Báo cáo hóa học: "Research Article Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay" pdf

Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2007, Article ID 68023, 15 pages
doi:10.1155/2007/68023
Research Article
Periodic and Almost Periodic Solutions of Functional
Difference Equations with Finite Delay
Yihong Song
Received 4 November 2006; Revised 29 January 2007; Accepted 29 January 2007
Recommended by John R. Graef
For periodic and almost periodic functional difference equations with finite delay, the ex-
istence of periodic and almost periodic solutions is obtained by using stability properties
of a bounded solution.
Copyright © 2007 Yihong Song. This is an open access article distributed under the Cre-
ative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we study periodic and almost periodic solutions of the following functional
difference equations with finite delay:
x(n +1)
= F

n,x
n

, n ≥ 0, (1.1)
under certain conditions for F(n,
·) (see below), where n, j,andτ are integers, and x
n
will
denote the function x(n + j), j

=−τ,−τ +1, ,0.
Equation (1.1) can be regarded as the discrete analogue of the following functional
differential equation with bounded delay:
dx
dt
= Ᏺ

t,x
t

, t ≥ 0, x
t
(0) = x(t +0)= φ(t), −σ ≤ t ≤ 0. (1.2)
Almost periodic solutions of (1.2) have been discussed in [1]. The aim of this paper is to
extend results in [1]to(1.1).
Delay difference equations or functional difference equations (no matter with finite or
infinite delay), inspired by the development of the study of delay differential equations,
have been studied extensively in the past few decades (see, [2–11], to mention a few, and
2AdvancesinDifference Equations
references therein). Recently, several papers [12–17] are devoted to study almost periodic
solutions of difference equations. To the best of our knowledge, little work has been done
on almost periodic solutions of nonlinear functional difference equations with finite de-
lay via uniform stability properties of a bounded solution. This motivates us to investigate
almost periodic solutions of (1.1).
This paper is organized as follows. In Section 2, we review definitions of almost pe-
riodic and asymptotically almost periodic sequences and present some related proper-
ties for our purposes and some stability definitions of a bounded solution of (1.1). In
Section 3, we discuss the existence of periodic solutions of (1.1). In Section 4, we discuss
the existence of almost periodic solutions of (1.1).
2. Preliminaries

We formalize our notation. Denote by
Z, Z
+
, Z
−
, respectively, the set of integers, the set
of nonnegative integers, and the set of nonpositive integers. For any a
∈ Z,letZ
+
a
={n :
n
≥ a, n ∈ Z}.Foranyintegersa<b,letdis[a,b] ={j : a ≤ j ≤ b, j ∈ Z} and dis(a,b] =
{
j : a<j≤ b, j ∈ Z} be discrete intervals of integers. Let E
d
denote either R
d
,thed-
dimensional real Euclidean space, or
C
d
,thed-dimensional complex Euclidean space. In
the following, we use
|·|to denote a norm of a vector in E
d
.
2.1. Almost periodic sequences. We review definitions of (uniformly) almost periodic
and asymptotically almost periodic sequences, which have been discussed by several au-
thors (see, e.g., [2, 18]), and present some related properties for our purposes. For almost

periodic and asymptotically almost periodic functions, we recommend [19, 18].
Let X and Y betwoBanachspaceswiththenorm
·
X
and ·
Y
, respectively. Let Ω
be a subset of X.
Definit ion 2.1. Le t f :
Z × Ω → Y and f (n,·) be continuous for each n ∈ Z.Then f is said
to be almost periodic in n
∈ Z uniformly for w ∈ Ω if for every ε>0andeverycompact
Σ
⊂ Ω corresponds an integer N
ε
(Σ) > 0suchthatamongN
ε
(Σ) consecutive integers there
is one, call it p,suchthat


f (n + p,w) − f (n,w)


Y
<ε ∀n ∈ Z, w ∈ Σ. (2.1)
Denote by Ꮽᏼ(
Z × Ω : Y) the set of all such functions. We may call f ∈ Ꮽᏼ(Z × Ω : Y)
a (uniformly) almost periodic sequence in Y.IfΩ is an empt y set and Y
= X,then f ∈

Ꮽᏼ(Z : X) is called an almost periodic sequence in X.
Almost periodic sequences can be also defined for any sequence
{ f (n)}
n≥a
,or f : Z
+
a
→
X by requiring that any N
ε
(Σ) consecutive integers is in Z
+
a
.
For unifor mly almost periodic sequences, we have the following results.
Theorem 2.2. Let f
∈ Ꮽᏼ(Z × Ω : Y) and le t Σ be any compact set in Ω. Then f (n,·) is
continuous on Σ uniformly for n
∈ Z and the range f (Z × Σ) is relatively compact, which
implies that f (
Z × Σ) is a bounded subset in Y.
Yihong Song 3
Theorem 2.3. Let f
∈ Ꮽᏼ(Z × Ω : Y). Then for any integer sequence {α

k
}, α

k
→∞as k →

∞
, there exists a subsequence {α
k
} of {α

k
}, α
k
→∞as k →∞,andafunctionξ : Z × Ω → Y
such that
f

n + α
k
,w

−→
ξ(n,w) (2.2)
uniformly on
Z × Σ as k →∞,whereΣ isanycompactsetinΩ.Moreover,ξ ∈ Ꮽᏼ(Z × Ω :
Y),thatis,ξ(n,w) is also almost periodic in n uniformly for w
∈ Ω.
If Ω is the empty set and Y
= X in Theorem 2.3,then{ξ(n)} is an almost periodic
sequence.
Theorem 2.4. If f
∈ Ꮽᏼ(Z × Ω : Y), then there exists a sequence {α
k
}, α
k

→∞as k →∞,
such that
f

n + α
k
,w

−→
f (n,w) (2.3)
uniformly on
Z × Σ as k →∞,whereΣ is any compact set in Ω.
Obviously,
{α
k
} in Theorem 2.4 can be chosen to be a positive integer sequence.
Definit ion 2.5. Asequence
{x(n)}
n∈Z
+
, x(n) ∈ X, or a function x : Z
+
→ X,iscalled
asymptotically almost periodic if x
= x
1
|
Z
+
+ x

