
1
(
t
0
,
α
1
)
>
0 such that the inequality |
u
0
|
<
δ
1 implies that
(29)
|
u
(
t
)
|
<
α
1
2
,
t
≥
t
0
,
where u
(
t
) is a solution of (5) with the initial condition u
(
t
0
)
=
u
0.
Sinceψ
3
∈
K, there exists a δ
2
=
δ
2
(
δ
1
)
>
0, δ
2
<
ρ
1 such that, for |
u
|
<
δ
2, the inequality
(30)
ψ
3
(
u
)
<
δ
1
holds.
From condition(
A
5
), it follows that there exist β
1
=
β
1
(
α
1
)
∈
K and b
(
ρ
)
>
β
1
≥
α
1 such that, for every solution ω
(
t
) of perturbed equation (7) with the initial condition ω
(
t
0
)
=
ω
0, the inequality
(31)
|
ω
(
t
)
|
<
β
1
,
t
≥
t
0
,
holds, provided that |
ω
0
|
<
α
1 and for every T
>
0
:
∫
t
0
t
0
+
T
‍
|
q
(
s
)
|
d
s
<
α
1.
Sinceb
∈
K, lim
⁡
s
→
∞
⁡
b
(
s
)
=
∞, and ψ
4
(
α
)
<
ψ
4
(
ρ
0
)
<
ρ
1
<
ρ, we choose β
=
β
(
β
1
)
>
0, ρ
>
β
>
α, β
>
ψ
4
(
α
) such that
(32)
b
(
β
)
≥
β
1
.
Sincea
,
ψ
4
∈
K and β
>
ψ
4
(
α
), we can find δ
3
=
δ
3
(
α
1
,
β
)
>
0
,
α
<
δ
3
<
min
⁡
(
δ
2
,
ρ
0
) such that the inequalities
(33)
a
(
δ
3
)
<
α
1
2
,
ψ
4
(
δ
3
)
<
β
hold.
Now let the initial functionϕ
1
,
ϕ
2
∈
C
(
[
-
r
,
0
]
,
R
n
) and perturbation G
1
(
t
,
x
,
y
), G
2
(
t
,
x
,
y
) of the right-hand side of differential systems (4) be such that
(34)
max
⁡
s
∈
[
-
r
,
0
]
h
0
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
<
α
,
and for every T
>
0
(35)
∫
t
0
t
0
+
T
‍
sup
⁡
(
x
-
y
,
x
1
-
y
2
)
∈
Ω
(
s
,
h
,
β
)
∥
G
1
(
s
,
x
,
x
1
)
-
G
2
(
s
,
y
,
y
2
)
∥
d
s
<
α
.
We will prove that(36)
h
(
t
,
x
1
(
t
)
-
y
2
(
t
)
)
<
β
,
t
≥
t
0
.
From (28) and the choice of β, it follows that h
0
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
<
α
<
ρ
0 implies that h
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
≤
ψ
4
(
h
0
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
)
<
ψ
4
(
α
)
<
β; that is,
(37)
h
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
<
β
,
s
∈
[
-
r
,
0
]
.
Suppose inequality (37) is not true. Therefore, there exists a point t
*
>
t
0 such that
(38)
h
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
=
β
,
h
(
t
,
x
1
(
t
)
-
y
2
(
t
)
)
<
β
,
t
∈
[
t
0
-
r
,
t
*
)
.
From inequality (38) and β
<
ρ, it follows the validity of the inclusions
(39)
(
t
,
x
1
(
t
)
,
y
2
(
t
)
)
∈
S
(
h
,
ρ
)
,
(
x
1
(
t
)
-
y
2
(
t
)
,
max
⁡
s
∈
[
t
-
r
,
t
]
(
x
1
(
s
)
-
y
2
(
s
)
)
)
∈
Ω
(
t
,
h
,
β
)
,
where t
∈
[
t
0
,
t
*
].
