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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 143175, 14 pages
doi:10.1155/2009/143175
Research Article
On the Spectrum of Almost Periodic Solution of
Second-Order Neutral Delay Differential Equations
with Piecewise Constant of Argument
Li Wang and Chuanyi Zhang
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Li Wang,
Received 16 December 2008; Accepted 10 April 2009
Recommended by Ondrej Dosly
The spectrum containment of almost periodic solution of second-order neutral delay differential
equations with piecewise constant of argument EPCA, for short of the form xtpxt − 1
qx2t 1/2 ft is considered. The main result obtained in this paper is different from that
given by some authors for ordinary differential equations ODE, for short and clearly shows the
differences between ODE and EPCA. Moreover, it is also different from that given for equation
xtpxt − 1
qxt ft because of the difference between t and 2t 1/2.
Copyright q 2009 L. Wang and C. Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction and Some Preliminaries
Differential equations with piecewise constant argument, which were firstly considered by
Cooke and Wiener 1 and Shah and Wiener 2, combine properties of both differential and
difference equations and usually describe hybrid dynamical systems and have applications
in certain biomedical models in the work of Busenberg and Cooke 3. Over the years, more
attention has been paid to the existence, uniqueness, and spectrum containment of almost
periodic solutions of this type of equations see, e.g., 4–12 and reference there in.
If g
1
t and g
2
t are almost periodic, then the module containment property
modg
1
⊂ modg
2
can be characterized in several ways see 13–16. For periodic function
this inclusion just means that the minimal period of g
1
t is a multiple of the minimal period
of g
2
t. Some properties of basic frequencies thebaseofspectrum were discussed for
almost periodic functions by Cartwright. In 17, Cartwright compared basic frequencies the
base of spectrum of almost periodic differential equations ODE ˙x ψx, t, x ∈ R
n
,with
those of its unique almost periodic solution. For scalar equation, n 1, Cartwright’s results
in 17 implied that the number of basic frequencies of ˙x ψx, t,x∈ R, is the same as that
of basic frequencies of its unique solution.
2 Advances in Difference Equations
The spectrum containment of almost periodic solution of equation xtpxt − 1
qxt ft was studied in 9, 10. Up to now, there have been no papers concerning the
spectrum containment of almost periodic solution of equation
xtpxt − 1
qx
2
t 1
2
f
t
, 1.1
where · denotes the greatest integer function, p, q are nonzero real constants, |p|
/
1,
q
/
− 2p
2
1,andft is almost periodic. In this paper, we investigate the existence,
uniqueness, and spectrum containment of almost periodic solutions of 1.1. T he main result
obtained in this paper is different from that given in 17 for ordinary differential equations
ODE, for short. This clearly shows differences between ODE and EPCA. Moreover, it is
also different from that given in 9, 10 for equation xtpxt − 1
qxt ft.
This is due to the difference between t and 2t 1/2. As well known, both solutions
of 1.1 and equation xtpxt − 1
qxt ft can be constructed by the solutions
of corresponding difference equations. However, noticing the difference b etween t and
2t 1/2, the solution of difference equation corresponding to the latter can be obtained
directly see 4, while the solution {x
n
} of difference equation corresponding to the former
i.e., 1.1 cannot be obtained directly. In fact, {x
n
} consists of two parts: {x
2n
} and {x
2n1
}.
We will first obtain {x
2n
} by solving a difference equation and then obtain {x
2n1
} from {x
2n
}.
Similar technology can be seen in 8. A detailed account will be given in Section 2.
Now, We give some preliminary notions, definitions, and theorem. Throughout this
paper Z, R,andC denote the sets of integers, real, and complex numbers, respectively. The
following preliminaries can be found in the books, for example, 13–16.
Definition 1.1. 1 AsubsetP of R is said to be relatively dense in R if there exists a number
p>0 such that P ∩t, t p
/
∅ for all t ∈ R.
2 A continuous function f : R → R is called almost periodic abbreviated as APR
if the -translation set of f
T
f,
τ ∈ R :
f
t τ
− f
t
<,
∀t ∈ R
1.2
is relatively dense for each >0.
Definition 1.2. Let f be a bounded continuous function. If the limit
lim
T →∞
1
2T
T
−T
f
t
dt 1.3
exists, then we call the limit mean of f and denote it by Mf.
If f ∈APR, then the limit
lim
T →∞
1
2T
Ts
−Ts
f
t
dt 1.4
exists uniformly with respect to s ∈ R. Furthermore, the limit is independent of s.
Advances in Difference Equations 3
For any λ ∈ R and f ∈APR since the function fe
−iλ·
is in APR, the mean exists
for this function. We write
a
λ; f
M
fe
−iλ·
, 1.5
then there exists at most a countable set of λ’s for which aλ; f
/
0. The set
Λ
f
λ : a
λ; f
/
0
1.6
is called the frequency set or spectrum of f. It is clear that if ft
n
k1
c
k
e
iλ
k
t
, then
aλ; fc
k
if λ λ
k
, for some k 1, ,n;andaλ; f0ifλ
/
λ
k
, for any k 1, ,n.
Thus, Λ
f
{λ
k
,k 1, ,n}.