2
,wherex
1
∈ Ꮽᏼ(Z,X)andx
2
: Z
+
→ X
satisfying
x
2
(n)
X
→ 0asn →∞. Denote by ᏭᏭᏼ(Z
+
,X) all such sequences.
Theorem 2.6. Let x :
Z
+
→ X. T hen the following statements are equivalent.
(1) x
∈ ᏭᏭᏼ(Z
+
,X).
(2) For any sequence
{α
k
}⊂Z
+
, α

k
> 0, and α
k
→∞as k →∞,thereisasubsequence
{β
k
}⊂{α
k
} such that β
k
→∞ as k →∞ and {x(n + β
k
)} converges uniformly on Z
+
as
k
→∞.
Similarly, asymptotically almost periodic sequence can be defined for any sequence
{x( n)}
n≥a
,orx : Z
+
a
→ X.
The proof of the above results is omitted here because it is not difficult for readers
giving proofs by the similar arguments in [19, 18] for continuous (uniformly) almost
periodic function φ :
R × Ω → X (see also [2]forthecasethatX = Y =
E
d

).
2.2. Some assumptions and stability definitions. We now present some definitions and
notations that will be used throughout this paper. For a given positive integer τ>0, we
define C tobeaBanachspacewithanorm
·by
C
=

φ | φ : dis[−τ,0] −→ E
d
,φ=max



φ( j)



for j ∈ dis[−τ,0]

. (2.4)
It is clear that C is isometric to the space
E
d×(τ+1)
.
Let n
0
∈ Z
+
and let {x(n)}, n ≥ n

0
− τ,beasequencewithx(n) ∈ E
d
.Foreachn ≥ n
0
,
we define x
n
: dis[−τ,0] → E
d
by the relation
x
n
( j) = x(n + j), j ∈ dis[−τ,0]. (2.5)
4AdvancesinDifference Equations
Let us return to system (1.1), that is,
x(n +1)
= F

n,x
n

, (2.6)
where F :
Z × C → E
d
and x
n
: dis[−τ,0] → C.
Definit ion 2.7. Let n

0
∈ Z
+
and let φ be a given vector in C.Asequencex ={x(n)}
n≥n
0
in
E
d
is said to be a solution of (2.6), passing through (n
0
,φ), if x
n
0
= φ, that is, x(n
0
+ j) =
φ( j)forj ∈ dis[−τ,0], x(n +1), and x
n
satisfy (2.6)forn ≥ n
0
,wherex
n
is defined by
(2.5). Denote by
{x( n, φ)}
n≥n
0
asolutionof(2.6)suchthatx
n

0
= φ.Nolossofclarity
arises i f we refer to the solution
{x( n, φ)}
n≥n
0
as x ={x(n)}
n≥n
0
.
We make the following assumptions on (2.6) throughout this paper.
(H1) F :
Z × C → E
d
and F(n,·)iscontinuousonC for each n ∈ Z.
(H2) System (2.6) has a bounded solution u
={u(n)}
n≥0
, passing through (0,φ
0
), φ
0
∈
C.
For this bounded solution
{u(n)}
n≥0
, there is an α>0suchthat|u(n)|≤α for all n ≥−τ,
which implies that
u

n
≤α and u
n
∈ S
α
={φ : φ≤α and φ ∈ C} for all n ≥ 0.
Definit ion 2.8. A bounded solution
x ={x(n)}
n≥0
of (2.6)issaidtobe
(i) uniformly stable, abbreviated to read “
x is ᐁ᏿,” i f f o r any ε>0 and any integer
n
0
≥ 0, there exists δ(ε) > 0suchthatx
n
0
− x
n
0
 <δ(ε) implies that x
n
− x
n
 <ε
for all n
≥ n
0
,where{x(n)}
n≥n

0
is any solution of (2.6);
(ii) uniformly asymptotically stable, abbreviated to read “
x is ᐁᏭ᏿,” if it is uni-
formly stable and there exists δ
0
> 0 such that for any ε>0, there is a positive
integer N
= N(ε) > 0suchthatifn
0
≥ 0andx
n
0
− x
n
0
 <δ
0
,thenx
n
− x
n
 <ε
for all n
≥ n
0
+ N,where{x( n)}
n≥n
0
is any solution of (2.6);

(iii) globally uniformly asymptotically stable, abbreviated to read “
x is ᏳᐁᏭ᏿,” if
it is uniformly stable and
x
n
− x
n
→0asn →∞,whenever{x(n)}
n≥n
0
is any
solution of (2.6).
Remark 2.9. It is easy to see that an equivalent definition for
x ={x(n)}
n≥0
being ᐁᏭ᏿ is
the following:
(ii
∗
) x ={x(n)}
n≥0
is ᐁᏭ᏿, if it is uniformly stable and there exists δ
0
> 0suchthat
if n
0
≥ 0andx
n
0
− x

n
0
 <δ
0
,thenx
n
− x
n
→0asn →∞,where{x(n)}
n≥n
0
is
any solution of (2.6).
3. Periodic systems
In this section, we discuss the existence of periodic solutions of (2.6), namely,
x(n +1)
= F

n,x
n

, n ≥ 0, (3.1)
under a periodic condition (H3) as follows.
(H3) The F(n,
·)in(3.1)isperiodicinn ∈ Z, that is, there exists a positive integer ω
such that F(n +ω,v)
= F(n,v)foralln ∈ Z and v ∈ C.
Yihong Song 5
We are now in a position to give our main results in this section. We first show that if
the bounded solution

{u(n)}
n≥0
of (3.1) is uniformly stable, then {u(n)}
n≥0
is an asymp-
totically almost periodic sequence.
Theorem 3.1. Suppose conditions (H1)–(H3) hold. If the bounded solution
{u(n)}
n≥0
of
(3.1)isᐁ᏿, the n
{u(n)}
n≥0
is an asymptotically almost periodic sequence in E
d
,equiva-
lently, (3.1) has an asymptotically almost periodic solution.
Proof. Since
u
n
≤α for n ∈ Z
+
, there is bounded (or compact) set S
α
⊂ C such that
u
n
∈ S
α
for all n ≥ 0. Let {n

k
}
k≥1
be any integer sequence such that n
k
> 0andn
k
→∞as
k
→∞.Foreachn
k
, there exists a nonnegative integer l
k
such that l
k
ω ≤ n
k
≤ (l
k
+1)ω.
Set n
k
= l
k
ω + τ
k
.Then0≤ τ
k
<ωfor all k ≥ 1. Since {τ
k

