# Existence of Almost Periodic Solutions for Impulsive Neutral Functional Differential Equations

Authors: Junwei Liu; Chuanyi Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The existence of piecewise almost periodic solutions for impulsive neutral
functional differential equations in Banach space is investigated. Our results are based on Krasnoselskii’s
fixed-point theorem combined with an exponentially stable strongly continuous operator semigroup. An
example is given to illustrate the theory.


---

## Body

## 1. Introduction

In this paper, we study the existence of piecewise almost periodic solutions for a class of abstract impulsive neutral functional differential equations with unbounded delay modeled in the form(1)
d
d
t
(
u
(
t
)
+
g
(
t
,
u
t
)
)
=
A
u
(
t
)
+
f
(
t
,
u
t
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
t
∈
R
,
t
≠
t
i
,
i
∈
Z
,
Δ
u
(
t
i
)
=
I
i
(
u
t
i
)
,
i
∈
Z
,
where A is the infinitesimal generator of an exponentially stable strongly continuous semigroup of linear operators {
T
(
t
)
}
t
≥
0 on a Banach space (
X
,
∥
·
∥
), the history u
t
:
(
-
∞
,
0
]
→
X, u
t
(
θ
)
=
u
(
t
+
θ
) belongs to an abstract phase space B defined axiomatically, f
(
·
)
,
g
(
·
)
,
I
i
(
·
)
(
i
∈
Z
) are appropriate functions, {
t
i
}
i
∈
Z is a discrete set of real numbers such that t
i
<
t
j when i
<
j, and the symbol Δ
ξ
(
t
) represents the jump of the function ξ
(
·
) at t, which is defined by Δ
ξ
(
t
)
=
ξ
(
t
+
)
-
ξ
(
t
-
).The existence of solutions to impulsive differential equations is one of the most attracting topics in the qualitative theory of impulsive differential equations due to their applications in mechanical, electrical engineering, ecology, biology, and others; see, for instance, [1–6] and the references therein. Some recent contributions on mild solutions to impulsive neutral functional differential equations have been established in [7–14]. On the other hand, the existence of almost periodic solutions for impulsive differential equations has been investigated by many authors; see, for example, [2–5, 15, 16]. However, the existence of almost periodic solutions for the impulsive neutral functional differential equations in the form (1) is an untreated topic in the literature and this fact is the motivation of the present work.The paper is organized as follows: in Section2, we recall some notations, concepts, and useful lemmas which are used in this paper. In Section 3, some criteria ensuring the existence of almost periodic solutions for impulsive neutral functional differential equations are obtained. In Section 4, we give an application.


## 2. Preliminaries

Let(
X
,
∥
·
∥
) be a Banach space. A
:
D
(
A
)
→
X is the infinitesimal generator of a strongly continuous semigroup of linear operators {
T
(
t
)
}
t
≥
0 on the Banach space X and M
1, δ are positive constants such that ∥
T
(
t
)
∥
≤
M
1
e
-
δ
t for t
≥
0. Let 0
∈
ρ
(
A
); it is possible to define the fractional power A
α, 0
<
α
<
1, as a closed linear operator with its domain D
(
A
α
). We denote by X
α a Banach space between D
(
A
) and X endowed with the norm ∥
x
∥
α
=
∥
A
α
x
∥, x
∈
D
(
A
α
); the following properties hold.Lemma 1 (see [17, 18]).
Let0
<
α
<
β
<
1; then X
β is continuously embedded into X
α with bounded K; that is,
(2)
∥
x
∥
α
≤
K
∥
x
∥
β
.
Moreover, the function s
→
A
α
T
(
s
) is continuous in the uniform operator topology on (
0
,
∞
) and there exists M
2
>
0 such that ∥
A
α
T
(
t
)
∥
≤
M
2
e
-
δ
t
t
-
α for every t
>
0.Throughout this paper, letT be the set consisting of all real sequences {
t
i
}
i
∈
Z such that γ
=
inf
⁡
i
∈
Z
(
t
i
+
1
-
t
i
)
>
0. For {
t
i
}
i
∈
Z
∈
T, let P
C
(
R
,
X
α
) be the space formed by all piecewise continuous functions ϕ
:
R
→
X
α such that ϕ
(
·
) is continuous at t for any t
∉
{
t
i
}
i
∈
Z and ϕ
(
t
i
)
=
ϕ
(
t
i
-
) for all i
∈
Z; let P
C
(
R
×
X
α
,
X
α
) be the space formed by all piecewise continuous functions ϕ
:
R
×
X
α
→
X
α such that, for any x
∈
X
α, ϕ
(
·
,
x
) is continuous at t for any t
∉
{
t
i
}
i
∈
Z and ϕ
(
t
i
,
x
)
=
ϕ
(
t
i
-
,
x
) for all i
∈
Z and for any t
∈
R, ϕ
(
t
,
·
) is continuous at x
∈
X
α.Definition 2.
A numberτ
∈
R is called an ϵ-translation number of the function ϕ
∈
P
C
(
R
,
X
α
) if
(3)
∥
ϕ
(
t
+
τ
)
-
ϕ
(
t
)
∥
α
<
ϵ
,
for all t
∈
R which satisfies |
t
-
t
i
|
>
ϵ, for all i
∈
Z. Denote by T
(
ϕ
,
ϵ
) the set of all ϵ-translation numbers of ϕ.Definition 3 (see [1]).
A functionϕ
∈
P
C
(
R
,
X
α
) is said to be piecewise almost periodic if the following conditions are fulfilled.(1)
{
t
i
j
=
t
i
+
j
-
t
i
}, i
∈
Z, j
=
0
,
±
1
,
±
2
,
…, are equipotentially almost periodic; that is, for any ϵ
>
0, there exists a relatively dense set of ϵ
-almost periods that are common to all the sequences {
t
i
j
}.
(2)
For anyϵ
>
0, there exists a positive number δ
=
δ
(
ϵ
) such that if the points t
′ and t
′′ belong to a same interval of continuity of ϕ and |
t
′
-
t
′′
|
<
δ, then ∥
ϕ
(
t
′
)
-
ϕ
(
t
′′
)
∥
α
<
ϵ.