Members of Λ
f
are called the Fourier exponents of f,andaλ; f’s are called the
Fourier coefficients of f. Obviously, Λ
f
is countable. Let Λ
f
{λ
k
} and A
k
aλ
k
; f.Thusf
can associate a Fourier series:
f
t
∼
∞
k1
A
k
e
iλ
k
t
. 1.7
The Approximation Theorem
Let f ∈APR and Λ
f
{λ
k
}. Then for any >0 there exists a sequence {σ
}of trigonometric
polynomials
σ
t
n
k1
b
k,
e
iλ
k
t
1.8
such that
σ
− f≤, 1.9
where b
k,
is the product of aλ
k
; f and certain positive number depending on and λ
k
and lim
→0
b
k,
aλ
k
; f.
Definition 1.3. 1 For a sequence {gn : n ∈ Z}, define gn,gnp {gn, ,gn p}
and call it sequence interval with length p ∈ Z.AsubsetP of Z is said to be relatively dense
in Z if there exists a positive integer p such that P ∩n, n p
/
∅ for all n ∈ Z.
2 A bounded sequence g : Z → R is called an almost periodic sequence abbreviated
as APSR if the -translation set of g
T
g,
τ ∈
Z :
g
n τ
− g
n
<, ∀n ∈ Z
1.10
is relatively dense for each >0.
4 Advances in Difference Equations
For an almost periodic sequence {gn}, it follows from the lemma in 13 that
a
z; g
lim
N →∞
1
2N
N
k−N
z
−k
g
k
, ∀z ∈ S
1
{
z ∈ C :
|
z
|
1
}
1.11
exists. The set
σ
b
g
z : a
z; g
/
0,z∈ S
1
1.12
is called the Bohr spectrum of {gn}. Obviously, for almost periodic sequence gn
m
k1
r
k
z
n
k
, az; gr
k
if z z
k
, for some k 1, ,m; az; g0ifz
/
z
k
, for any k 1, ,m.
So, σ
b
g{z
k
,k 1, ,m}.
2. The Statement of Main Theorem
We begin this section with a definition of the solution of 1.1.
Definition 2.1. A continuous function x : R → R is called a solution of 1.1 if the following
conditions are satisfied:
i xt satisfies 1.1 for t ∈ R, t
/
n ∈ Z;
ii the one-sided second-order derivatives xtpxt − 1
exist at n, n ∈ Z.
In 8, the authors pointed out that if xt is a solution of 1.1, then xtpxt − 1
are continuous at t ∈ R, which guarantees the uniqueness of solution of 1.1 and cannot be
omitted.
To study the spectrum of almost periodic solution of 1.1, we firstly study the solution
of 1.1.Let
f
1
n
n1
n
s
n
f
σ
dσ ds, f
2
n
n−1
n
s
n
f
σ
dσ ds, h
n
f
1
n
f
2
n
. 2.1
Suppose that xt is a solution of 1.1, then xtpxt − 1
exist and are continuous
everywhere on R. By a process of integrating 1.1 two times in t ∈ 2n − 1, 2n 1 or t ∈
2n, 2n 2 as in 7, 8, 18, we can easily get
x
2n 1
p − 2 − q
x
2n
1 − 2p
x
2n − 1
px
2n − 2
h
2n
,
1 −
q
2
x
2n 2
p − 2
x
2n 1
1 − 2p −
q
2
x
2n
px
2n − 1
h
2n1
.
2.2
These lead to the difference equations
px
2n−2
1 − 2p
x
2n−1
p − 2 − q
x
2n
x
2n1
h
2n
, 2.3
px
2n−1
1 − 2p −
q
2
x
2n
p − 2
x
2n1
1 −
q
2
x
2n2
h
2n1
. 2.4
Advances in Difference Equations 5
Suppose that |p|
/
1. First, multiply the two sides of 2.3 and 2.4 by p and 2p − 1,
respectively, then add the resulting equations to get
x
2n1
1
2p − 1
2
ph
2n
− p
p − 2 − q
x
2n
− p
2
x
2n−2
2p − 1
h
2n1
−
1
2p − 1
2
2p − 1
1 −
q
2
x
2n2
2p − 1
1 − 2p −
q
2
x
2n
.
2.5
Similarly, one gets
x
2n−1
1
2p − 1
2
2 − p
h
2n
−
2 − p
p − 2 − q
x
2n
−
2 − p
px
2n−2
1
2p − 1
2
h
2n1
−
1 −
q
2
x
2n2
−
1 − 2p −
q
2
x
2n
.
2.6
Replacing 2n by 2n 2 in 2.6 and comparing with 2.5,onegets
1 −
q
2
x
2n4
−
p
2
− 2pq 3q 2
x
2n2
2p
2
2pq −
q
2
1
x
2n
− p
2
x
2n−2
h
2n3
2 − p
h
2n2
1 − 2p
h
2n1
− ph
2n
.
2.7
The corresponding homogeneous equation is
1 −
q
2
x
2n4
−
p
2
− 2pq 3q 2
x
2n2
2p
2
2pq −
q
2
1
x
2n
− p
2
x
2n−2
0. 2.8
We can seek the particular solution as x
2n
ξ
n
for this homogeneous difference
equation. At this time, ξ will satisfy the following equation:
p
1
ξ
1 −
q
2
ξ
3
−
p
2
− 2pq 3q 2
ξ
2
2p
2
2pq −
q
2
1
ξ − p
2
0. 2.9
From the analysis above one sees that if xt is a solution of 1.1 and |p|
/
1, then one
gets 2.3 and 2.4. In fact, a solution of 1.1 is constructed by the common solution {x
n
} of
2.3 and 2.4. Moreover, it is clear that {x
n
} consists of two parts: {x
2n
} and {x
2n1
}. {x
2n
}
can be obtained by solving 2.7,and{x
2n1
} can be obtained by substituting {x
2n
} into 2.5
or 2.6. Without loss of generality, we consider 2.5 only. These will be shown in Lemmas
2.5 and 2.6.