}
k≥1
is bounded set, we can
assume that, taking a subsequence if necessary, τ
k
= j
∗
for all k ≥ 1, where 0 ≤ j
∗
<ω.
Now, set u
k
(n) = u(n + n
k
). Notice that u
n+n
k
( j) = u(n + n
k
+ j) = u
k
(n + j) = u
k
n
( j)and
hence, u
n+n
k
= u
k

n
.Thus,
u
k
(n +1)= u

n + n
k
+1

=
F

n + n
k
,u
n+n
k

=
F

n + n
k
,u
k
n

=
F


n + j
∗
,u
k
n

, (3.2)
which implies that
{u
k
(n)} is a solution of the system
x(n +1)
= F

n + j
∗
,x
n

(3.3)
through (0,u
n
k
). It is readily shown that if {u(n)}
n≥0
is ᐁ᏿,then{u
k
(n)}
n≥0

is also ᐁ᏿
with the same pair (ε,δ(ε)) as the one for
{u(n)}
n≥0
.
Since
{u(n + n
k
)} is bounded for al l n ≥−τ and n
k
, we can use the diagonal method to
get a subsequence
{n
k
j
} of {n
k
} such that u(n + n
k
j
)convergesforeachn ≥−τ as j →∞.
Thus, we can assume that the sequence u(n + n
k
)convergesforeachn ≥−τ as k →∞.
Notice that u
k
0
( j) = u
k
(0 + j) = u(j + n

k
). Then for any ε>0 there exists a positive integer
N
1
(ε)suchthatifk,m ≥ N
1
(ε), then


u
k
0
− u
m
0


<δ(ε), (3.4)
where δ(ε) is the number for the uniform stability of
{u(n)}
n≥0
. Notice that {u
m
(n) =
u(n + n
m
)}
n≥0
is also a solution of (3.3) and that {u
k

(n)}
n≥0
is uniformly stable. It follows
from Definition 2.8 and (3.4)that


u
k
n
− u
m
n


<ε ∀n ≥ 0, (3.5)
and hence,


u
k
(n) − u
m
(n)


<ε ∀n ≥ 0, k, m ≥ N
1
(ε). (3.6)
This implies that for any positive integer sequence n
k

, n
k
→∞as k →∞, there exists a
subsequence
{n
k
j
} of {n
k
} for which {u(n + n
k
j
)} converges uniformly on Z
+
as j →∞.
Thus,
{u(n)}
n≥0
is an asymptotically almost periodic sequence by Theorem 2.6 and the
proof is completed.

6AdvancesinDifference Equations
Lemma 3.2. Suppose that (H1)–(H3) hold and
{u(n)}
n≥0
, the bounded solution of (3.1), is
ᐁ᏿.Let
{n
k
}

k≥1
be an integer sequence such that n
k
> 0, n
k
→∞as k →∞, u(n + n
k
) →
η(n) for each n ∈ Z
+
and F(n + n
k
,v) → G(n,v) uniformly for n ∈ Z
+
and Σ as k →∞,
where Σ isanycompactsetinC. Then
{η(n)}
n≥0
is a solution of the system
x(n +1)
= G

n,x
n

, n ≥ 0, (3.7)
and is ᐁ᏿.Moreover,if
{u(n)}
n≥0
is ᐁᏭ᏿, then {η(n)}

n≥0
is also ᐁᏭ᏿.
Proof. Since u
k
(n) = u(n + n
k
)isuniformlyboundedforn ≥−τ and k ≥ 1, we can a s-
sume that, taking a subsequence if necessary, u(n + n
k
)alsoconvergesforeachn ∈ dis
[
−τ, −1]. Define η( j) = lim
k→∞
u( j + n
k
)for j ∈ dis[−τ,−1]. Then u(n + n
k
) → η(n)for
each n
∈ dis[−τ,∞), and hence, u
k
n
→ η
n
as k →∞for each n ≥ 0. Notice that u
k
n
∈ S
α
for all n ≥ 0, k ≥ 1, and η

n
∈ S
α
for n ≥ 0. It follows from Theorem 2.4 that there exists
asubsequence
{n
k
j
} of {n
k
}, n
k
j
→∞as j →∞,suchthatF(n + n
k
j
,v) → G(n,v)uni-
formly on
Z × S
α
as j →∞and G(n, ·)iscontinuousonS
α
uniformly for all n ∈ Z.Since
u(n + n
k
j
+1)= F(n + n
k
j
,u

k
j
n
)and
F

n + n
k
j
,u
k
j
n

−
G

n,η
n

=
F

n + n
k
j
,u
k
j
n


−
G

n,u
k
j
n

+ G

n,u
k
j
n

−
G

n,η
n

−→
0asj −→ ∞ ,
(3.8)
we have η(n +1)
= G(n,η
n
)(n ≥ 0). This shows that {η(n)}
n≥0

is a solution of (3.7).
To pro v e th at
{η(n)}
n≥0
is ᐁ᏿,wesetn
k
= l
k
ω + j
∗
as before, where 0 ≤ j
∗
<ω.Then
u
k
j
(n) = u(n + n
k
j
) → η(n)foreachn ∈ Z
+
as j →∞.SinceF(n + n
k
j
,v) = F(n + j
∗
,v) →
G(n,v)asj →∞,wehaveG(n, v) = F(n + j
∗
,v). For any ε>0, let δ(ε) > 0 be the one for

uniform stability of
{u(n)}
n≥0
.Foranyn
0
∈ Z
+
,let{x(n)}
n≥0
beasolutionof(3.7)such
that
η
n
0
− x
n
0
=μ<δ(ε). Since u
k
j
n
→ η
n
as j →∞ for each n ≥ 0, there is a positive
integer J
1
> 0suchthatif j ≥ J
1
,then



