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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 976865, 19 pages
doi:10.1155/2009/976865
Research Article
Almost Periodic Solutions of Prey-Predator
Discrete Models with Delay
Tomomi Itokazu and Yoshihiro Hamaya
Department of Information Science, Okayama University of Science, 1-1 Ridai-cho,
Okayama 700-0005, Japan
Correspondence should be addressed to Yoshihiro Hamaya,
Received 10 February 2009; Revised 18 May 2009; Accepted 9 July 2009
Recommended by Elena Braverman
The purpose of this article is to investigate the existence of almost periodic solutions of a
system of almost periodic Lotka-Volterra difference equations which are a prey-predator system
x
1
n1x
1
n exp{b
1
n−a
1
nx
1
n−c
2
n
n
s−∞
K
2
n−sx
2
s},x
2
n1x
2
n exp{−b
2
n−
a
2
nx
2
nc
1
n
n
s−∞
K
1
n − sx
1
s} and a competitive system x
i
n 1x
i
n exp{b
i
n −
a
ii
x
i
n −
l
j1,j
/
i
n
s−∞
K
ij
n − sx
j
s}, by using certain stability properties, which are referred
to as K, ρ-weakly uniformly asymptotic stable in hull and K, ρ-totally stable.
Copyright q 2009 T. Itokazu and Y. Hamaya. 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 Preliminary
For ordinary differential equations and functional differential equations, the existence of
almost periodic solutions of almost periodic systems has been studied by many authors. One
of the most popular methods is to assume the certain stability properties 1–4. Recently, Song
and Tian 5 have shown the existence of periodic and almost periodic solutions for nonlinear
Volterra difference equations by means of K, ρ-stability conditions. Their results are to
extend results in Hamaya 2 to discrete Volterra equations. To the best of our knowledge,
there are no relevant results on almost periodic solutions for discrete Lotka-Volterra models
by means of our approach, except for Xia and Cheng’s special paper 6. However, they
treated only nondelay case in 6. We emphasize that our results extend 3, 6, 7 as a discret
delay case.
In this paper, we will discuss the existence of almost periodic solutions for discrete
Lotka-Volterra difference equations with time delay.
In what follows, we denote by R
m
real Euclidean m-space, Z is the set of integers, Z
is
the set of nonnegative integers, and |·|will denote the Euclidean norm in R
m
. For any interval
2 Advances in Difference Equations
I ⊂ Z :−∞, ∞, we denote by BSI the set of all bounded functions mapping I into R
m
,
and set |φ|
I
sup{|φs| : s ∈ I}.
Now, for any function x : −∞,a → R
m
and n<a,define a function x
n
: Z
−
−∞, 0 → R
m
by x
n
sxn s for s ∈ Z
−
. Let BS be a real linear space of functions
mapping Z
−
into R
m
with sup-norm:
BS
φ | φ : Z
−
−→ R
m
with
φ
sup
s∈Z
−
φ
s
< ∞
. 1.1
We introduce an almost periodic function fn, x : Z × D → R
m
, where D is an open set in
R
m
.
Definition 1.1. It holds that fn, x is said to be almost periodic in n uniformly for x ∈ D, if for
any >0 and any compact set K in D, there exists a positive integer L
∗
, K such that any
interval of length L
∗
, K contains an integer τ for which
f
n τ, x
− f
n, x
≤ 1.2
for all n ∈ Z and all x ∈ K. Such a number τ in the above inquality is called an -translation
number of fn, x.
In order to formulate a property of almost periodic functions, which is equivalent to
the above difinition, we discuss the concept of the normality of almost periodic functions.
Namely, let fn, x be almost periodic in n uniformly for x ∈ D. Then, for any sequence
{h
k
}⊂Z, there exist a subsequence {h
k
} of {h
k
} and function gn, x such that
f
n h
k
,x
−→ g
n, x
1.3
uniformly on Z × K as k →∞, where K is a compact set in D. There are many properties
of the discrete almost periodic functions 5, 8, which are corresponding properties of the
continuous almost periodic functions ft, x ∈ CR × D, R
m
4. We will denote by Tf the
function space consisting of all translates of f, that is, f
τ
∈ Tf, where
f
τ
n, x
f
n τ, x
,τ∈ Z. 1.4
Let Hf denote the uniform closure of Tf in the sense of 1.4. Hf is called the hull of
f. In particular, we denote by Ωf the set of all limit functions g ∈ Hf such that for some
sequence {n
k
}, n
k
→∞as k →∞and fn n
k
,x → gn, x uniformly on Z × S for any
compact subset S in R
m
.By1.3,iff : Z × D → R
m
is almost periodic in n uniformly for
x ∈ D, so is a function in Ωf. The following concept of asymptotic almost periodicity was
introduced by Frechet in the case of continuous function cf. 4.
Definition 1.2. It holds that un is said to be asymptotically almost periodic if it is a sum of
an almost periodic function pn and a function qn defined on I
∗
a, ∞ ⊂ Z
0, ∞
which tends to zero as n →∞,thatis,
u
n
p
n
q
n
. 1.5
Advances in Difference Equations 3
However, un is asymptotically almost periodic if and only if for any sequence {n
k
} such
that n
k
→∞as k →∞, there exists a subsequence {n
k
} for which un n
k
converges
uniformly on n; a ≤ n<∞.