Assume thath
0
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
<
δ
3; then from the choice of δ
3 and inequality (28) it follows h
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
≤
ψ
4
(
h
0
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
)
≤
ψ
4
(
δ
3
)
<
β, which contradicts (38). Therefore
(40)
h
0
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
>
δ
3
,
h
0
(
t
0
+
s
,
ϕ
1
(
s
)
-
ϕ
2
(
s
)
)
<
α
<
δ
3
,
s
∈
[
-
r
,
0
]
,
and there exists a point t
0
*
∈
(
t
0
,
t
*
) such that δ
3
=
h
0
(
t
0
*
,
x
1
(
t
0
*
)
-
y
2
(
t
0
*
)
) and (
t
,
x
1
(
t
)
,
y
2
(
t
)
)
∈
S
(
h
,
β
)
∩
S
c
(
h
0
,
δ
3
) for t
∈
[
t
0
*
,
t
*
). Since β
<
ρ and δ
3
>
α, it follows that
(41)
(
t
,
x
1
(
t
)
,
y
2
(
t
)
)
∈
S
(
h
,
β
)
∩
S
c
(
h
0
,
α
)
,
t
∈
[
t
0
*
,
t
*
]
.
Letr
1
(
t
) be the maximal solution of differential equation (5) with the initial condition r
1
(
t
0
)
=
u
0, where u
0
=
max
⁡
s
∈
[
-
r
,
0
]
⁡
V
1
(
t
0
+
s
,
ϕ
1
(
t
0
+
s
)
,
ϕ
2
(
t
0
+
s
)
). From condition (
i
) of Theorem 6 and according to Lemma 5, we obtain
(42)
V
1
(
t
,
x
1
(
t
)
,
y
2
(
t
)
)
≤
r
1
(
t
)
,
t
∈
[
t
0
,
t
*
]
.
From inequality (30), we obtain
(43)
u
0
=
V
1
(
ξ
,
ϕ
1
(
ξ
)
,
ϕ
2
(
ξ
)
)
≤
ψ
3
(
h
0
(
ξ
,
ϕ
1
(
ξ
)
-
ϕ
2
(
ξ
)
)
)
<
ψ
3
(
α
)
<
ψ
3
(
δ
2
)
<
δ
1
,
where ξ
∈
[
t
0
-
r
,
t
0
].
From inequalities (29), (42), and (43), we have
(44)
V
1
(
t
,
x
1
(
t
)
,
y
2
(
t
)
)
≤
r
1
(
t
)
<
α
1
2
,
t
∈
[
t
0
,
t
*
]
or
(45)
max
⁡
s
∈
[
-
r
,
0
]
V
1
(
t
0
*
+
s
,
x
1
(
t
0
*
+
s
)
,
y
2
(
t
0
*
+
s
)
)
<
α
1
2
.
From inequality (33) and condition (
ii
) of Theorem 6, it follows that
(46)
V
2
(
α
)
(
t
0
*
+
s
,
x
1
(
t
0
*
+
s
)
,
y
2
(
t
0
*
+
s
)
)
≤
a
(
h
0
(
t
0
*
+
s
,
x
1
(
t
0
*
+
s
)
-
y
2
(
t
0
*
+
s
)
)
)
<
α
1
2
.
ConsiderV
∈
V
0 defined by
(47)
V
(
t
,
x
,
y
)
=
V
1
(
t
,
x
,
y
)
+
V
2
(
α
)
(
t
,
x
,
y
)
.
From inequalities (45) and (46) it follows that
(48)
max
⁡
s
∈
[
-
r
,
0
]
V
1
(
t
0
*
+
s
,
x
1
(
t
0
*
+
s
)
,
y
2
(
t
0
*
+
s
)
)
<
α
1
.
Lett
∈
[
t
0
*
,
t
*
] and ψ
1
,
ψ
2
∈
C
(
[
-
r
,
0
]
,
R
n
) be such that
(49)
(
t
,
ψ
1
(
0
)
,
ψ
2
(
0
)
)
∈
S
(
h
,
β
)
∩
S
c
(
h
0
,
α
)
,
(
ψ
1
(
0
)
-
ψ
2
(
0
)
,
max
⁡
s
∈
[
-
r
,
0
]
(
ψ
1
(
s
)
-
ψ
2
(
s
)
)
)
∈
Ω
(
t
,
h
,
β
)
,
and V
(
t
,
ψ
1
(
0
)
,
ψ
2
(
0
)
)
>
V
(
t
+
s
,
ψ
1
(
s
)
,
ψ
2
(
s
)
), s
∈
[
-
r
,
0
).
Using Lipschitz conditions forV
1, V
2
(
α
) and condition (
iii
) of Theorem 6, we obtain
(50)
D
(
4
)
V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
=
D
(
4
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
D
(
4
)
V
2
(
α
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
g
(
t
,
V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
+
(
M
1
+
M
2
)
×
sup
⁡
(
x
-
y
,
x
1
-
y
2
)
∈
Ω
(
t
,
h
,
β
)
∥
G
1
(
t
,
x
,
x
1
)
-
G
2
(
t
,
y
,
y
2
)
∥
=
g
(
t
,
V
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
+
q
(
t
)
.