u
k
j
n
0
− η
n
0


<δ(ε) − μ. (3.9)
Thus, for j
≥ J
1
,wehave


u
n
0
+j
∗
+l
k
j
ω
− x
n
0



≤


u
n
0
+j
∗
+l
k
j
ω
− η
n
0


+


η
n
0
− x
n
0



<δ(ε). (3.10)
Notice that
{u(n + j
∗
+ l
k
j
ω)} (n ≥ 0) is a uniformly stable solution of (3.7)withG(n,x
n
)
= F(n + j
∗
,x
n
). Then,


u
n+ j
∗
+l
k
j
ω
− x
n


<ε ∀n ≥ n
0

. (3.11)
Since
{η(n)} is also a solution of (3.7)andu
k
j
n
→ η
n
for each n ≥ 0asj →∞,foran
arbitrary ν > 0, there is J
2
> 0suchthatif j ≥ J
2
,then


η
n
0
− u
n
0
+j
∗
+l
k
j
ω



<δ(ν), (3.12)
Yihong Song 7
and hence,
η
n
− u
n+ j
∗
+l
k
j
ω
 < ν for all n ≥ n
0
,where(ν,δ(ν)) is a pair for the uniform
stability of u(n + j
∗
+ l
k
j
ω). This shows that if j ≥ max(J
1
,J
2
), then


η
n
− x

n


≤


η
n
− u
n+ j
∗
+l
k
j
ω


+


u
n+ j
∗
+l
k
j
ω
− x
n



<ε+ ν (3.13)
for all n
≥ n
0
, which implies that η
n
− x
n
≤ε for all n ≥ n
0
if η
n
0
− x
n
0
 <δ(ε) because
ν is arbitrary. This proves that
{η(n)}
n≥0
is uniformly stable.
To pr o ve t ha t
{η(n)}
n≥0
is ᐁᏭ᏿, we use definition (ii
∗
)inRemark 2.9.Let{x(n)} be
asolutionof(3.7)suchthat
η

n
0
− x
n
0
 <δ
0
,whereδ
0
is the number for the uniformly
asymptotic stability of
{u(n)}. Notice that u(n + j
∗
+ l
k
j
ω) = u
k
j
(n)isauniformlyasymp-
totically stable solution of (3.7)withG(n,φ)
= F(n + j
∗
,φ) and with the same δ
0
as the
one for
{u(n)}.Setη
n
0

− x
n
0
=μ<δ
0
.Again,forsufficient large j, we have the simi-
lar relations (3.10)and(3.12)with
u
n
0
+j
∗
+l
k
j
ω
− x
n
0
 <δ
0
and u
n
0
+j
∗
+l
k
j
ω

− η
n
0
 <δ
0
.
Thus,


η
n
− x
n


≤


η
n
− u
n+ j
∗
+l
k
j
ω


+



u
n+ j
∗
+l
k
j
ω
− x
n


−→
0 (3.14)
as n
→∞ if u
n
0
− x
n
0
 <δ
0
, because {u
k
j
(n)}, {x(n)},and{η(n)} satisfy (3.7)with
G(n,φ)
= F(n + j

∗
,φ). This completes the proof. 
Using Theorem 3.1 and Lemma 3.2, we can show that (3.1) has an almost periodic
solution.
Theorem 3.3. If the bounded solution
{u(n)}
n≥0
of (3.1)isᐁ᏿,thensystem(3.1) has an
almost periodic solution, which is also ᐁ᏿.
Proof. It follows from Theorem 3.1 that
{u(n)}
n≥0
is asymptotically almost periodic. Set
u(n)
= p(n)+q(n)(n ≥ 0), where {p(n)}
n≥0
is almost periodic sequence and q(n) → 0
as n
→∞. For positive integer sequence {n
k
ω}, there is a subsequence {n
k
j
ω} of {n
k
ω}
such that p(n +n
k
j
ω) → p

∗
(n)uniformlyonZ as j →∞and {p
∗
(n)} is almost periodic.
Then u( n + n
k
j
ω) → p
∗
(n) uniformly for n ≥−τ, and hence, u
n+n
k
j
ω
→ p
∗
n
for all n ≥ 0as
j
→∞.Since
u

n + n
k
j
ω +1

=
F


n + n
k
j
ω,u
n+n
k
j
ω

=
F

n,u
n+n
k
j
ω

−→
F

n, p
∗
n

(3.15)
as j
→∞,wehavep
∗
(n +1)= F(n, p

∗
n
)forn ≥ 0, that is, system (3.1) has an almost
periodic solution, which is also ᐁ᏿ by Lemma 3.2.

Now, we show that if the bounded solution {u(n)} is uniformly asymptotically stable,
then (3.1) has a periodic solution of period mω for some positive integer m.
Theorem 3.4. If the bounded solution
{u(n)}
n≥0
of (3.1)isᐁᏭ᏿,thensystem(3.1)hasa
periodic solution of period mω for some positive integer m,whichisalsoᐁᏭ᏿.
Proof. Set u
k
(n) = u(n + kω), k = 1, 2, BytheproofofTheorem 3.1, there is a subse-
quence
{u
k
j
(n)} converges to a solution {η(n)} of (3.3)foreachn ≥−τ and hence, u
k
j
0
→
η
0
as j →∞. Thus, there is a positive integer p such that u
k
p
0

− u
k
p+1
0
 <δ
0
(0 ≤ k
p
<k
p+1
),
where δ
0
is the one for uniformly asymptotic stability of {u(n)}
n≥0
.Letm = k
p+1
− k
p
8AdvancesinDifference Equations
and notice that u
m
(n) = u(n + mω) is a solution of (3.1). Since u
m
k
p
ω
( j) = u
m
(k

p
ω + j) =
u(k
p+1
ω + j) = u
k
p+1
ω
( j)for j ∈ dis[−τ,0], we have


u
m
k
p
ω
− u
k
p
ω


=


u
k
p+1
ω
− u

k
p
ω


=


u
k
p+1
0
− u
k
p
0


<δ
0
, (3.16)
and hence,


u
m
n
− u
n



−→
0asn −→ ∞ (3.17)
because
{u(n)}
n≥0
is ᐁᏭ᏿ (see also Remark 2.9). On the other hand, {u(n)}
n≥0
is asymp-
totically almost periodic by Theorem 3.1,then
u(n)
= p(n)+q(n), n ≥ 0, (3.18)
where
{p(n)}
n∈Z
is almost periodic and q(n) → 0asn →∞.Itfollowsfrom(3.17)and
(3.18)that


p(n) − p(n + mω)