2. Prey-Predator Model
We will consider the existence of a strictly positive component-wise almost periodic solution
of a system of Volterra difference equations:
x
1
n 1
x
1
n
exp
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
,
x
2
n 1
x
2
n
exp
−b
2
n
− a
2
n
x
2
n
c
1
n
n
s−∞
K
1
n − s
x
1
s
,
F
which describes a model of the dynamics of a prey-predator discrete system in mathematical
ecology. In F, setting a
i
n, b
i
n, and c
i
nR-valued bounded almost periodic in Z:
a
i
inf
n∈Z
a
i
n
,A
i
sup
n∈Z
a
i
n
,b
i
inf
n∈Z
b
i
n
,
B
i
sup
n∈Z
b
i
n
,c
i
inf
n∈Z
c
i
n
C
i
sup
n∈Z
c
i
n
i 1, 2
,
2.1
and K
i
: Z
0, ∞ → R
i 1, 2 denote delay kernels such that
K
i
s
≥ 0,
∞
s0
K
i
s
1,
∞
s0
sK
i
s
< ∞,
i 1, 2
.
2.2
Under the above assumptions, it follows that for any n
0
,φ
i
∈ Z
× BS, i 1, 2, there is a
unique solution unu
1
n,u
2
n of F through n
0
,φ
i
, i 1, 2, i f it remains bounded.
We set
α
1
exp
{
B
1
− 1
}
a
1
,α
2
exp
{
−b
2
C
1
α
1
− 1
}
a
2
,
β
1
min
exp
{
b
1
− A
1
α
1
− C
2
α
2
}
b
1
− C
2
α
2
A
1
,
{
b
1
− C
2
α
2
}
A
1
,
β
2
min
exp
−B
2
− A
2
α
2
c
1
β
1
−B
2
c
1
β
1
A
2
,
−B
2
c
1
β
1
A
2
2.3
cf. 6, Application 4.1.
4 Advances in Difference Equations
We now make the following assumptions:
i a
i
> 0,b
i
> 0 i 1, 2,c
1
> 0, c
2
≥ 0,
ii b
1
>C
2
α
2
,B
2
<c
1
β
1
,
iii there exists a positive constant m such that
a
i
>C
i
m
i 1, 2
. 2.4
Then, we have 0 <β
i
<α
i
for each i 1, 2.
We can show the following lemmas.
Lemma 2.1. If xnx
1
n,x
2
n is a solution of F through n
0
,φ
i
, i 1, 2 such that
β
i
≤ φ
i
s ≤ α
i
i 1, 2 for all s ≤ 0, then one has β
i
≤ x
i
n ≤ α
i
i 1, 2 for all n ≥ n
0
.
Proof. First, we claim that
lim sup
n →∞
x
1
n
≤ α
1
. 2.5
To do this, we first assume that there exists an l
0
≥ n
0
such that x
1
l
0
1 ≥ x
1
l
0
. Then, it
follows from the first equation of F that
b
1
l
0
− a
1
l
0
x
1
l
0
− c
2
l
0
l
0
s−∞
K
2
l
0
− s
x
2
s
≥ 0.
2.6
Hence,
x
1
l
0
≤
b
1
l
0
− c
2
l
0
l
0
s−∞
K
2
l
0
− s
x
2
s
a
1
l
0
≤
b
1
l
0
a
1
l
0
≤
B
1
a
1
.
2.7
It follows that
x
1
l
0
1
x
1
l
0
exp
b
1
l
0
− a
1
l
0
x
1
l
0
− c
2
l
0
l
0
s−∞
K
2
l
0
− s
x
2
s
≤ x
1
l
0
exp
{
B
1
− a
1
x
1
l
0
}
≤
exp
{
B
1
− 1
}
a
1
: α
1
,
2.8
where we use the fact that
max
x∈R
x exp
p − qx
exp
p − 1
q
, for p, q > 0.
2.9
Now we claim that
x
1
n
≤ α
1
, for n ≥ l
0
. 2.5
∗
Advances in Difference Equations 5
By way of contradiction, we assume that there exists a p
0
>l
0
such that x
1
p
0
>α
1
. Then,
p
0
≥ l
0
2. Let p
0
≥ l
0
2 be the smallest integer such that x
1
p
0
>α
1
. Then, x
1
p
0
− 1 ≤
x
1
p
0
. The above argument shows that x
1
p
0
≤ α
1
, which is a contradiction. This proves our
assertion. We now assume that x
1
n 1 <x
1
n for all n ≥ n
0
. Then lim
n →∞
x
1
n exists,
which is denoted by
x
1
. We claim that x
1
≤ expB
1
− 1/a
1
. Suppose to the contrary that
x
1
> expB
1
− 1/a
1
. Taking limits in the first equation in System F,wesetthat
0 lim
n →∞
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
≤ lim
n →∞
b
1
n
− a
1
n
x
1
n
≤ B
1
− a
1
x
1
< 0,
2.10
which is a contradiction. It follows that 2.5 holds, and then we have x
i
n ≤ α
1
for all n ≥ n
0
from 2.5
∗
. Next, we prove that
lim sup
n →∞
x
2
n
≤ α
2
. 2.11
We first assume that there exists an k
0
≥ n
0
such that k
0
≥ l
0
and x
2
k
0
1 ≥ x
2
k
0
. Then
−b
2
k
0
− a
2
k
0
x
2
k
0
c
1
k
0
k
0
s−∞
K
1
k
0
− s
x
1
s
≥ 0.
2.12
Hence,
x
2
k
0
≤
−b
2
k
0
c
1
k
0
k
0
s−∞
K
1
k
0
− s
x
1
s
a
2
k
0
≤
−b
2
C
1
α
1
a
2
.