(3)
For everyϵ
>
0,  T
(
ϕ
,
ϵ
) is a relatively dense set in R.We denote byA
P
T
(
R
,
X
α
) the space of all piecewise almost periodic functions. A
P
T
(
R
,
X
α
) endowed with the uniform convergence topology is a Banach space.Definition 4.
f
∈
P
C
(
R
×
X
α
,
X
α
) is said to be piecewise almost periodic in t uniformly in x
∈
X
α if for each compact set K
⊆
X
α, {
f
(
·
,
x
)
:
x
∈
K
} is uniformly bounded and, given ϵ
>
0, there exists a relatively dense set Ω
(
ϵ
) such that
(4)
∥
f
(
t
+
τ
,
x
)
-
f
(
t
,
x
)
∥
α
<
ϵ
,
for all x
∈
K, τ
∈
Ω
(
ϵ
), and t
∈
R, |
t
-
t
i
|
>
ϵ for all i
∈
Z. Denote by A
P
T
(
R
×
X
α
,
X
α
) the set of all such functions.Lemma 5 (see [15]).
Letϕ
∈
A
P
T
(
R
,
X
α
); then the range of ϕ, R
(
ϕ
), is a relatively compact subset of X
α.Lemma 6.
Suppose thatf
(
t
,
x
)
∈
A
P
T
(
R
×
X
α
,
X
α
) and f
(
t
,
·
) is uniformly continuous on each compact subset K
⊆
X
α uniformly for t
∈
R; that is, for every ϵ
>
0, there exists δ
>
0 such that x
,
y
∈
K and ∥
x
-
y
∥
α
<
δ implies that ∥
f
(
t
,
x
)
-
f
(
t
,
y
)
∥
α
<
ϵ for all t
∈
R. Then f
(
·
,
x
(
·
)
)
∈
A
P
T
(
R
,
X
α
) for any x
∈
A
P
T
(
R
,
X
α
).Proof.
Sincex
∈
A
P
(
R
,
X
α
), by Lemma 5, R
(
x
) is a relatively compact subset of X
α. Because f
(
t
,
·
) is uniformly continuous on each compact subset K
⊆
X
α uniformly for t
∈
R, then for any ϵ
>
0, there exist a number δ
:
0
<
δ
≤
ϵ
/
2, such that
(5)
∥
f
(
t
,
x
1
)
-
f
(
t
,
x
2
)
∥
α
<
ϵ
2
,
where x
1, x
2
∈
R
(
x
), and ∥
x
1
-
x
2
∥
α
<
δ, t
∈
R. By piecewise almost periodic of f and x, there exists a relatively dense set Ω of R such that the following conditions hold:
(6)
∥
f
(
t
+
τ
,
x
0
)
-
f
(
t
,
x
0
)
∥
α
<
ϵ
2
,
∥
x
(
t
+
τ
)
-
x
(
t
)
∥
α
<
ϵ
2
,
for x
0
∈
R
(
x
) and t
∈
R, |
t
-
t
i
|
>
ϵ, i
∈
Z, τ
∈
Ω. Note that
(7)
f
(
t
+
τ
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
)
)
=
f
(
t
+
τ
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
+
τ
)
)
+
f
(
t
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
)
)
.
We have
(8)
∥
f
(
t
+
τ
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
)
)
∥
α
≤
∥
f
(
t
+
τ
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
+
τ
)
)
∥
α
+
∥
f
(
t
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
)
)
∥
α
.
We deduce from (5) and (6) that the following formula holds:
(9)
∥
f
(
t
+
τ
,
x
(
t
+
τ
)
)
-
f
(
t
,
x
(
t
)
)
∥
α
≤
ϵ
,
h
t
∈
R
,
|
t
-
t
i
|
>
ϵ
,
i
∈
Z
,
τ
∈
Ω
.
That is, f
(
·
,
x
(
·
)
) is piecewise almost periodic. The proof is complete.Since the uniform continuity is weaker than the Lipschitz continuity, we obtain the following lemma as an immediate consequence of the previous lemma.Lemma 7.
Letf
(
t
,
x
)
∈
A
P
T
(
R
×
X
α
,
X
α
) and f is Lipschitz; that is, there is a positive number L such that
(10)
∥
f
(
t
,
x
)
-
f
(
t
,
y
)
∥
α
≤
L
∥
x
-
y
∥
α
,
for all t
∈
R and x
,
y
∈
X
α; if for any x
∈
A
P
T
(
R
,
X
α
), then f
(
·
,
x
(
·
)
)
∈
A
P
T
(
R
,
X
α
).In this paper, we assume that the phase spaceB is a linear space formed by functions mapping (
-
∞
,
0
] into X
α endowed with a norm ∥
·
∥
B and satisfying the following conditions.(1)
Ifx
∈
P
C
(
R
,
X
α
), then x
t
∈
B for all t
∈
R and ∥
x
t
∥
B
≤
L
sup
⁡
s
≤
t
∥
x
(
s
)
∥
α, where L
>
0 is a constant independent of x
(
·
) and t
∈
R.
(2)
The spaceB is complete.
(3)
If{
ϕ
n
}
n
∈
N
⊂
B is a uniformly bounded sequence in P
C
(
(
-
∞
,
0
]
,
X
α
) formed by functions with compact support and ϕ
n
→
ϕ in the compact open topology, then ϕ
∈
B and ∥
ϕ
n
-
ϕ
∥
B
→
0, as n
→
∞.Lemma 8 (see [1, 15]).
Assume thatf
∈
A
P
T
(
R
,
X
α
), the sequence {
x
i
:
i
∈
Z
} is almost periodic in X
α, and {
t
i
j
=
t
i
+
j
-
t
i
}, i
∈
Z, j
=
0
,
±
1
,
±
2
,
…, are equipotentially almost periodic. Then for each ϵ
>
0, there are relatively dense sets Ω
ϵ
,
f
,
x
i of R and Q
ϵ
,
f
,
x
i of Z such that the following conditions hold.(i)
∥
f
(
t
+
τ
)
-
f
(
t
)
∥
α
<
ϵ for all t
∈
R, |
t
-
t
i
|
>
ϵ, τ
∈
Ω
ϵ
,
f
,
x
i, and i
∈
Z.
(ii)
∥
x
i
+
q
-
x
i
∥
α
<
ϵ for all q
∈
Q
ϵ
,
f
,
x
i and i
∈
Z.
(iii)
For everyτ
∈
Ω
ϵ
,
f
,
x
i, there exists at least one number q
∈
Q
ϵ
,
f
,
x
i such that
(11)
|
t
i
q
-
τ
|
<
ϵ
,
i
∈
Z
.Definition 9.