Lemma 2.2. If f ∈APR,then{f
i
n
}, {h
n
}∈APSR, i 1, 2.
Lemma 2.3. Suppose that |p|
/
1 and q
/
−2p
2
1, then the roots of polynomial p
1
ξ are of moduli
different from 1.
6 Advances in Difference Equations
Lemma 2.4. Suppose that X is a Banach space, LX denotes the set of bounded linear operators from
X to X, A ∈LX, and A < 1,thenId −A is bounded invertible and
Id −A
−1
∞
n0
A
n
,
I −A
−1
≤
1
1 −
A
,
2.10
where A
0
Id, and Id is an identical operator.
The proofs of Lemmas 2.2, 2.3,and2.4 are elementary, and we omit the details.
Lemma 2.5. Suppose that |p|
/
1 and q
/
− 2p
2
1,then2.7 has a unique solution {x
2n
}∈
APSR.
Proof. As the proof of Theorem 9 in 8, define A : X → X by A{x
2n
} {x
2n2
}, where X
is the Banach space consisting of all bounded sequences {x
n
} in C with {x
n
} sup
n∈Z
|x
n
|.
It follows from Lemmas 2.2–2.4 that 2.7 has a unique solution {x
2n
} PA
−1
{h
2n5
2 −
ph
2n4
1 − 2ph
2n3
− ph
2n2
}∈APSR.
Substituting x
2n
into 2.5,weobtainx
2n1
. Easily, we can get {x
2n1
}∈APSR.
Consequently, the common solution {x
n
} of 2.3 and 2.4 can be obtained. Furthermore,
we have that {x
n
}∈APSR is unique.
Lemma 2.6. Suppose that |p|
/
1 and q
/
− 2p
2
1, f ∈APR.Let{x
n
}∈APSR be the
common solution of 2.3 and 2.4.Then1.1 has a unique solution xt ∈APR such that
xnx
n
,n∈ Z. In this case the solution xt is given for t ∈ R by
x
t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∞
k0
−p
k
ω
t − k
,
p
< 1,
−
∞
k1
−p
−k
ω
t k
,
p
> 1,
2.11
where
ω
t
x
2n
px
2n−1
y
2n
t − 2n
qx
2n
t − 2n
2
2
t
2n
s
2n
f
σ
dσ ds,
y
2n
x
2n1
p − 1 −
q
2
x
2n
− px
2n−1
− f
1
2n
,
2.12
for t ∈ 2n − 1, 2n 1,n∈ Z; {y
2n
}∈APSR,ωt ∈APR.
The proof is easy, we omit the details. Since the almost periodic solution xt of 1.1
is constructed by the common almost periodic solution of 2.3 and 2.4, easily, we have that
xtpxt − 1
are continuous at t ∈ R. It must be pointed out that in many works only one
of 2.3 and 2.4 is considered while seeking the unique almost periodic solution of 1.1,and
it is not true for the continuity of xtpxt − 1
on R, consequently, it is not true for the
uniqueness see 8.
Advances in Difference Equations 7
The expressions of x
2n
,x
2n1
,y
2n
,ωt, and xt are important in the process of
studying the spectrum containment of the almost periodic solution of 1.1. Before giving
the main theorem, we list the following assumptions which will be used later.
H
1
|p|
/
1, q
/
−2p
2
1.
H
2
kπ
/
∈Λ
f
, for all k ∈ Z.
H
3
If λ ∈ Λ
f
, then λ kπ
/
∈Λ
f
,0
/
k ∈ Z.
Our result can be formulated as follows.
Main Theorem
Let f ∈APR and H
1
be satisfied. Then 1.1 has a unique almost periodic solution xt
and Λ
x
⊂ Λ
f
{kπ : k ∈ Z}. Additionally, if H
2
and H
3
are also satisfied, then Λ
f
{kπ :
k ∈ Z}⊂Λ
x
, that is, the following spectrum relation Λ
x
Λ
f
{kπ : k ∈ Z} holds, where the
sum of sets A and B is defined as A B {a b : a ∈ A, b ∈ B}.
We postpone the proof of this theorem to the next section.
3. The Proof of Main Theorem
To show the Main Theorem, we need some more lemmas.
Lemma 3.1. Let f ∈APR,thenσ
b
f
i
2n
,σ
b
f
i
2n1
,σ
b
h
2n
,σ
b
h
2n1
⊂ e
i2Λ
f
, i 1, 2.If(H
3
)is
satisfied, then σ
b
f
i
2n
σ
b
f
i
2n1
e
i2Λ
f
, i 1, 2. Furthermore, if (H
3
) and (H
2
) are both satisfied,
then σ
b
h
2n
σ
b
h
2n1
e
i2Λ
f
.