2.13
It follows that
x
2
k
0
1
x
2
k
0
exp
−b
2
k
0
− a
2
k
0
x
2
k
0
c
1
k
0
k
0
s−∞
K
1
k
0
− s
x
1
s
≤ x
2
k
0
exp
{
−b
2
− a
2
x
2
k
0
C
1
α
1
}
≤
exp
−b
2
C
1
α
1
− 1
a
2
: α
2
,
2.14
where we also use the two facts which are used to prove 2.5. Now we claim that
x
2
n
≤ α
2
∀n ≥ k
0
. 2.11
∗
Suppose to the contrary that there exists a q
0
>k
0
such that x
2
q
0
>α
2
. Then q
0
≥ k
0
2. Let
q
0
≥ k
0
2 be the smallest integer such that x
2
q
0
>α
2
. Then x
2
q
0
− 1 <x
2
q
0
. Then the
above argument shows that x
2
q
0
≤ α
2
, which is a contradiction. This prove our claim from
2.11 and 2.11
∗
.
6 Advances in Difference Equations
Now, we assume that x
2
n 1 <x
2
n for all n ≥ n
0
. Then lim
n →∞
x
2
n exists, which
is denoted by
x
2
. We claim that x
2
≤ exp−b
2
C
1
α
1
− 1/a
2
. Suppose to the contrary that
x
2
> exp−b
2
C
1
α
1
− 1/a
2
. Taking limits in the first equation in System F,wesetthat
0 lim
n →∞
−b
2
n
− a
2
n
x
2
n
c
1
n
n
s−∞
K
1
n − s
x
1
s
≤−b
2
− a
2
x
2
C
1
α
1
< 0,
2.15
which is a contradiction. It follows that 2.11 holds.
We show that
lim inf
n →∞
x
1
n
≥ β
1
. 2.16
According to the above assertion, there exists a k
∗
≥ n
0
such that x
1
n ≤ α
1
and x
2
n ≤
α
2
, for all n ≥ k
∗
. We assume that there exists an l
0
≥ k
∗
such that x
1
l
0
1 ≤ x
1
l
0
.Note
that for n ≥ l
0
,
x
1
n 1
x
1
n
exp
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
≥ x
1
n
exp
{
b
1
− C
2
α
2
− A
1
x
1
n
}
.
2.17
In particular, with n l
0
, we have
b
1
− A
1
x
1
l
0
− C
2
α
2
≤ 0, 2.18
which implies that
x
1
l
0
≥
b
1
− C
2
α
2
A
1
.
2.19
Then,
x
1
l
0
1
≥
b
1
− C
2
α
2
A
1
exp
b
1
− C
2
α
2
− A
1
α
1
: x
1
.
2.20
We assert that
x
1
n
≥ x
1
, ∀n ≥ l
0
. 2.16
∗
By way of contradiction, we assume that there exists a p
0
≥ l
0
such that x
1
p
0
<x
1
. Then
p
0
≥ l
0
2. Let p
0
2 be the smallest integer such that x
1
p
0
<x
1
.Then x
1
p
0
≤ x
1
p
0
− 1.
The above argument yields x
1
p
0
≥ x
1
, which is a contradiction. This proves our claim. We
now assume that x
1
n 1 <x
1
n for all n ≥ n
0
. Then lim
n →∞
x
1
n exists, which is denoted
Advances in Difference Equations 7
by x
1
. We claim that x
1
≥ b
1
− C
2
α
2
/A
1
. Suppose to the contrary that x
1
< b
1
− C
2
α
2
/A
1
.
Taking the limits in the first equation in System F,wesetthat
0 lim
n →∞
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
≥ b
1
− A
1
x
1
− C
2
α
2
> 0,
2.21
which is a contradiction. It follows that 2.16 holds, and then β
1
≤ x
1
n for all n ≥ n
0
from 2.16 and 2.16
∗
. Finally, by using the inequality B
2
<c
1
β
1
, similar arguments lead to
lim inf
n →∞
x
2
≥ β
2
, and then x
2
n ≥ β
2
for all n ≥ k
0
. This proof is complete.
Lemma 2.2. Let K be the closed bounded set in R
2
such that
K
x
1
,x
2
∈ R
2
; β
i
≤ x
i
≤ α
i
for each i 1, 2
. 2.22
Then K is invariant for System F, that is, one can see that for any n
0
∈ Z and any ϕ
i
such that
ϕ
i
s ∈ K, s ≤ 0 i 1, 2, every solution of F through n
0
,ϕ
i
remains in K for all n ≥ n
0
and
i 1, 2.
Proof. From Lemma 2.1,itissufficient to prove that this K
/
φ.
To do this, by assumption of almost periodic functions, there exists a sequence
{n
k
},n
k
→∞as k →∞, such that b
i
n n
k
→ b
i
n,a
i
n n
k
→ a
i
n, and
c
i
n n
k
→ c
i
n as k →∞uniformly on Z and i 1, 2. Let xn be a solution of System F
through n
0
,ϕ that remains in K for all n ≥ n
0
, whose existence was ensured by Lemma 2.1.
Clearly, the sequence {xn n
k
} is uniformly bounded on bounded subset of Z. Therefore,
we may assume that the sequence {xn n
k
} converges to a function yny
1
n,y
2
n
as k →∞uniformly on each bounded subset of Z taking a subset of {xn n
k
} if necessary.
We may assume that n
k
≥ n
0
for all k. For n ≥ 0, we have
x
1
n n
k
1
x
1
n n
k
exp
b
1
n n
k
− a
1
n n
k
x
1
n n
k
− c
2
n n
k
nn
k
s−∞
K
2
n n
k
− s
x
2
s
,
x
2
n n
k
1
x
2
n n
k
exp
−b
2
n n
k
− a
2
n n
k
x
2
n n
k
c
1
n n
k
nn
k
s−∞
K
1
n n
k
− s
x
1
s
.
F
n
k
Since xn n
k
∈ K and yn ∈ K for all n ∈ Z, t here exists r>0 such that |xn n
k
|≤r and
|yn|≤r for all n ∈ Z. Then, by assumption of delay kernel K
i
,forthisr and any >0, there
exists an integer S S, r > 0 such that
n−S
s−∞
|
K
i
n n
k
− s
x
i
s
|
≤ ,
n−S
s−∞
K
i
n − s
y
i
s
≤ .