A bounded functionu
∈
P
C
(
R
,
X
α
) is a mild solution of (1) if the following holds: u
σ
=
ϕ
∈
B, the function A
T
(
t
-
s
)
g
(
s
,
u
s
) is integrable, and for any t
∈
R, t
i
<
t
≤
t
i
+
1,
(12)
u
(
t
)
=
T
(
t
-
σ
)
[
ϕ
(
0
)
+
g
(
σ
,
ϕ
)
]
-
g
(
t
,
u
t
)
-
∫
σ
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
σ
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
σ
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
.Since∥
T
(
t
-
σ
)
∥
≤
M
1
e
-
δ
(
t
-
σ
) for all t
≥
σ, let σ
→
-
∞; then we have ∥
T
(
t
-
σ
)
∥
→
0, and the above formula can be replaced by
(13)
u
(
t
)
=
-
g
(
t
,
u
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
.In fact, fort
>
σ,
(14)
-
∫
σ
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
σ
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
σ
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
=
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
-
∞
σ
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
-
∫
-
∞
σ
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
-
∞
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
-
∑
-
∞
<
t
i
<
σ
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
=
-
g
(
t
,
u
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
u
(
s
)
)
d
s
+
∑
-
∞
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
(
t
i
)
)
+
g
(
t
,
u
t
)
-
T
(
t
-
σ
)
×
[
∑
-
∞
<
t
i
<
σ
‍
-
g
(
σ
,
u
σ
)
-
∫
-
∞
σ
‍
A
T
(
σ
-
s
)
g
(
s
,
u
s
)
d
s
hhhhhhhh
+
∫
-
∞
σ
‍
T
(
σ
-
s
)
f
(
s
,
u
(
s
)
)
d
s
hhhhhhhh
+
∑
-
∞
<
t
i
<
σ
‍
T
(
σ
-
t
i
)
I
i
(
u
(
t
i
)
)
+
g
(
σ
,
u
σ
)
]
=
u
(
t
)
+
g
(
t
,
u
t
)
-
T
(
t
-
σ
)
[
u
(
σ
)
+
g
(
σ
,
u
σ
)
]
=
u
(
t
)
+
g
(
t
,
u
t
)
-
T
(
t
-
σ
)
[
u
σ
(
0
)
+
g
(
σ
,
u
σ
)
]
=
u
(
t
)
+
g
(
t
,
u
t
)
-
T
(
t
-
σ
)
[
ϕ
(
0
)
+
g
(
σ
,
ϕ
)
]
,
so that
(15)
u
(
t
)
=
T
(
t
-
σ
)
[
ϕ
(
0
)
+
g
(
σ
,
ϕ
)
]
-
g
(
t
,
u
t
)
-
∫
σ
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
σ
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
σ
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
.In order to obtain our results, we need to introduce some additional notations. Leth
:
R
→
R be a continuous function such that h
(
t
)
≥
1 for all t
∈
R and h
(
t
)
→
∞ as |
t
|
→
∞. We consider the space
(16)
(
P
C
)
h
0
(
R
,
X
α
)
=
{
u
∈
P
C
(
R
,
X
α
)
:
lim
⁡
|
t
|
→
∞
∥
u
(
t
)
∥
α
h
(
t
)
=
0
}
.
Endowed with the norm ∥
u
∥
h
=
sup
⁡
t
∈
R
(
∥
u
(
t
)
∥
α
/
h
(
t
)
), it is a Banach space.We recall here the following compactness criterion in these spaces, which we can refer to [7, 19–23].Lemma 10.
A setB
⊆
(
P
C
)
h
0
(
R
,
X
α
) is a relatively compact set if and only if(1)
lim
⁡
|
t
|
→
∞
(
∥
x
(
t
)
∥
α
/
h
(
t
)
)
=
0 uniformly for x
∈
B;
(2)
B
(
t
)
=
{
x
(
t
)
:
x
∈
B
} is relatively compact in X
α for every t
∈
R;
(3)
the setB is equicontinuous on each interval (
t
i
,
t
i
+
1
)
(
i
∈
Z
).Theorem 11 (Krasnoselskii’s fixed-point theorem [24]).
LetM be a closed convex nonempty subset of a Banach space X; suppose that A and B map M into X such that(i)
A
x
+
B
y
∈
M
(
∀
x
,
y
∈
M
),
(ii)
A is completely continuous,
(iii)
B is a contraction with constant L
<
1.
Then there is ay
∈
M with A
y
+
B
y
=
y.


## 3. Main Results

In this section, we discuss the existence of piecewise almost periodic solutions for impulsive neutral functional differential equation (1). To begin, let us list the following hypotheses.(A1)
The operatorA is the infinitesimal generator of an exponentially stable strongly continuous semigroup of linear operators {
T
(
t
)
}
t
≥
0; that is, there exist constants M
1
>
0, δ
>
0 such that ∥
T
(
t
)
∥
≤
M
1
e
-
δ
t for t
≥
0. Moreover, T
(
t
) is compact for  t
>
0.
(A2)
f
(
t
,
x
)
∈
A
P
T
(
R
×
B
,
X
α
) is uniformly continuous in x
∈
B uniformly in t
∈
R; I
i
(
x
) is almost periodic in i
∈
Z uniformly in x
∈
B and is a uniformly continuous function in x for all i
∈
Z. For every l
>
0, C
1
l
=
sup
⁡
t
∈
R
,
∥
x
∥
B
≤
l
L
∥
f
(
t
,
x
)
∥
α
<
∞, C
2
l
=
sup
⁡
i
∈
Z
,
∥
x
∥
B
≤
l
L
∥
I
i
(
x
)
∥
α
<
∞. Moreover, there exist a number L
0
>
0, such that M
1
(
C
1
/
δ
+
C
2
/
(
1
-
e
-
δ
γ
)
)
≤
L
0
/
2, where C
1
=
C
1
L
0, C
2
=
C
2
L
0.
(A3)
g
∈
A
P
T
(
R
×
B
,
X
β
), g
(
t
,
0
)
=
0, and there exist a number L
1
>
0 such that
(17)
∥
g
(
t
,
ϕ
1
)
-
g
(
t
,
ϕ
2
)
∥
β
≤
L
1
∥
ϕ
1
-
ϕ
2
∥
B
for allt
∈
R, ϕ
1, ϕ
2
∈
B.
(A4)
Let{
x
n
}
⊆
A
P
T
(
R
,
X
α
) be uniformly bounded in R and uniformly convergent in each compact set of R; then {
f
(
·
,
x
n
(
·
)
)
} is relatively compact in P
C
(
R
,
X
α
).Theorem 12.
Suppose that conditions (A1)–(A4) hold; then (1) has a piecewise almost periodic solution provided that K
L
L
1
+
(
M
2
π
/
δ
β
-
α
sin
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
)
L
1
L
≤
1
/
2.Proof.
Let(18)
B
=
{
u
∈
A
P
T
(
R
,
X
α
)
:
∥
u
∥
α
≤
L
0
}
.