−→
0asn −→ ∞ , (3.19)
which implies that p(n)
= p(n + mω)foralln ∈ Z because {p(n)} is almost periodic.
For integer sequence
{kmω}, k = 1,2, ,wehaveu(n + kmω) = p(n)+q(n + kmω).
Then u(n + kmω)
→ p(n) uniformly for all n ≥−τ as k →∞, and hence, u

n+kmω
→ p
n
for n ≥ 0ask →∞.Sinceu(n + kmω +1)= F(n,u
n+kmω
), we have p(n +1)= F(n, p
n
)
for n
≥ 0, which implies that (3.1) has a periodic solution {p(n)}
n≥0
of period mω.The
uniformly asymptotic stability of
{p(n)}
n≥0
follows from Lemma 3.2. 
Finally, we show that if the bounded solution {u(n)} is ᏳᐁᏭ᏿,then(3.1)hasaperi-
odic solution of period ω.
Theorem 3.5. If the bounded solution
{u(n)}
n≥0
of (3.1)isᏳᐁᏭ᏿,thensystem(3.1)has
a periodic solution of period ω.
Proof. By Theorem 3.1,
{u(n)}
n≥0
is asymptotically almost periodic. Then u(n) = p(n)+
q(n)(n
≥ 0), where {p(n)} (n ∈ Z) is an almost periodic sequence and q(n) → 0asn →∞.
Notice that u(n + ω)isalsoasolutionof(3.1) satisfying u

ω
∈ S
α
.Since{u(n)} is ᏳᐁᏭ᏿,
we have
u
n
− u
n+ω
→0asn →∞, which implies that p(n) = p(n + ω)foralln ∈ Z.By
the same technique in the proof of Theorem 3.4, we can show that
{p(n)} is an ω-periodic
solution of (3.1).

4. Almost periodic systems
In this section, we discuss the existence of asymptotically almost periodic solutions of
(2.6), that is,
x(n +1)
= F

n,x
n

, n ≥ 0, (4.1)
under the condition (H4) as fol lows.
(H4) F
∈ Ꮽᏼ(Z × C : E
d
), that is, F(n,v) is almost periodic in n ∈ Z uniformly for
v

∈ C.
Yihong Song 9
By H(F) we denote the uniform closure of F(n,v), that is, G
∈ H(F) if there is an integer
sequence
{α
k
} such that α
k
→∞and F(n + α
k
,v) → G(n,v)uniformlyonZ × Σ as k →∞,
where Σ is any compact set in C.NotethatH(F)
⊂ Ꮽᏼ(Z × C : E
d
)byTheorem 2.3 and
F
∈ H(F)byTheorem 2.4.
We first show that if (4.1) has a bounded asymptotically almost periodic solution,
then (4.1) has an almost periodic solution. In fact, we have a more general result in the
following.
Theorem 4.1. Suppose (H1), (H2), and (H4) hold. If the bounded solution
{u(n)}
n≥0
of
(4.1) is asymptotically almost periodic, then for any G
∈ H(F), the system
x(n +1)
= G


n,x
n

(4.2)
has an almost periodic solution for n
≥ 0. Consequently, (4.1) has an almost periodic solu-
tion.
Proof. Since the solution
{u(n)}
n≥0
is asymptotically almost periodic, it follows from
Theorem 2.6 that it has the decomposition u(n)
= p(n)+q(n)(n ≥ 0), where {p(n)}
n∈Z
is almost periodic and q(n) → 0asn →∞. Notice that {u(n)} is bounded. There is com-
pact set S
α
∈ C such that u
n
∈ S
α
and p
n
∈ S
α
for all n ≥ 0. For any G ∈ H(F), there is
an integer sequence
{n
k
}, n

k
> 0, such that n
k
→∞as k →∞and F(n + n
k
,v) → G(n,v)
uniformly on
Z × S
α
as k →∞. Taking a subsequence if necessary, we can also assume
that p(n + n
k
) → p
∗
(n)uniformlyonZ and {p
∗
(n)} is also an almost periodic sequence.
For any j
∈ dis[−τ,0], there is positive integer k
0
such that if k>k
0
,then j + n
k
≥ 0for
any j
∈ dis[−τ,0]. In this case, we see that u(n + n
k
) → p
∗

(n) uniformly for all n ≥−τ as
k
→∞, and hence, u
n+n
k
→ p
∗
n
in C uniformly for n ∈ Z
+
as k →∞.Since
u

n + n
k
+1

=
F

n + n
k
,u
n+n
k

=

F


n + n
k
,u
n+n
k

−
F

n + n
k
, p
∗
n

+

F

n + n
k
, p
∗
n

−
G

n, p
∗

n

+ G

n, p
∗
n

,
(4.3)
the first term of right-hand side of (4.3)tendstozeroask
→∞ by Theorem 2.2 and
F(n + n
k
, p
∗
n
) − G(n, p
∗
n
) → 0ask →∞,wehavep
∗
(n +1)= G(n, p
∗
n
)foralln ∈ Z
+
,
which implies that (4.2) has an almost periodic solution
{p

∗
(n)}
n≥0
, passing through
(0, p
∗
0
), where p
∗
0
( j) = p
∗
( j)for j ∈ dis[−τ,0]. 
To deal with almost periodic solutions of (4.1) in terms of uniform stability, we assume
that for each G
∈ H(F), system (4.2) has a unique solution for a given initial condition.
Lemma 4.2. Suppose (H1), (H2), and (H4) hold. Let
{u(n)}
n≥0
be the bounded s olution
of (4.1). Let
{n
k
}
k≥1
be a positive integer sequence such that n
k
→∞, u
n
k