Proof. Since f ∈APR,byLemma 2.2 we know that {f
i
2n
}, {f
i
2n1
}, {h
2n
}, {h
2n1
}∈APSR,
i 1, 2. It follows from The Approximation Theorem that, for any m>0,m ∈ Z, there exists
P
m
t
nm
k1
b
k,m
e
iλ
k
t
,λ
k
∈ Λ
f
such that P
m
−f≤1/m, where lim
m →∞
b
k,m
aλ
k
; f,and
we can assume that b
k,m
e
iλ
k
t
and b
k,m
e
−iλ
k
t
appear together in the trigonometric polynomial
P
m
t. Define
Q
1
m,2n
2n1
2n
s
2n
P
m
σ
dσ ds
nm
k1
c
1
k,m
e
i2λ
k
n
,
Q
2
m,2n
2n−1
2n
s
2n
P
m
σ
dσ ds
nm
k1
c
2
k,m
e
i2λ
k
n
,
Q
1
m,2n1
2n2
2n1
s
2n1
P
m
σ
dσ ds
nm
k1
c
1
k,m
e
iλ
k
e
i2λ
k
n
,
Q
2
m,2n1
2n
2n1
s
2n1
P
m
σ
dσ ds
nm
k1
c
2
k,m
e
iλ
k
e
i2λ
k
n
,
3.1
8 Advances in Difference Equations
where
c
1
k,m
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
b
k,m
2
,λ
k
0,
−b
k,m
e
iλ
k
− 1 − iλ
k
λ
2
k
,λ
k
/
0,
c
2
k,m
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
b
k,m
2
,λ
k
0,
−b
k,m
e
−iλ
k
− 1 iλ
k
λ
2
k
,λ
k
/
0.
3.2
Obviously, σ
b
Q
i
m,2n
,σ
b
Q
i
m,2n1
⊂ e
i2Λ
f
, i 1, 2, for all m ∈ Z. For any z ∈ S
1
, az; f
i
2n
lim
m →∞
az; Q
i
m,2n
, az; f
i
2n1
lim
m →∞
az; Q
i
m,2n1
, thus, we have σ
b
f
i
2n
,σ
b
f
i
2n1
⊂
e
i2Λ
f
, i 1, 2.
Since h
2n
f
1
2n
f
2
2n
and h
2n1
f
1
2n1
f
2
2n1
, for all n ∈ Z. For all z ∈ S
1
, we have
a
z; h
2n
a
z; f
1
2n
a
z; f
2
2n
, 3.3
a
z; h
2n1
a
z; f
1
2n1
a
z; f
2
2n1
. 3.4
Thus, σ
b
f
i
2n
⊂ e
i2Λ
f
and σ
b
f
i
2n1
⊂ e
i2Λ
f
imply σ
b
h
2n
⊂ e
i2Λ
f
and σ
b
h
2n1
⊂ e
i2Λ
f
,
respectively, i 1, 2.
If H
3
is satisfied, then for any λ
j
∈ Λ
f
, we have
a
e
i2λ
j
; f
1
2n
lim
m →∞
a
e
i2λ
j
; Q
1
m,2n
lim
m →∞
c
1
j,m
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
a
λ
j
; f
2
,λ
j
0,
−a
λ
j
; f
e
iλ
j
− 1 − iλ
j
λ
2
j
,λ
j
/
0,
a
e
i2λ
j
; f
2
2n
lim
m →∞
a
e
i2λ
j
; Q
2
m,2n
lim
m →∞
c
2
j,m
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
a
λ
j
; f
2
,λ
j
0,
−a
λ
j
; f
e
−iλ
j
− 1 iλ
j
λ
2
j
,λ
j
/
0,
a
e
i2λ
j
; f
1
2n1
lim
m →∞
a
e
i2λ
j
; Q
1
m,2n1
lim
m →∞
e
iλ
j
c
1
j,m
,
a
e
i2λ
j
; f
2
2n1
lim
m →∞
a
e
i2λ
j
; Q
2
m,2n1
lim
m →∞
e
iλ
j
c
2
j,m
.
3.5
Easily, we have ae
i2λ
j
; f
i
2n
/
0andae
i2λ
j
; f
i
2n1
/
0, that is, e
i2λ
j
⊂ σ
b
f
i
2n
,e
i2λ
j
⊂ σ
b
f
i
2n1
,
i 1, 2. By the arbitrariness of λ
j
,wegete
i2Λ
f
⊂ σ
b
f
i
2n
and e
i2Λ
f
⊂ σ
b
f
i
2n1
.So,e
i2Λ
f
σ
b
f
i
2n
σ
b
f
i
2n1
,i 1, 2.
Advances in Difference Equations 9
If H
3
and H
2
are both satisfied, suppose that there exists z
0
e
i2λ
j
∈ e
i2Λ
f
such that
az
0
; h
2n
0. H
2
implies e
iλ
j
/
± 1. Moreover, since H
3
holds, we have az
0
; f
i
2n
/
0,i
1, 2. az
0
; h
2n
az
0
; f
1
2n
az
0
; f
2
2n
leads to e
iλ
j
1, which contradicts with e
iλ
j
/
± 1. So,
e
i2Λ
f
⊂ σ
b
h
2n
. Noticing that σ
b
h
2n
⊂ e
i2Λ
f
, we have e
i2Λ
f
σ
b
h
2n
. Similarly, we can get
e
i2Λ
f
σ
b
h
2n1
. The proof is completed.
Lemma 3.2. Suppose that (H
1
) is satisfied, then σ
b
x
2n
⊂ e
i2Λ
f
.If(H
1
), (H
2
), and (H
3
)areall
satisfied, then σ
b
x
2n
e
i2Λ
f
,where{x
2n
} is the unique almost periodic sequence solution of 2.7.
Proof. Since H
1
holds, from Lemma 2.5 we know {x
2n
} p
1
A
−1
{g
n1
}∈APSR, where
g
n
h
2n3
2 − ph
2n2
1 − 2ph
2n1
− ph
2n
, for all n ∈ Z. For any z ∈ S
1
, it follows from
Lemma 2.3 that p
1
z
/
0. Noticing the expressions of {x
2n
} and g
n
,weobtain
za
z; g
n
p
1
z
a
z; x
2n
, 3.6
a
z; g
n
z 1 − 2p
a
z; h
2n1
2z − pz − p
a
z; h
2n
. 3.7
Those equalities and Lemma 3.1 imply that σ
b
x
2n
σ
b
g
n
and σ
b
x
2n
⊂ e
i2Λ
f
, when H
1
is satisfied. If H
1
, H
2
,andH
3
are all satisfied, we only need to prove e
i2Λ
f
⊂ σ
b
g
n
.
Suppose that there exists z
0
e
i2λ
j
∈ e
i2Λ
f
, obviously, e
iλ
j
/
± 1, such that az
0
; g
n
0.
From Lemma 3.1 , az
0
; h
2n
/
0,az
0
; h
2n1
/
0. Thus, 0 z
0
1−2paz
0
; h
2n1
2z
0
−pz
0
−
paz
0
; h
2n
, that is, e
i2λ
j
1−2pe
iλ
j
pe
i2λ
j
−2e
i2λ
j
p, which leads to e
iλ
j
p. This contradicts
with H
1
.Thus,e
i2Λ
f
⊂ σ
b
g
n
,thatis,e
i2Λ
f
⊂ σ
b
x
2n
. Noticing that σ
b
x
2n
⊂ e
i2Λ
f
,so,
e
i2Λ
f
σ
b
x
2n
. The proof is completed.
As mentioned above, the common almost periodic sequence solution {x
n
} of 2.3 and
2.4 consists of two parts: {x
2n
}and {x
2n1
}, where {x
2n
}∈APSR is the unique solution of
2.7,and{x
2n1
} is obtained by substituting {x
2n
} into 2.5. Obviously, {x
2n1
}∈APSR.
In the following, we give the spectrum containment of {x
2n1
}.
Lemma 3.3. Suppose t hat ( H
1
) is satisfied, then σ
b
x
2n1
⊂ e
i2Λ
f
.If(H
1
), (H
2
), and (H
3
)areall
satisfied, then σ
b
x
2n1
e
i2Λ
f
.
Proof. Since {x
2n
}, {h
2n
}, {h
2n1
}∈APSR, {x
2n1
}∈APSR. Noticing the expression of
x
2n1
, for any z ∈ S
1
, we have
2
p − 1
2
a
z, x
2n1
pa
z, h
2n
2p − 1
a
z, h
2n1
− z
−1
p
2
z
a
z, x
2n
, 3.8
where p
2
z2p − 11 − q/2z
2
−3p
2
2p − 1 − 2pq q/2z p
2
.IfH
1
is satisfied, it
follows from Lemmas 3.1 and 3.2 that σ
b
x
2n1
⊂ e
i2Λ
f
.
If H
1
, H
2
,andH
3
are all satisfied, supposing there exists z
0
e
i2λ
j
∈ e
i2Λ
f
,
obviously, e
iλ
j
/
± 1, such that az
0
; x
2n1
0, that is, z
−1
0
p
2
z
0
az
0
,x
2n
paz
0
,h
2n
2p −1az
0
,h
2n1
. Noticing 3.3–3.7, this equality is equivalent to p
2
e
i2λ
j
e
i2λ
j
1 −2p −
p
1
e
i2λ
j
2p −1e
iλ
j
p
2
e
i2λ
j
2e
i2λ
j
−pe
i2λ
j
−p − pp
1
e
i2λ
j
0, that is, q −2e
i3λ
j
2p −4 −
2qe
i2λ
j
4pq−2e
iλ
j
2p 0. Considering equation q−2x
3
2p−4−2qx
2
4pq−2x2p
0, its roots are x
1
, x
3
,andx
2
, obviously, x
i
/
±1, i 1, 2, 3. We claim that |x
i
|
/
1, i 1, 2, 3, that
is, this equation has no imaginary root. Otherwise, suppose that |x
1
| 1andx
3
x
1
, then by
the relationship between roots and coefficient of three-order equation, we know q 0, which
10 Advances in Difference Equations
leads to a contradiction. Thus q − 2e
i3λ
j
2p − 4 − 2qe
i2λ
j
4p q − 2e
iλ
j
2p
/
0; this
contradiction shows e
i2Λ
f
⊂ σ
b
x
2n1
. Noticing that σ
b
x
2n1
⊂ e
i2Λ
f
,thus,σ
b
x
2n1
e
i2Λ
f
.
The proof is completed.
Lemma 3.4. Suppose that (H
1
) is satisfied, then σ
b
y
2n
⊂ e
i2Λ
f
.If(H
1
), (H
2
), and (H
3
)areall
satisfied, then σ
b
y
2n
e
i2Λ
f
,where{y
2n
} is defined in Lemma 2.6.
Proof. From Lemma 2.6, we have y
2n
x
2n1
p − 1 − q/2x
2n
− px
2n−1
− f
1
2n
, for all n ∈ Z.
For any z ∈ S
1
a
z, y
2n
1 − pz
−1
a
z, x
2n1
p − 1 −
q
2
a
z, x
2n
− a
z, f
1
2n
. 3.9
Since H
1
holds, it follows from Lemmas 3.1–3.3 that we have σ
b
y
2n
⊂ e
i2Λ
f
.