2.23
8 Advances in Difference Equations
Then, we have
n
s−∞
K
i
n n
k
− s
x
i
s
−
n
s−∞
K
i
n − s
y
i
s
≤
n−S
s−∞
|
K
i
n n
k
− s
x
i
s
|
n−S
s−∞
K
i
n − s
y
i
s
n
sn−S
K
i
n n
k
− s
x
i
s
− K
i
n − s
y
i
s
≤ 2
n
sn−S
K
i
n n
k
− s
x
i
s
− K
i
n − s
y
i
s
.
2.24
Since x
i
n n
k
− s converges to y
i
n − s on discrete interval s ∈ n − S, n as k →∞, there
exists an integer k
0
>k
∗
, for some k
∗
> 0, such that
n
sn−S
K
i
n n
k
− s
x
i
s
− K
i
n − s
y
i
s
≤
i 1, 2
2.25
when k ≥ k
0
. Thus, we have
n
s−∞
K
i
n n
k
− s
x
i
s
−→
n
s−∞
K
i
n − s
y
i
s
2.26
as k →∞. Letting k →∞in F
n
k
, we have
y
1
n 1
y
1
n
exp
b
1
n
− a
1
n
y
1
n
− c
2
n
n
s−∞
K
2
n − s
y
2
s
,
y
2
n 1
y
2
n
exp
−b
2
n
− a
2
n
y
2
n
c
1
n
n
s−∞
K
1
n − s
y
1
s
,
2.27
for all n ≥ n
0
. Then, yny
1
n,y
2
n is a solution of System F on Z. It is clear that
yn ∈ K for all n ∈ Z.Thus,K
/
φ.
Advances in Difference Equations 9
We denote by ΩF the set of all limit functions G such that for some sequence {n
k
}
such that n
k
→∞as k →∞, b
i
n n
k
→ b
i
n,a
i
n n
k
→ a
i
n, and c
i
n n
k
→ c
i
n
uniformly on Z as k →∞. Here, the equation for G is
x
1
n 1
x
1
n
exp
b
1
n
−
a
1
n
x
1
n
−
c
2
n
n
s−∞
K
2
n − s
x
2
s
,
x
2
n 1
x
2
n
exp
−
b
2
n
−
a
2
n
x
2
n
c
1
n
n
s−∞
K
1
n − s
x
1
s
.
G
Moreover, we denote by v, G ∈ Ωu, F when for the same sequence {n
k
},un n
k
→ vn
uniformly on any compact subset in Z as k →∞. Then a system G is called a limiting
equation of F when G ∈ ΩF and vn is a solution of G when v, G ∈ Ωu, F.
Lemma 2.3. If a compact set K in R
2
of Lemma 2.2 is invariant for System F,thenK is invariant
also for every limiting equation of System F.
Proof. Let G be a limiting equation of system F. Since G ∈ ΩF, there exists a sequence
{n
k
} such that n
k
→∞as k →∞and that b
i
n n
k
→ b
i
n,a
i
n n
k
→ a
i
n, and
c
i
n n
k
→ c
i
n uniformly on Z as k →∞.Letn
0
≥ 0,φ ∈ BS such that φs ∈ K for all
s ≤ 0, and let yn be a solution of system G through n
0
,φ.Letx
k
n be the solution of
System F through n
0
n
k
,φ. Then x
k
n
0
n
k
sφs ∈ K for all s ≥ 0andx
k
n is defined
on n ≥ n
0
n
k
. Since K is invariant for System F, x
k
n ∈ K for all n ≥ n
0
n
k
.Ifweset
z
k
nx
k
n n
k
,k 1, 2, , then z
k
n is defined on n ≥ n
0
and is a solution of
x
1
n 1
x
1
n
exp
b
1
n n
k
− a
1
n n
k
x
1
n
− c
2
n n
k
nn
k
s−∞
K
2
n n
k
− s
x
2
s
,
x
2
n 1
x
2
n
exp
−b
2
n n
k
− a
2
n n
k
x
2
n
c
1
n n
k
nn
k
s−∞
K
1
n n
k
− s
x
1
s
,
2.28
such that z
k
n
0
sx
k
n
0
n
k
sφs ∈ K for all s ≤ 0. Since x
k
n ∈ K for all n ≥ n
0
n
k
,
z
k
n ∈ K for all n ≥ n
0
. Since the sequence {z
k
n} is uniformly bounded on n
0
, ∞ and
z
k
n
0
φ, {z
k
n} can be assumed to converge to the solution yn of G through n
0
,φ
uniformly on any compact set n
0
, ∞, because yn is the unique solution through n
0
,φ
and the same argument as in the proof of Lemma 2.2. Therefore, yn ∈ K for all n ≥ n
0
since
z
k
n ∈ K for all n ≥ n
0
and K is compact. This shows that K is invariant for limiting G.
10 Advances in Difference Equations
Let K be the compact set in R
m
such that un ∈ K for all n ∈ Z, where unφ
0
n
for n ≤ 0. For any θ, ψ ∈ BS, we set
ρ
θ, ψ
∞
j1
ρ
j
θ, ψ
2
j
1 ρ
j
θ, ψ
,
2.29
where
ρ
j
θ, ψ
sup
−j≤s≤0
θ
s
− ψ
s
.
2.30
Clearly, ρθ
n
,θ → 0asn →∞if and only if θ
n
s → θs uniformly on any compact subset
of −∞, 0 as n →∞.
In what follows, we need the following definitions of stability.
Definition 2.4. The bounded solution un of System F is said to be as follows:
iK, ρ-totally stable in short, K, ρ-TS if for any >0, there exists a δ > 0
such that if n
0
≥ 0, ρx
n
0
,u
n
0
<δ and h h
1
,h
2
∈ BSn
0
, ∞ which satisfies
|h|
n
0
,∞
<δ, then ρx
n
,u
n
<for all n ≥ n
0
, where xn is a solution of
x
1
n 1
x
1
n
exp
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
h
1
n
,
x
2
n 1
x
2
n
exp
−b
2
n
− a
2
n
x
2
n
c
1
n
n
s−∞
K
1
n − s
x
1
s
h
2
n
,
F h
through n
0
,φ such that x
n
0
sφs ∈ K for all s ≤ 0. In the case where hn ≡ 0,
this gives the definition of the K, ρ-US of un;
iiK, ρ-attracting in ΩFin short, K, ρ-A in ΩF if there exists a δ
0
> 0 such that
if n
0
≥ 0andanyv, G ∈ Ωu, F, ρx
n
0
,v
n
0
<δ
0
, then ρx
n
,v
n
→ 0asn →∞,
where xn is a solution of limiting equation of 2.5; G through n
0
,ψ such
that x
n
0
sψs ∈ K for all s ≤ 0;
iiiK, ρ-weakly uniformly asymptotically stable in ΩFin short, K, ρ-WUAS in
ΩF if it is K, ρ-US in ΩF, that is, if for any ε>0 there exists a δ > 0 such
that if n
0
≥ 0andanyv, G ∈ Ωu, F, ρx
n
0
,v
n
0
<δ, then ρx
n
,v
n
<for all
n ≥ n
0
, where xn is a solution of G through n
0
,ψ such that x
n
0
sψs ∈ K
for all s ≤ 0, and K, ρ-A in ΩF.
Advances in Difference Equations 11
Proposition 2.5. Under the assumption (i), (i), and (iii), if the solution un of System F is K, ρ-
WUAS in ΩF, then the solution un of System F is K, ρ-TS.
Proof. Suppose that un is not K, ρ-TS. T hen there exist a small >0, sequences {
k
}, 0 <
k
<and
k
→ 0ask →∞, sequences {s
k
}, {n
k
}, {h
k
}, {x
k
} such that s
k
→∞as k →∞,
0 <s
k
1 <n
k
, h
k
: Z → R
2
is bounded function satisfying |h
k
n| <
k
for n ≥ s
k
and such
that
ρ
u
s
k
,x
k
s
k
<
k
,ρ
u
n
k
,x
k
n
k
≥ , ρ
u
n
,x
k
n
<
s
k
,n
k
, 2.31
where x
k
n is a solution of
x
1
n 1
x
1
n
exp
b
1
n
− a
1
n
x
1
n
− c
2
n
n
s−∞
K
2
n − s
x
2
s
h
k1
n
,
x
2
n 1
x
2
n
exp
−b
2
n
− a
2
n
x
2
n
c
1
n
n
s−∞
K
1
n − s
x
1
s
h
k2
n
,
F h
k
such that x
k
s
k
s ∈ K for all s ≤ 0. We can assume that <δ
0
where δ
0
is the number for
K, ρ-A in ΩF of Definition 2.4. Moreover, by 2.31, we can chose sequence {τ
k
} such that
s
k
<τ
k
<n
k
,
ρ
u
τ
k
,x
k
τ
k
δ
/2
2
,
2.32
δ
/2
2
≤ ρ
u
n
,x
k
n
≤ for n ∈
τ
k
,n
k
,
2.33
where δ· is the number for K, ρ-US in ΩF. We may assume that un τ
k
→ vn as
k →∞on each bounded subset of Z for a function v, and for the sequence {τ
k
}, τ
k
→∞
as k →∞, taking a subsequence if necessary, there exists a v, G ∈ Ωu, F. Moreover, we
may assume that x
k
n τ
k
→ zn as k →∞uniformly on any bounded subset of Z for
function z, since the sequence {x
k
n τ
k
} is uniformly bounded on Z. Because, if we set
y
k
nx
k
n τ
k
, then y
k
n is defined on n ≥ n
0
τ
k
and y
k
n is a solution of
x
1
n 1
x
1
n
exp
b
1
nτ
k
−a
1
n τ
k
x
1
n
− c
2
n τ
k
n
s−∞
K
2
n n
k
− s
x
2
s
h
k1
n τ
k
,
12 Advances in Difference Equations
x
2
n 1
x
2
n
exp
−b
2
nτ
k
−a
2
n τ
k
x
2
n
c
1
n τ
k
n
s−∞
K
1
n n
k
− s
x
1
s
h
k2
n τ
k
,
2.34
such that y
k
0
sx
k
τ
k
s ∈ K for all s ≤ 0. Then we may show that taking a subsequence
if necessary, y
k
n converges to a solution zn of G such that z
0
s ∈ K for s ≤ 0, by
the same argument for Σ-calculations with condition of K
i
as in the proof of Lemma 2.2.
Then, the same argument as in the proof of Lemma 2.2 shows that z ∈ K. Now, suppose that
n
k
− τ
k
→∞as k →∞. Letting k →∞in 2.33, we have δ/2/2 ≤ ρv
n
,z
n
≤ on
n ≥ 0. Since <δ
0
and un is K, ρ-A in ΩF, we have δ/2/2 ≤ ρv
n
,z
n
→ 0as
n →∞, which is a contradiction. Thus n
k
−τ
k
∞as k →∞. Taking a subsequence again if
necessary, we can assum that n
k
− τ
k
→ r<∞ as k →∞. Letting k →∞in 2.32, we have
ρv
0
,z
0
δ/2/2 <δ/2 and hence ρv
n
,z
n
</2 for all n ≥ 0, because u is K, ρ-US
in ΩF. On the other hand, from 2.31, we have ρv
n
,z
n
≥ , which is a contradiction. This
shows that un is K, ρ-TS.
Now we will see that the existence of a strictly positive almost periodic solution of
System F can be obtained under conditions i, ii, and iii.
Theorem 2.6. one assumes conditions i, ii, and iii. Then System F has a unique almost
periodic solution pn in compact set K.
Proof. For System F, we first introduce the change of variables:
u
i
n
exp
{
v
i
n
}
,x
i
n
exp
y
i
n
,i 1, 2. 2.35
Then, System F can be written as
y
1
n 1
− y
1
n
b
1
n
− a
1
n
exp
y
1
n
− c
2
n
n
s−∞
K
2
n − s
exp
y
2
s
,
y
2
n 1
− y
2
n
−b
2
n
− a
2
n
exp
y
2
n
c
1
n
n
s−∞
K
1
n − s
exp
y
1
s
,
F
We now consider Liapunov functional:
V
v
n
,y
n
2
i1
v
i
n
− y
i
n
∞
s0
K
i
s
n−1
ln−s
c
i
s l
exp
{
v
i
l
}
− exp
y
i
l
,
2.36
Advances in Difference Equations 13
where yn and vn are solutions of
F which remains in K. Calculating t he differences, we
have
ΔV
v
n
,y
n
≤
|
v
1
n 1
− v
1
n
|
−
y
1
n 1
− y
1
n
∞
s0
K
1
s
c
1
s n
exp
{
v
1
n
}
− exp
y
1
n
− c
1
n
exp
{
v
1
n − s
}
− exp
y
1
n − s
|
v
2
n 1
− v
2
n
|
−
y
2
n 1
− y
2
n
∞
s0
K
2
s
c
2
s n
exp
{
v
2
n
}
− exp
y
2
n
− c
2
n
exp
{
v
2
n − s
}
− exp
y
2
n − s
b
1
n
− a
1
n
exp
{
v
1
n
}
− c
2
n
∞
s0
K
2
n − s
exp
{
v
2
s
}
−
b
1
n
− a
1
n
exp
y
1
n
− c
2
n
∞
s0
K
2
n − s
exp
y
2
s
∞
s0
K
1
s
c
1
s n
exp
{
v
1
n
}
− exp
y
1
n
−c
1
n
exp
{
v
1
n − s
}
− exp
y
1
n − s
−b
2
n
− a
2
n
exp
{
v
2
n
}
c
1
n
∞
s0
K
1
n − s
exp
{
v
1
s
}
−
−b
2
n
− a
2
n
exp
y
2
n
c
1
n
∞
s0
K
1
n − s
exp
y
1
s
∞
s0
K
2
s
c
2
s n
exp
{
v
2
n
}
− exp
y
2
n
−c
2
n
exp
{
v
2
n − s
}
− exp
y
2
n − s
≤
b
1
n
− a
1
n
exp
{
v
1
n
}
− c
2
n
exp
{
v
2
n − s
}
−
b
1
n
− a
1
n
exp
y
1
n
− c
2
n
exp
y
2
n − s
c
1
s n
exp
{
v
1
n
}
− exp
y
1
n
− c
1
n
exp
{
v
1
n − s
}
− exp
y
1
n − s
−b
2
n
− a
2
n
exp
{
v
2
n
}
c
1
n
exp
{
v
1
n − s
}
−
−b
2
n
− a
2
n
exp
y
2
n
c
1
n
exp
y
1
n − s
c
2
s n
exp
{
v
2
n
}
− exp
y
2
n
− c
2
n
exp
{
v
2
n − s
}
− exp
y
2
n − s
≤−a
1
n
exp
{
v
1
n
}
− exp
y
1
n
c
1
s n
exp
{
v
1
n
}
− exp
y
1
n
≤−a
2
n
exp
{
v
2
n
}
− exp
y
2
n
c
2
s n
exp
{
v
2
n
}
− exp
y
2
n
.
2.37
From the mean value theorem, we have
exp
{
v
i
n
}
− exp
y
i
n
exp
{
θ
i
n
}
v
i
n
− y
i
n
,i 1, 2, 2.38
14 Advances in Difference Equations
where θ
i
n lies between v
i
n and y
i
ni 1, 2. Then, by iii, we have
ΔV
v
n
,y
n
≤−mD
2
i1
v
i
n
− y
i
n
, 2.39
where set D max{exp{β
1
}, exp{β
2
}}, and let solutions x
i
n of System F be such that
x
i
n ≥ β
i
for n ≥ n
0
i 1, 2.Thus
2
i1
|v
i
n−y
i
n|→0asn →∞, and hence ρv
n
,y
n
→
0asn →∞. Moreover, we can show that vn is K, ρ-US in Ω of
F, by the same argument
as in 5. By using similar Liapunov functional to 2.36, we can show that vn is K, ρ-A in
Ω of
F. Therefore, vn is K, ρ-WUAS in Ω
F.Thus,fromProposition 2.5, vn is K, ρ-
TS, because K is invariant. By the equivalence between
F and F, solution un of System
F is K, ρ-TS. Therefore, it follows from Theorem 4.4 in 5 and
9 that System F has an
almost periodic solution pn such that β
i
≤ p
i
n ≤ α
i
, i 1, 2, f or all n ∈ Z.
3. Competitive System
We will consider the l-species almost periodic competitive Lotka-Volterra system:
x
i
n 1
x
i
n
exp
⎧
⎨
⎩
b
i
n
− a
ii
x
i
n
−
l
j1,j
/
i
a
ij
n
n
s−∞
K
ij
n − s
x
j
s
⎫
⎬
⎭
,i 1, 2, ,l,
H
where b
i
n,a
ij
n are positive almost periodic sequences on Z; a
ij
n are strictly positive,
and, moreover,
a
ij
inf
n∈Z
a
ij
n
,A
ij
sup
n∈Z
a
ij
n
,b
i
inf
n∈Z
b
i
n
,B
i
sup
n∈Z
b
i
n
,
K
ij
: Z
0, ∞
−→ R
i, j 1, 2, ,l
,
3.1
which can be seen as the discretization of the differential equation in 3.Weset
α
i
exp
{
B
i
− 1
}
a
ii
,
β
i
min
⎧
⎪
⎨
⎪
⎩
exp
b
i
− A
ii
α
i
−
l
j1,j
/
i
A
ij
α
j
b
i
−
l
j1,j
/
i
A
ij
α
j
A
ii
,
b
i
−
l
j1,j
/
i
α
j
A
ii
⎫
⎪
⎬
⎪
⎭
.
3.2
Now, we make the following assumptions:
iv K
ij
s ≥ 0, and
∞
s0
K
ij
s1,
∞
s0
sK
ij
s < ∞ i 1, 2, ,l;
v b
i
>
j1,j
/
i
A
ij
α
j
for i 1, 2, ,l;
Advances in Difference Equations 15
vi there exists a positive constant m such that
a
ii
>
l
j1,j
/
i
A
ij
m
i 1, 2, ,l
. 3.3
Then, we have 0 <β
i
<α
i
for each i 1, 2, ,l. Under the assumptions iv and v,it
follows that for any n
0
,φ ∈ Z
×BS, there is a unique solution unu
1
n,u
2
n, ,u
l
n
of H through n
0
,φ, if it remains bounded.
Then, we can show the similar lemmas to Lemma 2.1.
Lemma 3.1. If xnx
1
n,x
2
n, ,x
l
n is a solution of H through n
0
,φ such that
β
i
≤ φs ≤ α
i
i 1, 2, ,l for all s ≤ 0, then one has β
i
≤ x
i
n ≤ α
i
i 1, 2, ,l for all
n ≥ n
0
.
Proof. First, we claim that
lim sup
n →∞
x
i
n
≤ B
i
,i 1, 2, ,l. 3.4
Clearly, x
i
n > 0forn ≥ n
0
. To prove this, we first assume that there exists an l
0
≥ n
0
such
that x
i
l
0
1 ≥ x
i
l
0
. Then, it follows from the first equation of H that
b
i
l
0
− a
ii
l
0
x
i
l
0
−
l
j1,j
/
i
a
ij
l
0
l
0
s−∞
K
ij
l
0
− s
x
j
s
≥ 0. 3.5
Hence
x
i
l
0
≤
b
i
l
0
−
l
j1,j
/
i
a
ij
l
0
l
0
s−∞
K
ij
l
0
− s
x
j
s
a
ij
l
0
≤
b
i
l
0
a
ii
l
0
≤
B
i
a
ii
. 3.6
It follows that
x
i
l
0
1
x
i
l
0
exp
⎧
⎨
⎩
b
i
l
0
− a
ii
l
0
x
i
l
0
−
l
j1,j
/
i
a
ij
l
0
l
0
s−∞
K
ij
l
0
− s
x
j
s
⎫
⎬
⎭
≤ x
i
l
0
exp
{
B
i
− a
ii
x
i
l
0
}
≤
exp
{
B
i
− 1
}
a
ii
: α
i
.
3.7
Now we claim that
x
i
n
≤ B
i
, for n ≥ l
0
. 3.8
By way of contradiction, we assume that there exists a p
0
>l
0
such that x
i
p
0
>α
i
. Then, p
0
≥
l
0
2. Let p
0
≥ l
0
2 be the smallest integer such that x
i
p
0
>α
i
. Then, x
i
p
0
−1 ≤ x
i
p
0
.The
16 Advances in Difference Equations
above argument shows that x
i
p
0
≤ α
i
, which is a contradiction. This proves our assertion.
We now assume that x
i
n1 <x
i
n for all n ≥ n
0
. Then lim
n →∞
x
i
n exists, which is denoted
by
x
i
. We claim that x
i
≤ expB
i
− 1/a
ii
. Suppose to the contrary that x
i
> expB
i
− 1/a
ii
.
Taking limits in the first equation in System H,wesetthat
0 lim
n →∞
⎛
⎝
b
i
n
− a
ii
n
x
i
n
−
l
j1,j
/
i
a
ij
n
n
s−∞
K
ij
n − s
x
j
s
⎞
⎠
≤ lim
n →∞
b
i
n
− a
ii
n
x
i
n
≤ B
i
− a
ii
x
i
< 0,
3.9
which is a contradiction. It follows that 3.4 holds.
We first show that
lim inf
n →∞
x
i
n
≥ β
i
. 3.10
According to above assertion, there exists a k
∗
≥ n
0
such that x
i
n ≤ α
i
ε, for all n ≥ k
∗
.We
assume that there exists an l
0
≥ k
∗
such that x
i
l
0
1 ≤ x
i
l
0
. Note that for n ≥ l
0
,
x
i
n 1
x
i
n
exp
⎧
⎨
⎩
b
i
n
− a
ii
n
x
i
n
−
l
j1,j
/
i
a
ij
n
n
s−∞
K
ij
n − s
x
j
s
⎫
⎬
⎭
≥ x
i
n
exp
⎧
⎨
⎩
b
i
−
l
j1,j
/
i
A
ij
α
j
− A
ii
x
i
n
⎫
⎬
⎭
.
3.11
In particular, with n l
0
, we have
b
i
− A
ii
x
i
l
0
−
l
j1,j
/
i
A
ij
α
j
≤ 0
, 3.12
which implies that
x
i
l
0
≥
b
i
−
l
j1,j
/
i
A
ij
α
j
A
ii
.