Consider differential equation (7) where the perturbation on the right-hand side is given by
(51)
q
(
t
)
=
(
M
1
+
M
2
)
×
sup
⁡
(
x
-
y
,
x
1
-
y
2
)
∈
Ω
(
t
,
h
,
β
)
∥
G
1
(
t
,
x
,
x
1
)
-
G
2
(
t
,
y
,
y
2
)
∥
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
t
∈
[
t
0
*
,
t
*
]
.
Letr
*
(
t
) be the maximal solution of (7) with the initial condition r
*
(
t
0
*
)
=
ω
0
*, where ω
0
*
=
max
⁡
s
∈
[
-
r
,
0
]
⁡
V
(
t
0
*
+
s
,
x
1
(
t
0
*
+
s
)
,
y
2
(
t
0
*
+
s
)
). According to Lemma 5, the inequality
(52)
V
(
t
,
x
1
(
t
)
,
y
2
(
t
)
)
≤
r
*
(
t
)
,
t
∈
Ξ
∩
[
t
0
*
,
t
*
]
holds, where Ξ
⊆
[
t
0
*
,
∞
) is the interval of existence of r
*
(
t
).
Choose a pointT
*
>
t
* such that
(53)
∫
t
0
*
t
*
‍
q
(
s
)
d
s
+
1
2
(
T
*
-
t
*
)
q
(
t
*
)
<
α
1
.
Now define the continuous functionq
*
(
t
)
:
[
t
0
*
,
∞
)
→
R by
(54)
q
*
(
t
)
=
{
q
(
t
)
,
t
∈
[
t
0
*
,
t
*
]
,
q
(
t
*
)
t
*
-
T
*
,
t
∈
[
t
*
,
T
*
]
,
0
,
t
≥
T
*
.
From the choice of the perturbationG
1
(
t
,
x
,
y
), G
2
(
t
,
x
,
y
) it follows that for every T
>
0 the inequality
(55)
∫
t
0
*
t
0
*
+
T
‍
q
*
(
s
)
d
s
≤
(
M
1
+
M
2
)
,
∫
t
0
t
0
+
T
‍
sup
⁡
(
x
-
y
,
x
1
-
y
2
)
∈
Ω
(
s
,
h
,
β
)
∥
G
1
(
s
,
x
,
x
1
)
-
G
2
(
s
,
y
,
y
2
)
∥
d
s
<
α
1
holds.
Letr
1
(
t
) be the maximal solution of (7) with the initial condition r
1
(
t
0
*
)
=
ω
0
*, where the perturbation of the right-hand side is defined above function q
*
(
t
). Note that r
1
(
t
)
=
r
*
(
t
,
t
0
*
,
ω
0
*
), t
∈
[
t
0
*
,
t
*
).
From inequality (48) it follows that |
ω
0
*
|
<
α
1 and therefore inequality (31) holds; that is,
(56)
r
1
(
t
)
<
β
1
,
t
≥
t
0
*
.
From inequalities (52) and (56), the choice of the point t
*, and condition (
iii
), we obtain
(57)
b
(
β
)
≥
β
1
>
r
1
(
t
*
)
=
r
*
(
t
*
)
≥
V
(
t
*
,
x
1
(
t
*
)
,
y
2
(
t
*
)
)
≥
V
2
(
α
)
(
t
*
,
x
1
(
t
*
)
,
y
2
(
t
*
)
)
≥
b
(
h
(
t
*
,
x
1
(
t
*
)
-
y
2
(
t
*
)
)
)
=
b
(
β
)
.
The obtained contradiction proves the validity of inequality (37) for t
≥
t
0.
Inequality (37) proves uniform-relatively integral stability in terms of measures (
h
0
,
h
) of the considered differential systems with “maxima.”The following example is an application of Theorem6.Example 1.
Consider the two differential systems with “maxima”(58)
x
′
=
-
x
-
x
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
,
t
≥
t
0
,
x
(
t
+
t
0
)
=
ϕ
1
(
t
)
,
t
∈
[
-
r
,
0
]
,
y
′
=
-
y
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
,
t
≥
t
0
y
(
t
+
t
0
)
=
ϕ
2
(
t
)
,
t
∈
[
-
r
,
0
]
and the perturbed systems
(59)
x
′
=
-
x
-
x
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
x
(
s
)
e
-
t
,
t
≥
t
0
,
x
(
t
+
t
0
)
=
ϕ
1
(
t
)
,
t
∈
[
-
r
,
0
]
,
y
′
=
-
y
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
y
(
s
)
e
-
t
,
t
≥
t
0
,
y
(
t
+
t
0
)
=
ϕ
2
(
t
)
,
t
∈
[
-
r
,
0
]
.
LetV
1
(
t
,
x
,
y
)
=
x
2
+
y
2, V
2
(
μ
)
(
t
,
x
,
y
)
=
x
2
+
2
y
2, and h
0
(
t
,
x
-
y
)
=
2
(
∥
x
∥
+
∥
y
∥
), h
(
t
,
x
-
y
)
=
2
(
x
2
+
y
2
). Using the inequality x
2
+
y
2
≤
∥
x
∥
+
∥
y
∥, it is easy to check the validity of the conditions (
A
1
) and (ii) of Theorem 6 for a
(
u
)
=
u
2, b
(
u
)
=
(
1
/
2
)
u
2.
Lettingψ
1
,
ψ
2
∈
C
(
[
t
-
r
,
t
]
,
R
n
) be such that ψ
1
2
(
t
)
>
ψ
1
2
(
t
+
s
), ψ
2
2
(
t
)
>
ψ
2
2
(
t
+
s
), s
∈
[
-
r
,
0
), t
≥
t
0, then V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
>
V
1
(
t
+
s
,
ψ
1
(
t
+
s
)
,
ψ
2
(
t
+
s
)
) and
(60)
D
(
59
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
=
2
ψ
1
(
t
)
ψ
1
(
t
)
′
+
2
ψ
2
(
t
)
ψ
2
(
t
)
′
=
2
ψ
1
(
t
)
{
-
ψ
1
(
t
)
-
ψ
1
(
t
)
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
1
(
s
)
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
1
(
s
)
e
-
t
}
+
2
ψ
2
(
t
)
{
-
ψ
2
(
t
)
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
2
(
s
)
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
2
(
s
)
e
-
t
}
≤
(
-
2
e
t
-
t
0
+
1
+
e
-
t
)
(
ψ
1
2
(
t
)
+
ψ
2
2
(
t
)
)
≤
e
-
t
(
ψ
1
2
(
t
)
+
ψ
2
2
(
t
)
)
.
Lettingf
(
t
,
u
)
=
e
-
t
u, then D
(
57
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
f
(
t
,
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
).
Considering the comparison scalar differential system(61)
u
′
=
e
-
t
u
,
u
0
=
u
(
t
0
)
,
the solution is u
=
u
0
e
e
-
t
0
e
-
e
-
t, t
≥
t
0, and we can prove that the solution is equistable; that is, the conditions (
A
2
) and (
A
4
) of Theorem 6 hold.
Fors
∈
[
-
r
,
0
), t
≥
t
0, the inequality
(62)
D
(
58
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
D
(
58
)
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
=
4
ψ
1
(
t
)
ψ
1
(
t
)
′
+
6
ψ
2
(
t
)
ψ
2
(
t
)
′
=
4
ψ
1
(
t
)
{
-
ψ
1
(
t
)
-
ψ
1
(
t
)
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
1
(
s
)
}
+
6
ψ
2
(
t
)
{
-
ψ
2
(
t
)
e
t
-
t
0
+
1
2
max
⁡
s
∈
[
t
-
r
,
t
]
ψ
2
(
s
)
}
≤
(
-
2
e
t
-
t
0
+
1
)
(
2
ψ
1
2
(
t
)
+
3
ψ
2
2
(
t
)
)
≤
0
holds. From (62), the inequality
(63)
D
(
58
)
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
D
(
58
)
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
≤
g
(
t
,
V
1
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
+
V
2
(
μ
)
(
t
,
ψ
1
(
t
)
,
ψ
2
(
t
)
)
)
holds, where u
′
=
g
(
t
,
u
)
≡
0 and its perturbed differential equation ω
′
=
0
+
ω
0
e
-
t. We can prove that the differential equation g
(
t
,
u
)
=
0 is uniform-integrally stable. So the conditions (
A
3
) and (
A
5
) of Theorem 6 hold.
According to Theorem6, differential systems with “maxima” (58) are uniform-relatively integrally stable in terms of two measures (
h
0
,
h
).



---

*Source: 656290-2014-08-13.xml*