→ ψ, and F(n +
n
k
,v) → G(n,v) uniformly on Z × Σ as k →∞,whereΣ is any compact subset in C and
G
∈ H(F). If the bounded solution {u(n)}
n≥0
is ᐁ᏿, then the solution {η(n)}
n≥0
of (4.2),
through (0,ψ),isᐁ᏿. In addition, if
{u(n)}
n≥0
is ᐁᏭ᏿, then {η(n)}
n≥0
is also ᐁᏭ᏿.
Proof. Set u
k
(n) = u(n + n
k
). It is easy to see that u
k
(n) is a solution of
x(n +1)
= F

n + n
k
,x
n


, n ≥ 0, (4.4)
10 Advances in Difference Equations
passing though (0,u
n
k
)andu
k
n
∈ S
α
for all k,whereS
α
is compact subset of C such that
u
n
 <αfor all n ≥ 0. Since {u(n)}
n≥0
is ᐁ᏿, {u
k
(n)} is also ᐁ᏿ with the same pair
(ε,δ(ε)) as the one for
{u(n)}
n≥0
. Taking a subsequence if necessary, we can assume that
{u
k
(n)}
k≥1
converges to a vector η(n)foralln ≥ 0ask →∞.From(4.3)withp

∗
n
= η
n
,we
can see that
{η(n)}
n≥0
is the unique solution of (4.2), through (0,ψ).
To show that the solution
{η(n)}
n≥0
of (4.2)isᐁ᏿,weneedtoprovethatifforany
ε>0 and any integer n
0
≥ 0, there exists δ
∗
(ε) > 0suchthatη
n
0
− y
n
0
 <δ
∗
(ε) implies
that
η
n
− y

n
 <εfor all n ≥ n
0
,where{y(n)}
n≥n
0
is a solution of (4.2) passing through
(n
0
,φ)withy
n
0
= φ ∈ C.
For any given n
0
∈ Z
+
,ifk is sufficiently large, say k ≥ k
0
> 0, we have


u
k
n
0
− η
n
0



<
1
2
δ

ε
2

, (4.5)
where δ(ε) is the one for uniform stability of
{u(n)}
n≥0
.Letφ ∈ C such that


φ − η
n
0


<
1
2
δ

ε
2

(4.6)

and let
{x( n)}
n≥n
0
be the solution of (4.1)suchthatx
n
0
+n
k
= φ.Then{x
k
(n) = x(n + n
k
)}
is a solution of (4.4)withx
k
n
0
= φ.Since{u
k
(n)} is ᐁ᏿ and x
k
n
0
− u
k
n
0
 <δ(ε/2) for k ≥ k
0

,
we have


u
k
n
− x
k
n


<
ε
2
∀n ≥ n
0
, k ≥ k
0
. (4.7)
It follows from (4.7)that


x
k
n


≤



u
k
n


+
ε
2
<α+
ε
2
∀n ≥ n
0
, k ≥ k
0
. (4.8)
Then there exists a number α
∗
> 0suchthatx
k
n
∈ S
α
∗
for all n ≥ 0andk ≥ k
0
, which
implies that there is subsequence of
{x

k
(n)}
k≥k
0
for each n ≥ n
0
− τ, denoted by {x
k
(n)}
again, such that x
k
(n) → y(n)foreachn ≥ n
0
− τ, and hence, x
k
n
→ y
n
for all n ≥ n
0
as k →
∞
.Clearly,y
n
0
= φ and the set S
α
∗
is compact set in C.SinceF(n,v) is almost periodic in n
uniformly for v

∈ C, we can assume that, taking a subsequence if necessary, F(n + n
k
,v) →
G(n,v)uniformlyonZ × S
α
∗
as k →∞.Takingk →∞in x
k
(n +1)= F(n + n
n
k
,x
k
n
), we
have y(n +1)
= G(n, y
n
), namely, {y(n)} is the unique solution of (4.2), passing through
(n
0
,φ)withy
n
0
= φ ∈ C. On the other hand, for any integer N>0, there exists k
N
≥ k
0
such that if k ≥ k
N

,then


x
k
n
− y
n


<
ε
4
,


u
k
n
− η
n


<
ε
4
for n
0
≤ n ≤ n
0

+ N. (4.9)
From (4.7)and(4.9), we obtain


η
n
− y
n


<ε for n
0
≤ n ≤ n
0
+ N. (4.10)
Since N is arbitrary, we have
η
n
− y
n
 <εfor all n ≥ n
0
if φ − η
n
0
 < [δ(ε/2)]/2and
φ
∈ C, which implies that the solution {η(n)}
n≥0
of (4.2)isᐁ᏿.

Yihong Song 11
Now, we assume that
{u(n)}
n≥0
is ᐁᏭ᏿. Then the solution {u
k
(n)} of (4.4)isalso
ᐁᏭ᏿ with the same pair (δ
0
,ε,N(ε)) as the one for {u(n)}
n≥0
.Let(δ
∗
(ε),ε) be the pair
for uniform stability of
{η(n)}.
For any given n
0
∈ Z
+
,ifk is sufficiently large, say k ≥ k
0
> 0, we have


u
k
n
0
− η

n
0


<
1
2
δ
0
, (4.11)
where δ
0
is the one for uniformly asymptotic stability of {u(n)}
n≥0
.Letφ ∈ C such that
φ − η
n
0
 < (δ
0
/2) and let {x(n)}
n≥n
0
,foreachfixedk ≥ k
0
, be the solution of (4.1)such
that x
n
0
+n

k
= φ.Then{x
k
(n) = x(n + n
k
)} is a solution of (4.4)withx
k
n
0
= φ.Since{u
k
(n)}
is ᐁᏭ᏿ and x
k
n
0
− u
k
n
0
 < (δ
0
/2) for each fixed k ≥ k
0
,wehave


u
k
n

− x
k
n


<
ε
2
∀n ≥ n
0
+ N

ε
2

, k ≥ k
0
. (4.12)
By the same argument as the above, we can assume that, taking a subsequence if neces-
sary,
{x
k
(n)} converges to the solution {y(n)} of (4.2)through(n
0
,φ)andF(n + n
k
,v) →
G(n,v)uniformlyonZ × S
α
∗