Note that B is a closed convex set of A
P
T
(
R
,
X
α
). By (A3) and Lemma 1, we have
(19)
∥
A
T
(
t
-
s
)
g
(
s
,
u
s
)
∥
α
≤
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
∥
g
(
s
,
u
s
)
∥
β
≤
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
∥
g
(
s
,
u
s
)
-
g
(
s
,
0
)
∥
β
≤
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
L
1
∥
u
s
∥
B
≤
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
L
1
L
sup
⁡
r
≤
s
∥
u
(
r
)
∥
α
≤
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
L
1
L
L
0
,
and we infer that s
→
A
T
(
t
-
s
)
g
(
s
,
u
s
) is integrable on (
-
∞
,
t
].
Define the operatorΥ on (
P
C
)
h
0
(
R
,
X
α
) by
(20)
Υ
u
(
t
)
=
-
g
(
t
,
u
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
+
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
.
In order to prove that Υ has a fixed point in A
P
T
(
R
,
X
α
), we introduce the decomposition Υ
=
Υ
1
+
Υ
2, where
(21)
Υ
1
u
(
t
)
=
-
g
(
t
,
u
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
u
s
)
d
s
,
Υ
2
u
(
t
)
=
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
.
Our proof will be split into the following three steps.
Step  1. We claim that Υ
1
x
+
Υ
2
y
∈
B
(
for
all
x
,
y
∈
B
).
For anyx
,
y
∈
B, by (A3) and Lemma 1, we have
(22)
∥
Υ
1
x
(
t
)
∥
α
=
∥
-
g
(
t
,
x
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
x
s
)
d
s
∥
α
≤
∥
g
(
t
,
x
t
)
∥
α
+
∫
-
∞
t
‍
∥
A
T
(
t
-
s
)
g
(
s
,
x
s
)
∥
α
d
s
≤
K
∥
g
(
t
,
x
t
)
∥
β
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
∥
g
(
s
,
x
s
)
∥
β
d
s
=
K
∥
g
(
t
,
x
t
)
-
g
(
t
,
0
)
∥
β
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
h
h
h
h
×
∥
g
(
s
,
x
s
)
-
g
(
s
,
0
)
∥
β
d
s
≤
K
L
1
∥
x
t
∥
B
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
L
1
∥
x
s
∥
B
d
s
≤
K
L
1
L
sup
⁡
r
≤
t
∥
x
(
r
)
∥
α
+
∫
-
∞
t
‍
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
h
h
h
h
h
h
h
h
h
h
×
L
1
L
sup
⁡
r
≤
s
∥
x
(
r
)
∥
α
d
s
≤
K
L
1
L
∥
x
∥
α
+
∫
-
∞
t
‍
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
×
L
1
L
∥
x
∥
α
d
s
≤
K
L
1
L
∥
x
∥
α
+
M
2
π
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
×
L
1
L
∥
x
∥
α
,
∥
Υ
2
y
(
t
)
∥
α
=
∥
∑
t
i
<
t
‍
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
y
s
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
y
t
i
)
∥
α
≤
∫
-
∞
t
‍
∥
T
(
t
-
s
)
∥
∥
f
(
s
,
y
s
)
∥
α
d
s
+
∑
t
i
<
t
‍
∥
T
(
t
-
t
i
)
∥
∥
I
i
(
y
t
i
)
∥
α
≤
∫
-
∞
t
‍
M
1
e
-
δ
(
t
-
s
)
C
1
d
s
+
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
C
2
≤
M
1
δ
C
1
+
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
C
2
.
In order to estimate the last part of the second term on the right hand side of the above formula, we assume that, for every t
∈
R, there exists j
∈
Z, such that t
j
≤
t
<
t
j
+
1,
(23)
t
-
t
i
=
(
t
-
t
j
)
+
(
t
j
-
t
i
)
≥
(
j
-
i
)
γ
,
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
C
2
≤
∑
-
∞
<
i
≤
j
‍
M
1
e
-
δ
(
j
-
i
)
γ
C
2
=
∑
0
≤
k
=
j
-
i
<
∞
‍
M
1
e
-
δ
k
γ
C
2
=
M
1
1
-
e
-
δ
γ
C
2
,
so
(24)
∥
Υ
2
y
(
t
)
∥
α
≤
M
1
δ
C
1
+
M
1
1
-
e
-
δ
γ
C
2
.
Then,
(25)
∥
Υ
1
x
(
t
)
+
Υ
2
y
(
t
)
∥
α
≤
∥
Υ
1
x
(
t
)
∥
α
+
∥
Υ
2
y
(
t
)
∥
α
≤
K
L
1
L
∥
x
∥
α
+
M
2
π
δ
β
-
α
sin
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
L
1
L
∥
x
∥
α
+
M
1
δ
C
1
+
M
1
1
-
e
-
δ
γ
C
2
,
from K
L
1
L
+
(
M
2
π
/
δ
β
-
α
sin
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
)
L
1
L
≤
1
/
2, and we obtain that
(26)
∥
Υ
1
x
(
t
)
+
Υ
2
y
(
t
)
∥
α
≤
L
0
,
x
,
y
∈
B
.
By (A2) and Lemma6, f
(
t
,
y
t
)
∈
A
P
T
(
R
,
X
α
), and, by (A3) and Lemma 7, g
(
t
,
x
t
)
∈
A
P
T
(
R
,
X
β
). By (A2) and [1, Lemma 37], {
I
i
(
y
t
i
)
} is almost periodic. From [1, Theorem 73], for the two almost periodic functions f
(
t
,
y
t
), g
(
t
,
x
t
), there exists a relatively dense set of their common ϵ-translation numbers. Then by Lemma 8, for every ϵ
>
0, there exist relatively dense sets Ω
ϵ
,
f
,
g
,
I
i of R and Q
ϵ
,
f
,
g
,
I
i of Z such that for τ
∈
Ω
ϵ
,
f
,
g
,
I
i
,
∃
q
∈
Q
ϵ
,
f
,
g
,
I
i,
(27)
∥
g
(
t
+
τ
,
x
t
+
τ
)
-
g
(
t
,
x
t
)
∥
β
≤
ϵ
,
∥
f
(
t
+
τ
,
y
t
+
τ
)
-
f
(
t
,
y
t
)
∥
α
≤
ϵ
,
Hi
∥
I
i
+
q
(
y
t
i
+
q
)
-
I
i
(
y
t
i
)
∥
α
≤
ϵ
,
|
t
i
q
-
τ
|
≤
ϵ
,
where t
∈
R, |
t
-
t
i
|
>
ϵ, i
∈
Z. So for τ
∈
Ω
ϵ
,
f
,
g
,
I
i, we know that
(28)
Υ
1
x
(
t
+
τ
)
-
Υ
1
x
(
t
)
=
-
g
(
t
+
τ
,
x
t
+
τ
)
-
∫
-
∞
t
+
τ
‍
A
T
(
t
+
τ
-
s
)
g
(
s
,
x
s
)
d
s
+
g
(
t
,
x
t
)
+
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
x
s
)
d
s
=
-
g
(
t
+
τ
,
x
t
+
τ
)
+
g
(
t
,
x
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
[
g
(
s
+
τ
,
x
s
+
τ
)
-
g
(
s
,
x
s
)
]
d
s
,
so
(29)
∥
Υ
1
x
(
t
+
τ
)
-
Υ
1
x
(
t
)
∥
α
≤
∥
g
(
t
+
τ
,
x
t
+
τ
)
-
g
(
t
,
x
t
)
∥
α
+
∫
-
∞
t
‍
∥
A
T
(
t
-
s
)
[
g
(
s
+
τ
,
x
s
+
τ
)
-
g
(
s
,
x
s
)
]
∥
α
d
s
≤
K
∥
g
(
t
+
τ
,
x
t
+
τ
)
-
g
(
t
,
x
t
)
∥
β
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
h
h
h
h
h
h
h
h
h
×
∥
g
(
s
+
τ
,
x
s
+
τ
)
-
g
(
s
,
x
s
)
∥
β
d
s
≤
K
ϵ
+
∫
-
∞
t
‍
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
ϵ
d
s
≤
ϵ
(
K
+
M
2
π
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
)
.
That is,
(30)
∥
Υ
1
x
(
t
+
τ
)
-
Υ
1
x
(
t
)
∥
α
≤
ϵ
(
Γ
(
1
+
α
-
β
)
)
-
1
K
+
(
M
2
π
)
×
(
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
×
Γ
(
1
+
α
-
β
(
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
)
)
-
1
)
,
(31)
Υ
2
y
(
t
+
τ
)
-
Υ
2
y
(
t
)
=
∫
-
∞
t
+
τ
‍
T
(
t
+
τ
-
s
)
f
(
s
,
y
s
)
d
s
+
∑
t
i
<
t
+
τ
‍
T
(
t
+
τ
-
t
i
)
I
i
(
y
t
i
)
-
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
y
s
)
d
s
-
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
y
t
i
)
=
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
+
τ
,
y
s
+
τ
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
+
q
(
y
t
i
+
q
)
-
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
y
s
)
d
s
-
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
y
t
i
)
=
∫
-
∞
t
‍
T
(
t
-
s
)
[
f
(
s
+
τ
,
y
s
+
τ
)
HHHHHhhhhhhhHHH
-
f
(
s
,
y
s
)
]
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
[
I
i
+
q
(
y
t
i
+
q
)
-
I
i
(
y
t
i
)
]
,
so we have
(32)
∥
∫
-
∞
t
‍
T
(
t
-
s
)
[
f
(
s
+
τ
,
y
s
+
τ
)
-
f
(
s
,
y
s
)
]
d
s
∥
α
≤
∫
-
∞
t
‍
∥
T
(
t
-
s
)
∥
∥
f
(
s
+
τ
,
y
s
+
τ
)
-
f
(
s
,
y
s
)
∥
α
d
s
≤
∫
-
∞
t
‍
M
1
e
-
δ
(
t
-
s
)
ϵ
d
s
≤
M
1
δ
ϵ
,
∥
∑
t
i
<
t
‍
T
(
t
-
t
i
)
[
I
i
+
q
(
y
t
i
+
q
)
-
I
i
(
y
t
i
)
]
∥
α
≤
∑
t
i
<
t
‍
∥
T
(
t
-
t
i
)
∥
∥
I
i
+
q
(
y
t
i
+
q
)
-
I
i
(
y
t
i
)
∥
α
≤
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
ϵ
≤
M
1
1
-
e
-
δ
γ
ϵ
,
so
(33)
∥
Υ
2
y
(
t
+
τ
)
-
Υ
2
y
(
t
)
∥
α
≤
(
M
1
δ
+
M
1
1
-
e
-
δ
γ
)
ϵ
.
Combining (26), (30), and (33), it follows that Υ
1
x
+
Υ
2
y
∈
B (for all x
,
y
∈
B).
Step  2.
Υ
1 is a contraction.
Letx
,
y
∈
B; by (A3) and Lemma 1, we have
(34)
∥
Υ
1
x
(
t
)
-
Υ
1
y
(
t
)
∥
α
=
∥
-
g
(
t
,
x
t
)
-
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
x
s
)
d
s
hhhh
+
g
(
t
,
y
t
)
+
∫
-
∞
t
‍
A
T
(
t
-
s
)
g
(
s
,
y
s
)
d
s
∥
α
≤
∥
g
(
t
,
x
t
)
-
g
(
t
,
y
t
)
∥
α
+
∫
-
∞
t
‍
∥
A
T
(
t
-
s
)
[
g
(
s
,
x
s
)
-
g
(
s
,
y
s
)
]
∥
α
d
s
≤
K
∥
g
(
t
,
x
t
)
-
g
(
t
,
y
t
)
∥
β
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
∥
g
(
s
,
x
s
)
-
g
(
s
,
y
s
)
∥
β
d
s
≤
K
L
1
∥
x
t
-
y
t
∥
B
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
L
1
∥
x
s
-
y
s
∥
B
d
s
≤
K
L
1
L
sup
⁡
r
≤
t
∥
x
(
r
)
-
y
(
r
)
∥
α
+
∫
-
∞
t
‍
∥
A
1
+
α
-
β
T
(
t
-
s
)
∥
L
1
L
sup
⁡
r
≤
s
∥
x
(
r
)
-
y
(
r
)
∥
α
d
s
≤
K
L
1
L
∥
x
-
y
∥
α
+
∫
-
∞
t
‍
M
2
e
-
δ
(
t
-
s
)
(
t
-
s
)
-
(
1
+
α
-
β
)
L
1
L
∥
x
-
y
∥
α
d
s
≤
K
L
1
L
∥
x
-
y
∥
α
+
M
2
π
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
×
L
1
L
∥
x
-
y
∥
α
.
Therefore,
(35)
∥
Υ
1
x
-
Υ
1
y
∥
α
≤
[
K
L
1
L
+
M
2
π
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
L
1
L
]
×
∥
x
-
y
∥
α
.
Since K
L
1
L
+
(
M
2
π
/
δ
β
-
α
sin
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
)
L
1
L
≤
1
/
2
<
1, it follows that Υ
1 is a contraction.
Step  3. Υ
2 is completely continuous.
Claim   1. Υ
2 is continuous.
Let{
x
n
}
⊆
A
P
T
(
R
,
X
α
), x
n
→
x in A
P
T
(
R
,
X
α
) as n
→
∞; by Lemma 5, we may find a compact subset B
0
⊆
X
α such that x
n
(
t
)
,
x
(
t
)
∈
B
0 for all t
∈
R, n
∈
N; here we assume B
⊆
B
0. By (A2), for the given ϵ, there exist δ
>
0 such that x
,
y
∈
B, ∥
x
-
y
∥
B
≤
δ, implies that
(36)
∥
f
(
t
,
x
t
)
-
f
(
t
,
y
t
)
∥
α
≤
ϵ
,
∥
I
i
(
x
t
i
)
-
I
i
(
y
t
i
)
∥
α
≤
ϵ
.
For the above δ, there exists n
0
∈
N such that
(37)
∥
x
n
(
t
)
-
x
(
t
)
∥
α
≤
δ
L
for n
>
n
0 and t
∈
R; then
(38)
h
h
i
h
h
h
h
h
∥
(
x
n
)
t
-
x
t
∥
B
≤
δ
,
∥
f
(
t
,
(
x
n
)
t
)
-
f
(
t
,
x
t
)
∥
α
≤
ϵ
,
h
i
∥
I
i
(
(
x
n
)
t
i
)
-
I
i
(
x
t
i
)
∥
α
≤
ϵ
,
for n
>
n
0 and t
∈
R. Hence,
(39)
∥
Υ
2
(
x
n
)
(
t
)
-
Υ
2
x
(
t
)
∥
α
=
∥
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
x
t
i
)
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
(
x
n
)
s
)
d
s
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
(
x
n
)
t
i
)
-
∫
-
∞
t
‍
T
(
t
-
s
)
f
(
s
,
x
s
)
d
s
-
∑
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
x
t
i
)
∥
α
≤
∫
-
∞
t
‍
∥
T
(
t
-
s
)
∥
∥
f
(
s
,
(
x
n
)
s
)
-
f
(
s
,
x
s
)
∥
α
d
s
+
∑
t
i
<
t
‍
∥
T
(
t
-
t
i
)
∥
∥
I
i
(
(
x
n
)
t
i
)
-
I
i
(
x
t
i
)
∥
α
≤
∫
-
∞
t
‍
M
1
e
-
δ
(
t
-
s
)
ϵ
d
s
+
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
ϵ
=
M
1
δ
ϵ
+
M
1
1
-
e
δ
γ
ϵ
,
for n
>
n
0 and t
∈
R, from which it follows that Υ
2 is continuous.
Claim  2. {
Υ
2
u
(
t
)
:
u
∈
B
} is a relatively compact subset of X
α for each t
∈
R.
For anyϵ
>
0, let
(40)
Υ
2
ϵ
u
(
t
)
=
∫
-
∞
t
-
ϵ
‍
T
(
t
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
-
ϵ
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
=
T
(
ϵ
)
[
∑
t
i
<
t
-
ϵ
‍
∫
-
∞
t
-
ϵ
‍
T
(
t
-
ϵ
-
s
)
f
(
s
,
u
s
)
d
s
h
h
h
h
h
+
∑
t
i
<
t
-
ϵ
‍
T
(
t
-
ϵ
-
t
i
)
I
i
(
u
t
i
)
]
=
T
(
ϵ
)
Υ
2
u
(
t
-
ϵ
)
,
where  {
Υ
2
u
(
t
-
ϵ
)
:
u
∈
B
} is uniformly bounded in X
α and T
(
ϵ
) is compact, so {
Υ
2
ϵ
u
(
t
)
:
u
∈
B
} is relatively compact in X
α. Moreover,
(41)
∥
Υ
2
u
(
t
)
-
Υ
2
ϵ
u
(
t
)
∥
α
=
∥
∑
t
-
ϵ
<
t
i
<
t
‍
∫
t
-
ϵ
t
‍
T
(
t
-
s
)
f
(
s
,
x
s
)
d
s
+
∑
t
-
ϵ
<
t
i
<
t
‍
T
(
t
-
t
i
)
I
i
(
u
t
i
)
∥
α
≤
∫
t
-
ϵ
t
‍
∥
T
(
t
-
s
)
∥
∥
f
(
s
,
x
s
)
∥
α
d
s
+
∑
t
-
ϵ
<
t
i
<
t
‍
∥
T
(
t
-
t
i
)
∥
∥
I
i
(
u
t
i
)
∥
α
≤
∫
t
-
ϵ
t
‍
M
1
e
-
δ
(
t
-
s
)
∥
f
(
s
,
x
s
)
∥
α
d
s
+
∑
t
-
ϵ
<
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
∥
I
i
(
u
t
i
)
∥
α
≤
ϵ
M
1
C
1
+
ϵ
γ
M
1
C
2
,
so {
Υ
2
u
(
t
)
:
u
∈
B
} is a relatively compact subset of X
α for each t
∈
R.
Claim  3.  {
Υ
2
u
:
u
∈
B
} is equicontinuous on each interval (
t
i
,
t
i
+
1
)
(
i
∈
Z
).
Lett
′′
<
t
′
,
t
′′
,
t
′
∈
(
t
i
,
t
i
+
1
)
,
i
∈
Z
,
u
∈
B,
(42)
Υ
2
u
(
t
′
)
-
Υ
2
u
(
t
′′
)
=
∫
-
∞
t
′
‍
T
(
t
′
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
′
‍
T
(
t
′
-
t
i
)
I
i
(
u
t
i
)
-
∫
-
∞
t
′′
‍
T
(
t
′′
-
s
)
f
(
s
,
u
s
)
d
s
-
∑
t
i
<
t
′′
‍
T
(
t
′′
-
t
i
)
I
i
(
u
t
i
)
=
∫
-
∞
t
′′
‍
[
T
(
t
′
-
s
)
-
T
(
t
′′
-
s
)
]
f
(
s
,
u
s
)
d
s
+
∫
t
′′
t
′
‍
T
(
t
′
-
s
)
f
(
s
,
u
s
)
d
s
+
∑
t
i
<
t
′′
‍
[
T
(
t
′
-
t
i
)
-
T
(
t
′′
-
t
i
)
]
I
i
(
u
t
i
)
+
∑
t
′′
<
t
i
<
t
′
‍
T
(
t
′
-
t
i
)
I
i
(
u
t
i
)
.
Moreover,
(43)
∫
-
∞
t
′′
‍
[
T
(
t
′
-
s
)
-
T
(
t
′′
-
s
)
]
f
(
s
,
u
s
)
d
s
=
∫
0
∞
‍
[
T
(
t
′
-
t
′′
+
s
)
-
T
(
s
)
]
f
(
t
′′
-
s
,
u
t
′′
-
s
)
d
s
=
∫
0
∞
‍
[
T
(
t
′
-
t
′′
)
-
I
]
T
(
s
)
f
(
t
′′
-
s
,
u
t
′′
-
s
)
d
s
.
By (A1), for the given ϵ
>
0, there exists μ
(
ϵ
)
<
ϵ
/
4
M
1
(
C
1
+
C
2
/
γ
), such that if 0
<
t
′
-
t
′′
<
μ, then
(44)
∥
T
(
t
′
-
t
′′
)
-
I
∥
<
min
⁡
{
δ
ϵ
4
M
1
C
1
,
(
1
-
e
-
δ
γ
)
ϵ
4
M
1
C
2
}
.
So
(45)
∥
∫
-
∞
t
′′
‍
[
T
(
t
′
-
s
)
-
T
(
t
′′
-
s
)
]
f
(
s
,
u
(
s
)
)
d
s
∥
α
≤
∫
0
∞
‍
∥
T
(
t
′
-
t
′′
)
-
I
∥
∥
T
(
s
)
∥
×
∥
f
(
t
′′
-
s
,
u
(
t
′′
-
s
)
)
∥
α
d
s
≤
∫
0
∞
‍
δ
ϵ
4
M
1
C
1
M
1
e
-
δ
s
C
1
d
s
<
ϵ
4
,
∥
∫
t
′′
t
′
‍
T
(
t
′
-
s
)
f
(
s
,
u
(
s
)
)
d
s
∥
α
≤
∫
t
′′
t
′
‍
∥
T
(
t
′
-
s
)
∥
∥
f
(
s
,
u
(
s
)
)
∥
α
d
s
<
μ
M
1
C
1
<
ϵ
4
.
Similarly,
(46)
∥
∑
t
i
<
t
′′
‍
[
T
(
t
′
-
t
i
)
-
T
(
t
′′
-
t
i
)
]
I
i
(
u
(
t
i
)
)
∥
α
=
∥
∑
t
i
<
t
′′
‍
[
T
(
t
′
-
t
′′
)
T
(
t
′′
-
t
i
)
-
T
(
t
′′
,
t
i
)
]
I
i
(
u
(
t
i
)
)
∥
α
=
∥
∑
t
i
<
t
′′
‍
[
T
(
t
′
-
t
′′
)
-
I
]
T
(
t
′′
-
t
i
)
I
i
(
u
(
t
i
)
)
∥
α
≤
∑
t
i
<
t
′′
‍
∥
T
(
t
′
-
t
′′
)
-
I
∥
∥
T
(
t
′′
-
t
i
)
∥
∥
I
i
(
u
(
t
i
)
)
∥
α
≤
∑
t
i
<
t
′′
‍
(
1
-
e
-
δ
γ
)
ϵ
4
M
1
C
2
M
1
e
-
δ
(
t
′′
-
t
i
)
C
2
≤
ϵ
4
,
∥
∑
t
′′
<
t
i
<
t
′
‍
T
(
t
′
-
t
i
)
I
i
(
u
(
t
i
)
)
∥
α
≤
∑
t
′′
<
t
i
<
t
′
‍
∥
T
(
t
′
,
t
i
)
∥
∥
I
i
(
u
(
t
i
)
)
∥
α
<
μ
γ
M
1
C
2
<
ϵ
4
.
So that, for u
∈
B
,
t
′
,
t
′′
∈
(
t
i
,
t
i
+
1
)
,
i
∈
Z, for any ϵ
>
0, there exists a positive number μ
(
ϵ
), μ
<
ϵ
/
4
M
1
(
C
1
+
C
2
/
γ
); if 0
<
t
′
-
t
′′
<
μ,
(47)
∥
Υ
2
u
(
t
′
)
-
Υ
2
u
(
t
′′
)
∥
α
≤
ϵ
.
That is, {
Υ
2
u
:
u
∈
B
} is equicontinuous on each interval (
t
i
,
t
i
+
1
)
(
i
∈
Z
).
Since{
Υ
2
u
:
u
∈
B
}
⊆
(
P
C
)
h
0
(
R
,
X
α
) and {
Υ
2
u
:
u
∈
B
} satisfies the conditions of Lemma 10, Υ
2 is completely continuous.
By Krasnoselskii’s fixed-point theorem (Theorem11), we know that Υ has a fixed point u
∈
B; that is, (1) has a piecewise almost periodic solution u
(
t
). The proof is complete.Note that the condition of uniformly continuous is weaker than that of Lipschitz continuous, so if assumption (A2) is replaced by the following assumption:(A2′)f
(
t
,
x
)
∈
A
P
T
(
R
×
B
,
X
α
)
,
f
(
t
,
0
)
=
0
,
I
i
(
0
)
=
0
,
I
i
(
x
) is almost periodic in i
∈
Z uniformly in x
∈
B and
(48)
∥
f
(
t
,
x
)
-
f
(
t
,
y
)
∥
α
≤
L
2
∥
x
-
y
∥
B
,
h
h
i
∥
I
i
(
x
)
-
I
i
(
y
)
∥
α
≤
L
3
∥
x
-
y
∥
B
,
for all x
,
y
∈
B.We can get the almost periodic solution of (1) by means of contraction mapping principle.Corollary 13.
Suppose that conditions (A1), (A2′), and (A3)-(A4) hold; (1) has a piecewise almost periodic solution provided that
(49)
M
1
L
2
L
δ
+
M
1
L
3
L
1
-
e
-
δ
r
+
(
K
+
M
2
π
δ
β
-
α
sin
⁡
(
π
(
1
+
α
-
β
)
)
Γ
(
1
+
α
-
β
)
)
×
L
1
L
<
1
.Proof.