If H
1
, H
2
,andH
3
are all satisfied, supposing there exists z
0
e
i2λ
j
∈ e
i2Λ
f
such
that az
0
; y
2n
0, it follows from H
2
that e
iλ
j
/
± 1. Notice that 3.3–3.8, az
0
; y
2n
0
is equivalent to pz
0
−paz
0
,h
2n
z
0
−p2p −1az
0
,h
2n1
−2p − 1
2
z
0
az
0
,f
1
2n
p −1−
q/22p − 1
2
z
2
0
−z
0
−pp
2
z
0
p
1
z
0
−1
z
0
1 −2paz
0
,h
2n1
2z
0
−pz
0
−paz
0
,h
2n
0. This equality is equivalent to e
iλ
j
− 1 − iλ
j
e
iλ
j
e
−iλ
j
− 2p
1
e
i2λ
j
−1
1 − q/2e
i6λ
j
1 q/2e
i5λ
j
pq − p
2
− 1 − 3q/2e
i4λ
j
− p
2
1 q/2e
i3λ
j
p
2
pqe
i2λ
j
p
2
e
iλ
j
. Since
λ
j
∈ R,thatis,λ
j
λ
j
, this leads to e
−i5λ
j
e
iλ
j
− 1
2
e
iλ
j
1
2
e
i2λ
j
1−pe
i4λ
j
p
2
1 − 2p −
q/2e
i3λ
j
2p
2
− 2p 2 qe
i2λ
j
p
2
1 − 2p − q/2e
iλ
j
− p0. We firstly claim that the
equation −px
4
p
2
1 − 2p − q/2x
3
2p
2
− 2p 2 qx
2
p
2
1 − 2p − q/2x − p 0
has no imaginary root, that is, equations x
2
a/2 −
a
2
/4 − b 2x 1 −
√
1 − a 0and
x
2
a/2
a
2
/4 − b 2x 1
√
1 − a 0 both have no imaginary roots, where a q/2 −
1−p
2
2p/p, b 2p −q −2−2p
2
/p. If these two equations have imaginary roots, then a 1,
b 4 −4p 1/p. Since p
/
0, |p|
/
1, then b<−4orb>12. If the first equation has imaginary
roots, then −4 <b≤ 9/4, which contradicts with b<−4orb>12. If the second equation has
imaginary roots, then 0 <b≤ 9/4, which also contradicts with b<−4orb> 12. The claim
follows. Thus −pe
i4λ
j
p
2
1−2p−q/2e
i3λ
j
2p
2
−2p2qe
i2λ
j
p
2
1−2p−q/2e
iλ
j
−p
/
0,
and e
iλ
j
±i. Substituting e
iλ
j
±i into e
iλ
j
−1 −iλ
j
e
iλ
j
e
−iλ
j
−2p
1
e
i2λ
j
−1
1 −q/2e
i6λ
j
1q/2e
i5λ
j
pq−p
2
−1−3q/2e
i4λ
j
−p
2
1q/2e
i3λ
j
p
2
pqe
i2λ
j
p
2
e
iλ
j
,wegetλ
j
0. This
is impossible. Thus, for any z
0
e
i2λ
j
∈ e
i2Λ
f
, we have az
0
; y
2n
/
0, that is, e
i2Λ
f
⊂ σ
b
y
2n
.
Noticing that σ
b
y
2n
⊂ e
i2Λ
f
, we have σ
b
y
2n
e
i2Λ
f
. The proof has finished.
In Lemma 2.6, we have given the expression of the almost periodic solution of 1.1
explicitly by a known function ω. This brings more convenience to study the spectrum
containment of almost periodic solution of 1.1. Now, we are in the position to show the
Main Theorem.
The proof of Main Theorem
Since H
1
is satisfied, by Lemma 2.6, 1.1 has a unique almost periodic solution xt
satisfying xtpxt−1ωt. Thus, for any λ ∈ R, we have aλ; ωt 1pe
−iλ
aλ; xt.
Since H
1
holds, then Λ
x
Λ
ω
. We only need to prove Λ
ω
⊂ Λ
f
{kπ, k ∈ Z} when H
1
is
satisfied, and Λ
f
{kπ, k ∈ Z} Λ
ω
when H
1
–H
3
are all satisfied.
Advances in Difference Equations 11
When H
1
is satisfied, we prove Λ
ω
⊂ Λ
f
{kπ, k ∈ Z} firstly. For any
λ
/
∈Λ
f
{kπ, k ∈ Z}, it follows from Lemmas 3.1–3.4 that e
i2λ
/
∈σ
b
x
2n
, e
i2λ
/
∈σ
b
x
2n1
and
e
i2λ
/
∈σ
b
y
2n
,thatis,ae
i2λ
; x
2n
ae
i2λ
; y
2n
ae
i2λ
; x
2n1
0. From the expression of ωt
given in Lemma 2.6,weknow
a
λ; ω
t
lim
N →∞
1
4N
N
j−N
2j1
2j−1
t
2j
s
2j
f
σ
e
−iλt
dσ ds dt. 3.10
As mentioned in Lemma 3.1, for any m>0,m∈ Z, there exists P
m
t
nm
k1
b
k,m
e
iλ
k
t
,λ
k
∈
Λ
f
such that P
m
t → ft, as m →∞. By simple calculation, we have
0 lim
N →∞
1
4N
N
j−N
2j1
2j−1
t
2j
s
2j
P
m
σ
e
−iλt
dσ ds dt, ∀m>0,m∈ Z. 3.11
Therefore, aλ; ωt 0, that is, λ
/
∈Λ
ω
, which implies Λ
ω
⊂ Λ
f
{kπ, k ∈ Z}.
Additionally, if H
2
and H
3
are also satisfied, to show the equality Λ
ω
Λ
f
{kπ, k ∈
Z}, we only need to show the inverse inclusion, that is, Λ
f
{kπ, k ∈ Z}⊂Λ
ω
. For any
m>0,m,n ∈ Z, define
h
m,2n
Q
1
m,2n
Q
2
m,2n
,
h
m,2n1
Q
1
m,2n1
Q
2
m,2n1
, {x
m,2n
}
p
1
A
−1
{g
m,n1
}, where g
m,n
h
m,2n3
2 − p
h
m,2n2
1 − 2p
h
m,2n1
− p
h
m,2n
, and define
y
m,2n
x
m,2n1
p − 1 −
q
2
x
m,2n
− p x
m,2n−1
− Q
1
m,2n
,
2p − 1
2
x
m,2n1
p
h
m,2n
− p
p − 2 − q
x
m,2n
− p
2
x
m,2n−2
2p − 1
h
m,2n1
−
2p − 1
1 −
q
2
x
m,2n2
−
2p − 1
1 − 2p −
q
2
x
m,2n
,
ω
m
t
x
m,2n
p x
m,2n−1
y
m,2n
t − 2n
q x
m,2n
t − 2n
2
2
t
2n
s
2n
P
m
σ
dσ ds,
3.12
t ∈ 2n − 1, 2n 1, then, {
h
m, 2n
}, {
h
m,2n1
}, {g
m,n
}, {x
m,2n
}, {x
m,2n1
}, {y
m,2n
}∈
APSR, ω
m
t ∈APR,andasm →∞, {
h
m,2n
}→{h
2n
}, {
h
m,2n1
}→{h
2n1
}, {g
m,n
}→
{g
n
}, {x
m,2n
}→{x
2n
}, {x
m,2n1
}→{x
2n1
}, {y
m, 2n
}→{y
2n
} in APSR, ω
m
t → ωt
in APR, which implies f or any λ ∈ R, aλ; ω
m
t → aλ; ωt as m →∞, where
12 Advances in Difference Equations
Q
i
m,2n
,Q
i
m,2n1
,andP
m
t are as in Lemma 3.1, i 1, 2. Similarly as above, for all z ∈ S
1
,
we can get
a
z;
h
m,2n
a
z; Q
1
m,2n
a
z; Q
2
m,2n
,
a
z;
h
m,2n1
a
z; Q
1
m,2n1
a
z; Q
2
m,2n1
,
a
z, y
m,2n
1 − pz
−1
a
z, x
m,2n1
p − 1 −
q
2
a
z, x
m,2n
− a
z, Q
1
m,2n
,
2p − 1
2
a
z, x
m,2n1
pa
z,
h
m,2n
2p − 1
a
z,
h
m,2n1
− z
−1
p
2
z
a
z, x
m,2n
,
a
z; g
m,n
z
−1
p
1
z
a
z; x
m,2n
z 1 − 2p
a
z;
h
m,2n1
2z − pz − p
a
z;
h
m,2n
.
3.13
We claim: Λ
f
⊂ Λ
ω
. Suppose that the claim is false, then there would exist λ
j
∈ Λ
f
such that
λ
j
/
∈Λ
ω
. Noticing H
2
and H
3
, an elementary calculation leads to
2a
λ
j
; ω
m
2λ
2
j
qλ
2
j
− 2q
e
iλ
j
− e
−iλ
j
i2qλ
j
e
iλ
j
e
−iλ
j
2iλ
3
j
a
e
i2λ
j
; x
m,2n
pe
−2iλ
j
e
iλ
j
− e
−iλ
j
iλ
j
a
e
i2λ
j
; x
m,2n1
iλ
j
e
iλ
j
e
−iλ
j
λ
2
j
a
e
i2λ
j
; y
m,2n
e
−iλ
j
− e
iλ
j
λ
2
j
a
e
i2λ
j
; y
m,2n
−2iλ
j
2
e
iλ
j
− e
−iλ
j
− iλ
j
e
iλ
j
e
−iλ
j
iλ
3
j
b
j,m
.
3.14
From the above equality, we have −λ
2
j
p
1
e
i2λ
j
2aλ
j
; ω
m
b
j,m
c
1
, where, c
1
2p
1
e
i2λ
j
−
qe
iλ
j
− 1
3
e
iλ
j
1
3
e
iλ
j
− p/iλ
3
j
. So, −λ
2
j
p
1
e
i2λ
j
2aλ
j
; ωt aλ
j
; fc
1
. Since λ
j
/
∈Λ
ω
,
aλ
j
; ω0, we have c
1
0, which is equivalent to 2iλ
3
j
qe
iλ
j
− 1
3
e
iλ
j
1
3
e
iλ
j
−
p/p
1
e
i2λ
j
. Since λ
j
∈ R,thatis,λ
j
λ
j
, this leads to pe
iλ
j
− 1
3
e
iλ
j
1pe
i4λ
j
e
i3λ
j
q/2−1−
p
2
2pe
i2λ
j
−q−2−2p
2
2pe
iλ
j
q/2−1−p
2
2pp0. Noticing λ
j
∈ Λ
f
, it follows from
H
2
that e
iλ
j
/
±1, that is, e
iλ
j
− 1
3
e
iλ
j
1
/
0. From Lemma 3.4, we know that the equation
px
4
x
3
q/2 −1 −p
2
2px
2
−q −2 −2p
2
2pxq/2 −1−p
2
2pp 0 has no imaginary
root. Thus pe
i4λ
j
e
i3λ
j
q/2 −1 −p
2
2pe
i2λ
j
−q −2 −2p
2
2pe
iλ
j
q/2 −1 −p
2
2pp
/
0,
which leads to a contradiction. The claim follows.
Advances in Difference Equations 13
NowweareabletoproveΛ
f
{kπ, k ∈ Z}⊂Λ
ω
. For any λ
j
0
∈ Λ
f
let λ
j
λ
j
0
kπ, 0
/
k ∈ Z. Noticing H
2
and H
3
, e
iλ
j
0
/
±1,e
iλ
j
0
/
p, and we have
2a
λ
j
; ω
m
2λ
2
j
qλ
2
j
− 2q
e
iλ
j
− e
−iλ
j
i2qλ
j
e
iλ
j
e
−iλ
j
2iλ
3
j
a
e
i2λ
j
; x
m,2n
pe
−2iλ
j
e
iλ
j
− e
−iλ
j
iλ
j
a
e
i2λ
j
; x
m,2n1
iλ
j
e
iλ
j
e
−iλ
j
λ
2
j
a
e
i2λ
j
; y
m,2n
e
−iλ
j
− e
iλ
j
λ
2
j
a
e
i2λ
j
; y
m,2n
e
−iλ
j
− e
iλ
j
−iλ
j
λ
2
j
0
−
iλ
j
e
iλ
j
e
−iλ
j
e
−iλ
j
− e
iλ
j
iλ
2
j
λ
j
0
b
j
0
,m
.
3.15
The above equality is equivalent to −λ
2
j
0
p
1
e
i2λ
j
2aλ
j
; ω
m
t −1
k
−qe
iλ
j
0
− 1
3
e
iλ
j
0
1
3
e
iλ
j
0
− p/iλ
3
j
b
j
0
,m
.So,−λ
2
j
0
p
1
e
i2λ
j
2aλ
j
; ωt −1
k
−qe
iλ
j
0
− 1
3
e
iλ
j
0
1
3
e
iλ
j
0
−
p/iλ
3
j
aλ
j
0
; f
/
0, which implies that λ
j
∈ Λ
ω
,thatis,Λ
f
{kπ, k
/
0}⊂Λ
ω
.Fromthe
claim above, we get Λ
f
{kπ, k ∈ Z}⊂Λ
ω
. This completes the proof.
References
1 K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” Journal
of Mathematical Analysis and Applications, vol. 99, no. 1, pp. 265–297, 1984.
2 S. M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument
deviations,” International J ournal of Mathematics and Mathematical Sciences, vol. 6, no. 4, pp. 671–703,
1983.
3 S. Busenberg and K. L. Cooke, “Models of vertically transmitted diseases with sequential-continuous
dynamics,” in Nonlinear Phenomena in Mathematical Sciences, V. Lakshmikantham, Ed., pp. 179–187,
Academic Press, New York, NY, USA, 1982.
4 G. Seifert, “Second-order neutral delay-differential equations with piecewise constant time depen-
dence,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 1–9, 2003.
5 D. X. Piao, “Almost periodic solutions of neutral differential difference equations with piecewise
constant arguments,” Acta Mathematica Sinica, vol. 18, no. 2, pp. 263–276, 2002.
6 H X. Li, “Almost periodic solutions of second-order neutral delay-differential equations with
piecewise constant arguments,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp.
693–709, 2004.
7 H X. Li, “Almost periodic weak solutions of neutral delay-differential equations with piecewise
constant argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 3, pp. 530–545,
2006.
8 E. A. Dads and L. Lhachimi, “New approach for the existence of pseudo almost periodic solutions
for some second order differential equation with piecewise constant argument,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 64, no. 6, pp. 1307–1324, 2006.
9 R. Yuan, “On the spectrum of almost periodic solution of second order scalar functional differential
equations with piecewise constant argument,” Journal of Mathematical Analysis and Applications, vol.
303, no. 1, pp. 103–118, 2005.
10 L. Wang, R. Yuan, and C. Zhang, “Corrigendum to: “On the spectrum of almost periodic solution of
second order scalar functional differential equations with piecewise constant argument”,” Journal of
Mathematical Analysis and Applications, vol. 349, no. 1, p. 299, 2009.
11 X. Yang and R. Yuan, “On the module containment of the almost periodic solution for a class
of differential equations with piecewise constant delays,” Journal of Mathematical Analysis and
Applications, vol. 322, no. 2, pp. 540–555, 2006.
14 Advances in Difference Equations
12 R. Yuan, “On the spectrum of almost periodic solution of second-order differential equations with
piecewise constant argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 8, pp.
1189–1205, 2004.
13 C. Corduneanu, Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no.
22, John Wiley & Sons, New York, NY, USA, 1968.
14 A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics,Springer,
Berlin, Germany, 1974.
15 B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge
University Press, Cambridge, UK, 1982.
16 C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, China; Kluwer
Academic Publishers, Dordrecht, The Netherlands, 2003.
17 M. L. Cartwright, “Almost periodic differential equations and almost periodic flows,” Journal of
Differential Equations, vol. 5, no. 1, pp. 167–181, 1969.
18 R. Yuan, “Pseudo-almost periodic solutions of second-order neutral delay differential equations with
piecewise constant argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 41, no. 7-8, pp.
871–890, 2000.