3.13
Then,
x
i
l
0
1
≥
b
i
−
l
j1,j
/
i
A
ij
α
j
A
ii
exp
⎛
⎝
b
i
−
l
j1,j
/
i
A
ij
α
j
− A
ii
α
i
⎞
⎠
: x
i
.
3.14
We assert that
x
i
n
≥ x
i
, ∀n ≥ l
0
. 3.15
Advances in Difference Equations 17
By way of contradiction, we assume that there exists a p
0
≥ l
0
such that x
i
p
0
<x
iε
. Then p
0
≥
l
0
2. Let p
0
2 be the smallest integer such that x
i
p
0
<x
iε
.Then x
i
p
0
≤ x
i
p
0
−1. The above
argument yields x
i
p
0
≥ x
iε
, which is a contradiction. This proves our claim. We now assume
that x
i
n1 <x
i
n for all n ≥ n
0
. Then lim
n →∞
x
i
n exists, which is denoted by x
i
. We claim
that x
i
≥ b
i
−
l
j1,j
/
i
A
ij
α
j
/A
ii
. Suppose to the contrary that x
i
< b
i
−
l
j1,j
/
i
A
ij
α
j
/A
ii
.
Taking limits in the first equation in System F,wesetthat
0 lim
n →∞
⎛
⎝
b
i
n
− a
ii
n
x
i
n
−
l
j1,j
/
i
a
ij
n
l
s−∞
K
ij
n − s
x
j
s
⎞
⎠
≥ b
i
− A
ii
x
i
−
l
j1,j
/
i
A
ij
α
j
> 0,
3.16
which is a contradiction. It follows that 3.10 holds. This proof is complete.
By the same arguments of Lemmas 2.2, 2.3 and Proposition 2.5, we obtain Lemmas 3.2,
3.3 and Proposition 3.4. So, we will omit to these proofs.
Lemma 3.2. Let K be the closed bounded set in R
l
such that
K
x
1
,x
2
, ,x
l
∈ R
l
; β
i
≤ x
i
≤ α
i
for each i 1, 2, ,l
. 3.17
Then K is invariant for System H, that is, one can see that for any n
0
∈ Z and any ϕ such that
ϕs ∈ K, s ≤ 0, every solution of H through n
0
,ϕ remains in K for all n ≥ n
0
and i 1, 2, ,l.
Lemma 3.3. If a compact set K in R
l
of Lemma 3.2 is invariant for System H,thenK is invariant
also for every limiting equation of System H.
Proposition 3.4. Under the assumption (iv), (v), and (vi), if the solution un of System H is
K, ρ-WUAS in ΩH, then the solution un of System H is K, ρ-TS.
By making changes of the variables x
i
nexp{y
i
n} and defining the Liapunov functional
V by
V
v
n
,y
n
l
i1
v
i
n
− y
i
n
∞
s0
K
ij
s
n−1
ln−s
c
i
s l
exp
{
v
i
l
}
− exp
y
i
l
,
3.18
where yn and vn are solutions of changing equation
H for H which remains in K,the
arguments similar to the Theorem 2.6 lead to the following results.
Theorem 3.5. one assumes conditions (iv), (v), and (vi). Then System H has a unique almost
periodic solution pn in compact set K.
From Theorem 3.5, one obtains the following result, which was proved by Gopalsamy in [10]
when System H is continuous case.
18 Advances in Difference Equations
Corollary 3.6. Under the assumption (iv), (v), and (vi), s uppose that b
i
n and a
ij
n are positive ω-
periodic sequences ω ∈ Z for all i, j 1, 2, ,l. Then System (H) has a unique ω-periodic solution
in K.
4. Examples
For simplicity, we consider the following prey-predator system with finite delay:
x
1
n 1
x
1
n
exp
1.5
sin n
2
− 0.5x
1
n
−
1 sin n
16
n
s−∞
1
2
s1
x
2
n − s
,
x
2
n 1
x
2
n
exp
−
1 0.2sinn
50
− 0.5x
2
n
1
4
n
s−∞
1
2
s1
x
1
n − s
.
E
1
Then, we have
b
1
1,B
1
2,a
1
A
1
0.5,c
1
C
1
1
4
,K
1
s
1
2
s1
,
b
2
0.016,B
2
0.024,a
2
A
2
0.5,c
2
0,C
2
1
8
,K
2
s
1
2
s1
4.1
for System F.Thus,
α
1
≈ 5.436,α
2
≈ 2.818,β
1
≈ 0.163. 4.2
It is easy to verify that System E
1
satisfies all the assumptions in our Theorem 2.6.Thus,
System E
1
has an almost periodic solution.
We next consider the following competitive system with finite delay:
x
1
n 1
x
1
n
exp
2
sin
√
2n
2
− x
1
n
−
1
16
n
s−∞
1
2
s1
x
2
n − s
,
x
2
n 1
x
2
n
exp
2 cos 2n − x
2
n
−
1
4
n
s−∞
1
2
s1
x
1
n − s
.
E
2
Then, we have
b
1
1.5,B
1
2.5,a
11
A
11
1,a
12
A
12
1
8
,K
12
s
1
2
s1
,
b
2
1,B
2
3,a
22
A
22
1,a
21
A
21
1
4
,K
21
s
1
2
s1
4.3
Advances in Difference Equations 19
for System E
2
.Thus,
α
1
≈ 4.077,α
2
≈ 7.389. 4.4
It is easy to verify that System E
2
satisfies all the assumptions in Theorem 3.5. Thus, System
E
2
has an almost periodic solution.
Acknowledgment
The authors would like to express their gratitude to the referees for their many helpful
comments.
References
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Journal, vol. 41, no. 1, pp. 105–116, 1989.
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pp. 859–870, 2007.
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