As the same discussion as Step 2 of Theorem12, we can prove that Υ
1 is a contraction, and it remains to show that Υ
2 is a contraction. By Step 1 of Theorem 12, Υ
2
B
⊆
B. For x, y
∈
B,
(50)
∥
Υ
2
x
(
t
)
-
Υ
2
y
(
t
)
∥
α
=
∥
∑
t
i
<
t
‍
∫
-
∞
t
‍
T
(
t
-
s
)
[
f
(
s
,
x
s
)
-
f
(
s
,
y
s
)
]
d
s
h
h
+
∑
t
i
<
t
‍
T
(
t
-
t
i
)
[
I
i
(
x
t
i
)
-
I
i
(
y
t
i
)
]
∥
α
≤
∫
-
∞
t
‍
∥
T
(
t
-
s
)
∥
∥
f
(
s
,
x
s
)
-
f
(
s
,
y
s
)
∥
α
d
s
+
∑
t
i
<
t
‍
∥
T
(
t
-
t
i
)
∥
∥
I
i
(
x
t
i
)
-
I
i
(
y
t
i
)
∥
α
≤
∫
-
∞
t
‍
M
1
e
-
δ
(
t
-
s
)
L
2
∥
x
s
-
y
s
∥
B
d
s
+
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
L
3
∥
x
t
i
-
y
t
i
∥
B
≤
∫
-
∞
t
‍
M
1
e
-
δ
(
t
-
s
)
L
2
L
sup
⁡
r
≤
s
∥
x
(
r
)
-
y
(
r
)
∥
α
d
s
+
∑
t
i
<
t
‍
M
1
e
-
δ
(
t
-
t
i
)
L
3
L
sup
⁡
r
≤
t
i
∥
x
(
r
)
-
y
(
r
)
∥
α
≤
M
1
L
(
L
2
δ
+
L
3
1
-
e
-
δ
γ
)
∥
x
-
y
∥
α
.
The proof is complete.


## 4. Application

Consider the following impulsive neutral differential system:(51)
∂
∂
t
[
u
(
t
,
ξ
)
+
∫
-
∞
0
‍
∫
0
π
‍
b
1
(
s
,
η
,
ξ
)
u
(
t
+
s
,
η
)
d
η
d
s
]
=
∂
2
∂
ξ
2
u
(
t
,
ξ
)
+
b
2
(
ξ
)
u
(
t
,
ξ
)
+
∫
-
∞
0
‍
b
3
(
s
)
u
(
t
+
s
,
ξ
)
d
s
+
b
4
(
t
,
ξ
)
,
hhhhhhhhhhhhhhhhhhi
(
t
,
ξ
)
∈
R
×
[
0
,
π
]
,
u
(
t
,
0
)
=
u
(
t
,
π
)
=
0
,
t
∈
R
,
Δ
u
(
t
i
)
(
ξ
)
=
∫
0
π
‍
a
i
(
t
i
-
s
)
u
(
s
,
ξ
)
d
s
,
h
h
h
h
h
h
h
h
h
h
h
h
h
i
∈
Z
,
ξ
∈
[
0
,
π
]
,
where {
t
i
j
=
t
i
+
j
-
t
i
}, i
∈
Z, j
=
0
,
±
1
,
±
2
,
…, are equipotentially almost periodic such that γ
=
inf
⁡
i
∈
Z
{
t
i
+
1
-
t
i
}
>
0. The system (51) arises, for example, in control systems described by abstract retarded functional differential equations with feedback control governed by proportional integrodifferential law; see [25, 26] for details.LetX
=
L
2
(
[
0
,
π
]
) and A be the infinitesimal generator of an analytic semigroup {
T
(
t
)
}
t
≥
0 which is compact for t
>
0 and given by A
u
=
u
′′ with domain D
(
A
)
=
{
u
∈
L
2
(
[
0
,
π
]
)
:
u
′′
∈
L
2
(
[
0
,
π
]
)
,
u
(
0
)
=
u
(
π
)
=
0
}. The semigroup {
T
(
t
)
}
t
≥
0 is defined for u
∈
L
2
(
[
0
,
π
]
) by
(52)
T
(
t
)
u
=
∑
n
=
1
∞
‍
e
-
n
2
t
〈
u
,
ϕ
n
〉
ϕ
n
,
where {
ϕ
n
,
n
∈
Z
} is an orthonormal basis of X; then |
T
(
t
)
u
|
≤
e
-
t
|
u
|, 0
≠
u
∈
L
2
(
[
0
,
π
]
). For u
∈
L
2
(
[
0
,
π
]
), α
∈
(
0,1
), (
-
A
)
-
α
u
=
∑
n
=
1
∞
‍
n
-
2
α
〈
u
,
ϕ
n
〉
ϕ
n,  X
α is the Banach space endowed with the norm ∥
·
∥
α between X and D
(
A
).In this paper, we assumeα
=
1
/
4, β
=
3
/
4.To study the system (51), we make the following assumptions.(i) The functions(
∂
i
/
∂
ξ
i
)
b
1
(
τ
,
η
,
ξ
)
,
i
=
0,1 are Lebesgue measurable, b
1
(
τ
,
η
,
0
)
=
b
1
(
τ
,
η
,
π
)
=
0 and
(53)
L
1
≔
max
⁡
{
∫
0
π
‍
∫
-
∞
0
‍
∫
0
π
‍
(
∂
i
∂
ξ
i
b
1
(
τ
,
η
,
ξ
)
)
2
d
η
d
τ
d
ξ
:
i
=
0,1
}
<
∞
.(ii) The functionsb
i
(
i
=
2,3
,
4
), a
i
(
i
∈
Z
) are continuous, and the sequence of functions {
a
i
,
i
∈
Z
} is almost periodic.Under these conditions, we definef
,
g
:
R
×
B
→
X
,
I
i
:
B
→
X
(
i
∈
Z
) by
(54)
g
(
t
,
ϕ
)
=
∫
-
∞
0
‍
∫
0
π
‍
b
1
(
s
,
η
,
ξ
)
ϕ
(
t
+
s
,
η
)
d
η
d
s
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
t
∈
R
,
ξ
∈
[
0
,
π
]
,
f
(
t
,
ϕ
)
=
b
2
(
ξ
)
ϕ
(
t
,
ξ
)
+
∫
-
∞
0
‍
b
3
(
s
)
ϕ
(
t
+
s
,
ξ
)
d
s
+
b
4
(
t
,
ξ
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
t
∈
R
,
ξ
∈
[
0
,
π
]
,
I
i
(
ϕ
)
=
∫
0
π
‍
a
i
(
t
i
-
s
)
ϕ
(
s
,
ξ
)
d
s
,
i
∈
Z
,
ξ
∈
[
0
,
π
]
.We assume further thatf
,
g satisfy the following condition.(iii)g
∈
A
P
T
(
R
×
B
,
X
β
) and f
∈
A
P
T
(
R
×
B
,
X
α
) are uniformly continuous in x
∈
B uniformly in t
∈
R.Under the above assumptions, we can rewrite (51) as the abstract form (1) and verify that the assumptions of Theorem 12 hold; then we can get the next result, which is a consequence of Theorem 12.Proposition 14.
Assume that the previous conditions are verified, if(55)
L
1
L
(
K
+
π
)
<
1
.
The system (51) has a piecewise almost periodic solution.



---

*Source: 782018-2014-08-12.xml*