as k →∞,whereS
α
∗
is compact set in C with |x
k
(n)|≤α
∗
for all k ≥ k
0
and n ≥ n
0
− τ.Then{y(n)} is the unique solution of (4.2), passing through
(n
0
,φ)withy
n
0
= φ ∈ C. On the other hand, for any integer N>0 there exists k
N
≥ k
0
such that if k ≥ k
N
,then


x
k
n
− y

n


<
ε
4
,


u
k
n
− η
n


<
ε
4
for n
0
+ N

ε
2

≤
n ≤ n
0
+ N


ε
2

+ N (4.13)
and hence,
y
n
− η
n
 <εfor n
0
+ N(ε/2) ≤ n ≤ n
0
+ N(ε/2) + N.SinceN is arbitrary, we
have


y
n
− η
n


<ε ∀n ≥ n
0
+ N

ε
2


(4.14)
if
φ − η
n
0
 < (δ
0
/2) and φ ∈ C.Theproofiscompleted. 
Before dealing with the asymptotic almost periodicity of {u(n)}, we need the following
lemma.
Lemma 4.3. Suppose that assumptions (H1), (H2), and (H4) hold, the bounded solution
{u(n) = u(n,ψ
0
)} of (4.1)isᐁᏭ᏿ and for each G ∈ H(F), the solution of (4.2)isuniquefor
any given initial data. Let S
⊇ S
α
be a given compact set in C. Then for any ε>0,thereexists
δ
= δ(ε) > 0 such that if n
0
≥ 0, u
n
0
− x
n
0
 <δ, and {h(n)} isasequencewith|h(n)|≤δ
for n

≥ n
0
, one has u
n
− x
n
 <εfor all n ≥ n
0
,where{x(n)} is any bounded solut ion of the
system
x(n +1)= F

n,x
n

+ h(n), n ≥ n
0
, (4.15)
passing through (n
0
,x
n
0
) such that x
n
∈ S for all n ≥ n
0
.
12 Advances in Difference Equations
Proof. Suppose that the bounded solution

{u(n)}
n≥0
of (4.1)isᐁᏭ᏿ with the triple
(δ(
·),δ
0
,N(·)). In order to establish Lemma 4.3 by a contradiction, we assume that
Lemma 4.3 is not true. Then for some compact set S
∗
⊇ S
α
, there exist an ε,0<ε<δ
0
,
sequences
{n
k
}⊂Z
+
, {r
k
}⊂Z
+
,mappingsequencesh
k
:dis[n
k
,∞) → E
d
, ϕ

k
: dis[n
k
−
τ, n
k
] → E
d
,and


u
n
k
− x
k
n
k


<
1
k
,


h
k
(n)



≤
1
k
for n
≥ n
k
,


u
n
− x
k
n


≤
ε for n
k
≤ n ≤ n
k
+ r
k
− 1,


u
n
k

+r
k
− x
k
n
k
+r
k


≥
ε
(4.16)
for sufficiently large k,where
{x
k
(n)} is a solution of
x(n +1)
= F

n,x
n

+ h
k
(n), n ≥ n
k
, (4.17)
passing through (n
k

,ϕ
k
)suchthatx
k
n
∈ S
∗
for all n ≥ n
k
and k ≥ 1. Since S
∗
is bounded
subset of C, it follows that
{x
k
(n
k
+ r
k
+ n)}
k≥1
and {x
k
(n
k
+ n)}
k≥1
are uniformly
bounded for al l n
k

and n ≥−τ.Wefirstconsiderthecasewhere{r
k
}
k≥1
contains an
unbounded subsequence. Set N
= N(ε) > 1. Taking a subsequence if necessary, we may
assume that there is G
∈ H(F)suchthatF(n + n
k
+ r
k
− N,v) → G(n, v)uniformlyon
Z
+
× S
∗
, x
k
(n + n
k
+ r
k
− N) → z(n), and u(n + n
k
+ r
k
− N) → w(n)forn ∈ Z
+
as k →∞,

where z,w :
Z
+
→ E
d
are some bounded functions. Since
x
k

n + n
k
+ r
k
− N +1

=
F

n + n
k
+ r
k
− N,x
k
n+n
k
+r
k
−N


+ h
k

n + n
k
+ r
k
− N

,
(4.18)
passing to limit as k
→∞, by the similar arguments in the proof of Theorem 4.1,we
conclude that
{z(n)}
n≥0
is the solution of the following equation:
x(n +1)
= G

n,x
n

, n ∈ Z
+
. (4.19)
Similarly,
{w(n)}
n∈Z
+

is also a solution of (4.19). Since x
k
n
k
+r
k
−N
( j) = x
k
(n
k
+r
k
− N + j) →
z( j) = z
0
( j)andu
n
k
+r
k
−N
( j) = u(n
k
+ r
k
− N + j) → w( j) = w
0
( j)ask →∞ for all j ∈
dis[−τ, 0], it follows from (4.16)thatw

0
− z
0
≤lim
k→∞
w
n
k
+r
k
−N
− z
n
k
+r
k
−N
≤ε<δ
0
.
Notice that
{w(n)}
n∈Z
+
is a s olution of (4.19), passing through (0,w
0
), and is ᐁᏭ᏿ by
Lemma 4.2.Wehave
w
N

− z
N
 <ε. On the other hand, since
u
n
k
+r
k
( j) = u

N + j + n
k
+ r
k
− N

−→
w(N + j) = w
N
( j),
x
k
n
k
+r
k
( j) = x
k

N + j + n

k
+ r
k
− j

−→
z(N + j) = z
N
( j)
(4.20)
as k
→∞for each j ∈ dis[−τ,0], it follows from (4.16)that


w
N
− z
N


=
lim
k→∞


u
n
k
+r
k

− x
k
n
k
+r
k


≥
ε. (4.21)
This is a contradiction. Thus, the sequence
{r
k
} must be bounded. Taking a subsequence
if necessary, we can assume that 0 <r
k
≡ r
0
< ∞. Moreover, we may assume that x
k
(n
k
+
n)
→ z(n)andu(n
k
+ n) → w(n)foreachn ≥−τ,andF(n + n
k
,v) →


G(n,v)uniformly
Yihong Song 13
on
Z × S
∗
, for some functions z(n), w(n)onZ
+
,and

G ∈ H(F). Since u
n
k
( j) = u(n
k
+
j)
→ w(j) =

w
0
( j)andx
k
n
k
( j) = x
k
(n
k
+ j) → z(j) =


z
0
( j)ask →∞for all j ∈ dis[−τ,0],
we have
 w
0
− z
0
=lim
k→∞
u
n
k
− x
k
n
k
=lim
k→∞
u
n
k
− ϕ
k
=0by(4.16), and hence,
w
0
≡ z
0
, that is, w(j) =


z( j)forall j ∈ dis[−τ,0]. Moreover, z(n)and w(n) satisfy the
same relation
x(n +1)
=

G

n,x
n

, n ∈ Z
+
. (4.22)
The uniqueness of the solutions for the initial value problems implies that
z(n) ≡ w(n)
for n
∈ Z
+
, and hence,  w
r
0
− z
r
0
=0. On the other hand, since u
n
k
+r
0

( j) = u(n
k
+ r
0
+
j)
→ w(r
0
+ j) =

w
r
0
( j)andx
k
n
k
+r
0
( j) = x
k
(n
k
+ r
0
+ j) → z(r
0
+ j) =

z

r
0
( j)ask →∞for
all j
∈ dis[−τ,0], from (4.16)wehave



w
r
0
− z
r
0


=
lim
k→∞


u
n
k
+r
k
− x
k
n
k

+r
k


≥
ε (4.23)
This is a contradiction. This contradiction shows that Lemma 4.3 is true.

We are now in a position to prove the following result.
Theorem 4.4. Suppose that for each G
∈ H(F), the solution of (4.2)isuniquefortheinitial
condition. If the bounded solution
{u(n)}
n≥0
of (4.1)isᐁᏭ᏿, then {u(n)}
n≥0
is asymptot-
ically almost periodic. Consequently, (4.1) has an almost periodic solut ion which is ᐁᏭ᏿.
Proof. Let the bounded solution
{u(n)} of (4.1)beᐁᏭ᏿ with the triple (δ(·),δ
0
,N(·)).
Let
{n
k
}
k≥1
be any positive integer such that n
k
→∞as k →∞.Setu

k
(n) = u(n + n
k
).
Then u
k
(n) is a solution of
x(n +1)
= F

n + n
k
,x
n

(4.24)
and
{u
k
(n)} is ᐁᏭ᏿ with the same triple (δ(·), δ
0
,N(·)). By Lemma 4.3,forthesetS
α
and any 0 <ε<1 there exists δ
1
(ε) > 0suchthat|h(n)| <δ
1
(ε)andx
k
n

0
− x
n
0
 <δ
1
(ε)
for some n
0
≥ 0implythatx
k
n
− x
n
 < (ε/2) for all n ≥ n
0
,where{x(n)} (n ≥ n
0
)isa
solution of
x(n +1)
= F

n + n
k
,x
n

+ h(n), (4.25)
through (n

0
,x
n
0
)andx
n
∈ S
α
for n ≥ n
0
.Since{u
k
( j) = u(n
k
+ j)} is uniformly bounded
for all k
≥ 1and j ≥−τ, taking a subsequence if necessary, we can assume that {u
k
( j)} is
convergent for each j
∈ dis[−τ,∞), F(n + n
k
,v) → G(n,v)uniformlyonZ
+
× S
α
. In this
case, there is a positive integer k
1
(ε)suchthatifm,k ≥ k

1
(ε), then


u
k
0
− u
m
0


<δ
1
(ε). (4.26)
On the other hand,
{u
m
(n) = u(n + n
m
)}, u
m
n
∈ S
α
for n ∈ Z
+
, is a solution of (4.25)with
h(n)
= h

k,m
(n), that is,
x(n +1)
= F

n + n
k
,x
n

+ h
k,m
(n), (4.27)
14 Advances in Difference Equations
where h
k,m
(n)isdefinedbytherelation
h
k,m
(n) = F

n + n
m
,u
m
n

−
F


n + n
k
,u
m
n

, n ∈ Z
+
. (4.28)
To app ly Lemma 4.3 to (4.24) and its associated (4.27), we will discuss the properties
of the sequence
{h
k,m
(n)}
n≥0
.SinceF(n + n
k
,v) → G(n,v)uniformlyonZ
+
× S
α
,forthe
above δ
1
(ε) > 0,thereisapositiveintegerk
2
(ε) >k
1
(ε)suchthatifk,m ≥ k
2

(ε), then


F

n + n
m
,v

−
F

n + n
k
,v



<δ
1
(ε) ∀n ∈ Z
+
, v ∈ S
α
, (4.29)
which implies that
|h
k,m
(n)|=|F(n + n
m

,u
m
n
) − F(n + n
k
,u
m
n
)| <δ
1
(ε)foralln ∈ Z.Ap-
plying Lemma 4.3 to (4.24) and its associated (4.27), with the above arguments and con-
dition (4.26), we conclude that for any positive integer sequence
{n
k
}
k≥1
, n
k
→∞ as
k
→∞,andε>0, there is a positive integer k
2
(ε) > 0suchthat


u
k
n
− u

m
n


<ε ∀n ≥ 0ifk,m>k
2
(ε), (4.30)
and hence,
|u
k
(n) − u
m
(n)|=|u
k
n
(0) − u
m
n
(0)| <εfor all n ≥ 0ifk,m>k
2
(ε). This implies
that the bounded solution
{u(n)}
n≥0
of (4.1) is asymptotically almost periodic sequence
by Theorem 2.6.Furthermore,(4.1) has an almost periodic solution, which is ᐁᏭ᏿ by
Theorem 4.1. This ends the proof.

Acknowledgments
This work is supported in part by NNSF of China (no. 10471102). We thank the referees

for helpful comments on our presentation.
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Yihong Song: Department of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, China
Email address: