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,[email protected]
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
x1n1 x1nexp{b1n−a1nx1n−c2n n
s−∞K2n−sx2s}, x2n1 x2nexp{−b2n−
a2nx2n c1nns−∞K1n−sx1s}and a competitive systemxin1 xinexp{bin−
aiixin−lj1,j /ins−∞Kijn−sxjs}, by using certain stability properties, which are referred to asK, ρ-weakly uniformly asymptotic stable in hull andK, ρ-totally stable.
Copyrightq2009 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 properties1–4. Recently, Song and Tian5have 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 Hamaya2to 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 in6. We emphasize that our results extend3,6,7as 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.
I ⊂Z : −∞,∞,we denote by BSIthe set of all bounded functions mappingI intoRm, and set|φ|I sup{|φs|:s∈I}.
Now, for any functionx : −∞, a → Rm and n < a, define a functionxn : Z− −∞,0 → Rm byxns xns fors ∈ Z−.Let BS be a real linear space of functions
mappingZ−intoRmwith sup-norm:
BS
φ|φ:Z−−→Rm withφsup
s∈Z−
φs<∞
. 1.1
We introduce an almost periodic functionfn, x:Z×D → Rm,whereDis an open set in Rm.
Definition 1.1. It holds thatfn, xis said to be almost periodic innuniformly forx∈D,if for any >0 and any compact setKinD,there exists a positive integerL∗, Ksuch that any interval of lengthL∗, Kcontains an integerτfor which
fnτ, x−fn, x≤ 1.2
for alln∈Zand allx∈K. Such a numberτin the above inquality is called an-translation number offn, 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 innuniformly for x ∈ D. Then, for any sequence {h
k} ⊂Z, there exist a subsequence{hk}of{hk}and functiongn, xsuch that
fnhk, x−→gn, x 1.3
uniformly onZ×Kask → ∞, whereK is a compact set inD. There are many properties of the discrete almost periodic functions 5,8, which are corresponding properties of the continuous almost periodic functionsft, x∈CR×D, Rm 4. We will denote byTfthe function space consisting of all translates off,that is,fτ ∈Tf,where
fτn, x fnτ, x, τ ∈Z. 1.4
LetHfdenote the uniform closure ofTfin the sense of1.4.Hfis called the hull of f.In particular, we denote byΩfthe set of all limit functionsg∈Hfsuch that for some sequence{nk},nk → ∞ask → ∞andfnnk, x → gn, xuniformly onZ×Sfor any
compact subsetSinRm. By1.3, iff :Z×D → Rmis almost periodic innuniformly for x∈D,so is a function inΩf. The following concept of asymptotic almost periodicity was introduced by Frechet in the case of continuous functioncf.4.
Definition 1.2. It holds thatunis said to be asymptotically almost periodic if it is a sum of an almost periodic functionpnand a functionqndefined onI∗ a,∞ ⊂ Z 0,∞ which tends to zero asn → ∞, that is,
However,unis asymptotically almost periodic if and only if for any sequence{nk}such
thatnk → ∞ask → ∞,there exists a subsequence{nk} for whichunnkconverges
uniformly onn;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:
x1n1 x1nexp
We now make the following assumptions:
iai>0, bi>0i1,2, c1>0,c2 ≥0,
iib1> C2α2, B2< c1β1,
iiithere exists a positive constantmsuch that
ai> Cim i1,2. 2.4
Then, we have 0< βi< αifor eachi1,2.
We can show the following lemmas.
Lemma 2.1. If xn x1n, x2nis a solution of Fthrough n0, φi,i 1,2such that
βi≤φis≤αi i1,2for alls≤0, then one hasβi≤xin≤αi i1,2for alln≥n0.
Proof. First, we claim that
lim sup
n→ ∞ x1n≤α1. 2.5
To do this, we first assume that there exists anl0 ≥ n0such thatx1l01≥ x1l0. Then, it
follows from the first equation ofFthat
b1l0−a1l0x1l0−c2l0 l0
s−∞
K2l0−sx2s≥0. 2.6
Hence,
x1l0≤
b1l0−c2l0
l0
s−∞K2l0−sx2s
a1l0 ≤
b1l0
a1l0 ≤
B1
a1
. 2.7
It follows that
x1l01 x1l0exp
b1l0−a1l0x1l0−c2l0 l0
s−∞
K2l0−sx2s
≤x1l0exp{B1−a1x1l0} ≤
exp{B1−1}
a1
:α1,
2.8
where we use the fact that
max
x∈R xexp
p−qx exp
p−1
q , forp, q >0. 2.9
Now we claim that
By way of contradiction, we assume that there exists ap0 > l0 such thatx1p0 > α1.Then,
p0 ≥ l02. Letp0 ≥ l02 be the smallest integer such that x1p0 > α1. Then,x1p0−1 ≤
x1p0. The above argument shows thatx1p0≤α1, which is a contradiction. This proves our
assertion. We now assume thatx1n1 < x1nfor alln ≥ n0. Then limn→ ∞x1nexists,
which is denoted byx1. We claim thatx1 ≤ expB1 −1/a1. Suppose to the contrary that
x1>expB1−1/a1. Taking limits in the first equation in SystemF, we set that
0 lim
n→ ∞
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
≤ lim
n→ ∞b1n−a1nx1n≤B1−a1x1<0,
2.10
which is a contradiction. It follows that2.5holds, and then we havexin≤α1for alln≥n0
from2.5∗. Next, we prove that
lim sup
n→ ∞ x2n≤α2. 2.11
We first assume that there exists ank0≥n0such thatk0≥l0andx2k01≥x2k0. Then
−b2k0−a2k0x2k0 c1k0 k0
s−∞
K1k0−sx1s≥0. 2.12
Hence,
x2k0≤
−b2k0 c1k0
k0
s−∞K1k0−sx1s
a2k0 ≤
−b2C1α1
a2
. 2.13
It follows that
x2k01 x2k0exp
−b2k0−a2k0x2k0 c1k0 k0
s−∞
K1k0−sx1s
≤x2k0exp{−b2−a2x2k0 C1α1}
≤ exp−b2aC1α1−1
2
:α2,
2.14
where we also use the two facts which are used to prove2.5. Now we claim that
x2n≤α2 ∀n≥k0. 2.11∗
Suppose to the contrary that there exists aq0 > k0such thatx2q0> α2. Thenq0≥k02. Let
q0 ≥ k02 be the smallest integer such thatx2q0> α2. Thenx2q0−1 < x2q0. Then the
above argument shows thatx2q0≤α2, which is a contradiction. This prove our claim from
Now, we assume thatx2n1< x2nfor alln≥n0. Then limn→ ∞x2nexists, which
is denoted byx2. We claim thatx2 ≤ exp−b2C1α1−1/a2. Suppose to the contrary that
x2>exp−b2C1α1−1/a2. Taking limits in the first equation in SystemF, we set that
0 lim
n→ ∞
−b2n−a2nx2n c1n n
s−∞
K1n−sx1s
≤ −b2−a2x2C1α1<0,
2.15
which is a contradiction. It follows that2.11holds. We show that
lim inf
n→ ∞ x1n≥β1. 2.16
According to the above assertion, there exists ak∗ ≥n0such thatx1n≤α1andx2n≤
α2, for alln≥k∗. We assume that there exists anl0≥k∗such thatx1l01≤x1l0. Note
that forn≥l0,
x1n1 x1nexp
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
≥x1nexp{b1−C2α2−A1x1n}.
2.17
In particular, withnl0, we have
b1−A1x1l0−C2α2≤0, 2.18
which implies that
x1l0≥ b1−AC2α2 1
. 2.19
Then,
x1l01≥ b1−AC2α2 1
expb1−C2α2−A1α1:x1. 2.20
We assert that
x1n≥x1, ∀n≥l0. 2.16∗
By way of contradiction, we assume that there exists ap0 ≥ l0 such thatx1p0 < x1. Then
p0 ≥l02. Letp02 be the smallest integer such thatx1p0< x1.Thenx1p0≤x1p0−1.
The above argument yieldsx1p0≥x1, which is a contradiction. This proves our claim. We
byx1. We claim thatx1≥b1−C2α2/A1. Suppose to the contrary thatx1 <b1−C2α2/A1.
Taking the limits in the first equation in SystemF, we set that
0 lim
n→ ∞
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
≥b1−A1x1−C2α2>0,
2.21
which is a contradiction. It follows that2.16holds, and then β1 ≤ x1n for all n ≥ n0
from2.16and2.16∗. Finally, by using the inequalityB2 < c1β1, similar arguments lead to
lim infn→ ∞x2≥β2, and thenx2n≥β2for alln≥k0. This proof is complete.
Lemma 2.2. LetKbe the closed bounded set inR2such that
Kx1, x2∈R2;βi ≤xi ≤αi for eachi1,2
. 2.22
ThenKis invariant for SystemF, that is, one can see that for any n0 ∈Z and anyϕi such that
ϕis ∈K,s ≤0 i 1,2, every solution of Fthroughn0, ϕiremains inK for alln≥n0and
i1,2.
Proof. FromLemma 2.1, it is sufficient to prove that thisK /φ.
To do this, by assumption of almost periodic functions, there exists a sequence {nk}, nk → ∞ as k → ∞, such that bin nk → bin, ain nk → ain, and
cinnk → cinask → ∞uniformly onZandi1,2. Letxnbe a solution of SystemF
throughn0, ϕthat remains inKfor alln≥n0, whose existence was ensured byLemma 2.1.
Clearly, the sequence{xnnk}is uniformly bounded on bounded subset ofZ. Therefore,
we may assume that the sequence{xnnk}converges to a functionyn y1n, y2n
ask → ∞uniformly on each bounded subset ofZtaking a subset of{xnnk}if necessary.
We may assume thatnk≥n0for allk. Forn≥0, we have
x1nnk1
x1nnkexp
b1nnk−a1nnkx1nnk−c2nnk nnk
s−∞
K2nnk−sx2s
,
x2nnk1
x2nnkexp
−b2nnk−a2nnkx2nnk c1nnk nnk
s−∞
K1nnk−sx1s
.
Fnk
Sincexnnk∈Kandyn∈Kfor alln∈Z, there existsr >0 such that|xnnk| ≤rand
|yn| ≤rfor alln∈Z. Then, by assumption of delay kernelKi, for thisrand any >0, there
exists an integerSS, r>0 such that
n−S
s−∞|
Kinnk−sxis| ≤,
n−S
s−∞
Then, we have
n
s−∞
Kinnk−sxis− n
s−∞
Kin−syis
≤ n−S
s−∞|Ki
nnk−sxis| n−S
s−∞
Kin−syis
n
sn−S
Kinnk−sxis−Kin−syis
≤2
n
sn−S
Kinnk−sxis−Kin−syis.
2.24
Sincexinnk−sconverges toyin−son discrete intervals∈n−S, nask → ∞, there
exists an integerk0> k∗, for somek∗>0, such that
n
sn−S
Kinnk−sxis−Kin−syis≤ i1,2 2.25
whenk≥k0. Thus, we have
n
s−∞
Kinnk−sxis−→ n
s−∞
Kin−syis 2.26
ask → ∞. Lettingk → ∞inFnk, we have
y1n1 y1nexp
b1n−a1ny1n−c2n n
s−∞
K2n−sy2s
,
y2n1 y2nexp
−b2n−a2ny2n c1n n
s−∞
K1n−sy1s
,
2.27
for alln ≥ n0. Then,yn y1n, y2nis a solution of SystemFonZ. It is clear that
We denote byΩFthe set of all limit functionsGsuch that for some sequence{nk}
such thatnk → ∞ask → ∞,binnk → bin, ainnk → ain,andcinnk → cin
uniformly onZask → ∞. Here, the equation forGis
x1n1 x1nexp
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
,
x2n1 x2nexp
−b2n−a2nx2n c1n n
s−∞
K1n−sx1s
.
G
Moreover, we denote byv, G∈Ωu, Fwhen for the same sequence{nk}, unnk → vn
uniformly on any compact subset inZ ask → ∞. Then a systemGis called a limiting equation ofFwhenG∈ΩFandvnis a solution ofGwhenv, G∈Ωu, F.
Lemma 2.3. If a compact setKinR2ofLemma 2.2is invariant for SystemF, thenKis invariant also for every limiting equation of SystemF.
Proof. LetGbe a limiting equation of systemF. SinceG∈ΩF, there exists a sequence {nk}such thatnk → ∞ask → ∞and thatbinnk → bin, ainnk → ain,and
cinnk → cinuniformly onZask → ∞. Letn0 ≥ 0, φ∈BSsuch thatφs∈Kfor all
s ≤ 0, and letynbe a solution of systemGthroughn0, φ. Letxknbe the solution of
SystemFthroughn0nk, φ. Thenxkn0nks φs∈Kfor alls≥0 andx
knis defined
onn ≥ n0nk. SinceKis invariant for SystemF,xkn∈ Kfor alln≥ n0nk. If we set
zkn xknn
k, k1,2, . . ., thenzknis defined onn≥n0and is a solution of
x1n1 x1nexp
b1nnk−a1nnkx1n−c2nnk nnk
s−∞
K2nnk−sx2s
,
x2n1 x2nexp
−b2nnk−a2nnkx2n c1nnk nnk
s−∞
K1nnk−sx1s
,
2.28
such thatzkn0s xkn0nks φs ∈ K for alls ≤ 0. Sincexkn ∈ K for alln ≥ n0nk,
zkn ∈ K for alln ≥ n
0. Since the sequence{zkn}is uniformly bounded onn0,∞and
zk
n0 φ, {z
kn} can be assumed to converge to the solutionyn of G through n 0, φ
uniformly on any compact setn0,∞, becauseynis the unique solution throughn0, φ
and the same argument as in the proof ofLemma 2.2. Therefore,yn∈Kfor alln≥n0since
zkn∈Kfor alln≥n
LetKbe the compact set inRmsuch thatun∈Kfor alln∈Z, whereun φ0n
forn≤0. For anyθ, ψ∈BS,we set
ρθ, ψ∞ j1
ρjθ, ψ
2j1ρjθ, ψ, 2.29
where
ρjθ, ψ sup −j≤s≤0
θs−ψs. 2.30
Clearly,ρθn, θ → 0 asn → ∞if and only ifθns → θsuniformly on any compact subset
of−∞,0asn → ∞.
In what follows, we need the following definitions of stability.
Definition 2.4. The bounded solutionunof SystemFis said to be as follows:
i K, ρ-totally stable in short,K, ρ-TSif for any > 0,there exists aδ > 0 such that ifn0 ≥0,ρxn0, un0< δandh h1, h2∈BSn0,∞which satisfies
|h|n0,∞< δ, thenρxn, un< for alln≥n0, wherexnis a solution of
x1n1 x1nexp
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
h1n,
x2n1 x2nexp
−b2n−a2nx2n c1n n
s−∞
K1n−sx1s
h2n,
Fh
throughn0, φsuch thatxn0s φs∈Kfor alls≤0. In the case wherehn≡0,
this gives the definition of theK, ρ-US ofun;
ii K, ρ-attracting inΩF in short,K, ρ-A inΩFif there exists aδ0>0 such that
ifn0 ≥0 and anyv, G∈Ωu, F,ρxn0, vn0< δ0, thenρxn, vn → 0 asn → ∞,
where xn is a solution oflimiting equation of 2.5;Gthrough n0, ψsuch
thatxn0s ψs∈Kfor alls≤0;
iii K, ρ-weakly uniformly asymptotically stable inΩF in short,K, ρ-WUAS in
ΩFif it isK, ρ-US inΩF, that is, if for anyε >0 there exists aδ>0 such that ifn0 ≥0 and anyv, G∈Ωu, F,ρxn0, vn0< δ, thenρxn, vn < for all
n≥n0, wherexnis a solution ofGthroughn0, ψsuch thatxn0s ψs∈K
Proposition 2.5. Under the assumption (i), (i), and (iii), if the solutionunof SystemFisK, ρ -WUAS inΩF, then the solutionunof SystemFisK, ρ-TS.
Proof. Suppose thatunis notK, ρ-TS. Then there exist a small >0, sequences{k},0 <
k< andk → 0 ask → ∞, sequences{sk},{nk},{hk},{xk}such thatsk → ∞ask → ∞,
0< sk1< nk,hk :Z → R2is bounded function satisfying|hkn|< kforn≥skand such
that
ρusk, xksk
< k, ρ
unk, xknk
≥, ρun, xnk
< sk, nk, 2.31
wherexknis a solution of
x1n1 x1nexp
b1n−a1nx1n−c2n n
s−∞
K2n−sx2s
hk1n,
x2n1 x2nexp
−b2n−a2nx2n c1n n
s−∞
K1n−sx1s
hk2n,
Fhk
such thatxksks ∈ K for alls ≤ 0. We can assume that < δ0 whereδ0 is the number for
K, ρ-A inΩFofDefinition 2.4. Moreover, by2.31, we can chose sequence{τk}such that
sk< τk< nk,
ρuτk, xτkk
δ/2
2 , 2.32
δ/2 2 ≤ρ
un, xnk
≤ forn∈τk, nk, 2.33
whereδ·is the number forK, ρ-US inΩF. We may assume thatunτk → vnas
k → ∞on each bounded subset ofZfor a functionv, and for the sequence{τk},τk → ∞
ask → ∞, taking a subsequence if necessary, there exists av, G∈ Ωu, F. Moreover, we may assume thatxknτk → znask → ∞uniformly on any bounded subset ofZfor
function z, since the sequence{xknτk}is uniformly bounded onZ. Because, if we set
ykn xknτk, thenyknis defined onn≥n
0τkandyknis a solution of
x1n1
x1nexp
b1nτk−a1nτkx1n−c2nτk n
s−∞
K2nnk−sx2s
x2n1
x2nexp
−b2nτk−a2nτkx2n c1nτk n
s−∞
K1nnk−sx1s
hk2nτk,
2.34
such thatyk0s xkτks ∈ K for all s ≤ 0. Then we may show that taking a subsequence if necessary, yknconverges to a solution zn of G such that z0s ∈ K fors ≤ 0, by
the same argument for Σ-calculations with condition ofKi as in the proof of Lemma 2.2.
Then, the same argument as in the proof ofLemma 2.2shows thatz∈K. Now, suppose that nk−τk → ∞ask → ∞. Letting k → ∞in2.33, we haveδ/2/2 ≤ ρvn, zn ≤ on
n ≥ 0. Since < δ0 and unisK, ρ-A in ΩF, we haveδ/2/2 ≤ ρvn, zn → 0 as
n → ∞, which is a contradiction. Thusnk−τk∞ask → ∞. Taking a subsequence again if
necessary, we can assum thatnk−τk → r <∞ask → ∞. Lettingk → ∞in2.32, we have
ρv0, z0 δ/2/2< δ/2and henceρvn, zn< /2 for alln≥0, becauseuisK, ρ-US
inΩF. On the other hand, from2.31, we haveρvn, zn≥, which is a contradiction. This
shows thatunisK, ρ-TS.
Now we will see that the existence of a strictly positive almost periodic solution of SystemFcan be obtained under conditionsi,ii,andiii.
Theorem 2.6. one assumes conditions i,ii, and iii. Then System F has a unique almost periodic solutionpnin compact setK.
Proof. For SystemF, we first introduce the change of variables:
uin exp{vin}, xin exp yin, i1,2. 2.35
Then, SystemFcan be written as
y1n1−y1n b1n−a1nexp y1n
−c2n
n
s−∞
K2n−sexp y2s
,
y2n1−y2n −b2n−a2nexp y2n
c1n n
s−∞
K1n−sexp y1s
,
F
We now consider Liapunov functional:
Vvn, yn2 i1
vin−yin ∞
s0
Kis n−1
ln−s
cislexp{vil} −exp yil
,
whereynandvnare solutions ofFwhich remains inK. Calculating the differences, we
From the mean value theorem, we have
whereθinlies betweenvinandyin i1,2. Then, byiii, we have
ΔVvn, yn≤ −mD2
i1
vin−yin, 2.39
where set D max{exp{β1},exp{β2}}, and let solutions xinof System F be such that
xin≥βiforn≥n0 i1,2. Thus2i1|vin−yin| → 0 asn → ∞, and henceρvn, yn →
0 asn → ∞. Moreover, we can show thatvnisK, ρ-US inΩofF, by the same argument as in5. By using similar Liapunov functional to2.36, we can show thatvnisK, ρ-A in
ΩofF. Therefore,vnisK, ρ-WUAS inΩF. Thus, fromProposition 2.5,vnisK, ρ -TS, becauseKis invariant. By the equivalence betweenFandF, solutionunof System
FisK, ρ-TS. Therefore, it follows from Theorem 4.4 in5and9that SystemFhas an almost periodic solutionpnsuch thatβi≤pin≤αi,i1,2, for alln∈Z.
3. Competitive System
We will consider thel-species almost periodic competitive Lotka-Volterra system:
xin1 xinexp
⎧ ⎨
⎩bin−aiixin−
l
j1,j /i
aijn n
s−∞
Kijn−sxjs
⎫ ⎬
⎭, i1,2, . . . , l,
H
wherebin, aijnare positive almost periodic sequences on Z;aijnare strictly positive,
and, moreover,
aijinf
n∈Zaijn, Aij supn∈Zaijn, bi ninf∈Zbin, Bisupn∈Zbin,
Kij:Z 0,∞−→R i, j 1,2, . . . , l,
3.1
which can be seen as the discretization of the differential equation in3. We set
αiexp{Bia−1} ii ,
βimin
⎧ ⎪ ⎨ ⎪ ⎩
expbi−Aiiαi−
l
j1,j /iAijαj
bi−
l
j1,j /iAijαj
Aii ,
bi−
l j1,j /iαj
Aii
⎫ ⎪ ⎬ ⎪ ⎭.
3.2
Now, we make the following assumptions:
ivKijs≥0,and∞s0Kijs 1,
∞
s0sKijs<∞i1,2, . . . , l;
vithere exists a positive constantmsuch that
aii > l
j1,j /i
Aijm i1,2, . . . , l. 3.3
Then, we have 0< βi < αifor eachi1,2, . . . , l. Under the assumptionsivandv, it
follows that for anyn0, φ∈Z×BS,there is a unique solutionun u1n, u2n, . . . , uln
ofHthroughn0, φ, if it remains bounded.
Then, we can show the similar lemmas toLemma 2.1.
Lemma 3.1. If xn x1n, x2n, . . . , xln is a solution of H through n0, φ such that
βi ≤ φs ≤ αi i 1,2, . . . , lfor all s ≤ 0, then one hasβi ≤ xin ≤ αi i 1,2, . . . , lfor all
n≥n0.
Proof. First, we claim that
lim sup
n→ ∞ xin≤Bi, i1,2, . . . , l. 3.4
Clearly,xin >0 forn ≥n0. To prove this, we first assume that there exists anl0 ≥ n0such
thatxil01≥xil0. Then, it follows from the first equation ofHthat
bil0−aiil0xil0− l
j1,j /i
aijl0 l0
s−∞
Kijl0−sxjs≥0. 3.5
Hence
xil0≤
bil0−
l
j1,j /iaijl0
l0
s−∞Kijl0−sxjs
aijl0 ≤
bil0
aiil0 ≤
Bi
aii. 3.6
It follows that
xil01 xil0exp
⎧ ⎨
⎩bil0−aiil0xil0− l
j1,j /i
aijl0 l0
s−∞
Kijl0−sxjs
⎫ ⎬ ⎭
≤xil0exp{Bi−aiixil0} ≤
exp{Bi−1}
aii :αi.
3.7
Now we claim that
xin≤Bi, forn≥l0. 3.8
By way of contradiction, we assume that there exists ap0 > l0such thatxip0> αi.Then,p0≥
above argument shows thatxip0 ≤αi, which is a contradiction. This proves our assertion.
We now assume thatxin1< xinfor alln≥n0. Then limn→ ∞xinexists, which is denoted
byxi. We claim thatxi ≤ expBi−1/aii. Suppose to the contrary thatxi >expBi−1/aii.
Taking limits in the first equation in SystemH, we set that
0 lim
By way of contradiction, we assume that there exists ap0≥l0such thatxip0< xiε. Thenp0≥
l02. Letp02 be the smallest integer such thatxip0< xiε.Thenxip0≤xip0−1. The above
argument yieldsxip0≥xiε, which is a contradiction. This proves our claim. We now assume
thatxin1< xinfor alln≥n0. Then limn→ ∞xinexists, which is denoted byxi. We claim
thatxi ≥bi−lj1,j /iAijαj/Aii. Suppose to the contrary thatxi <bi−lj1,j /iAijαj/Aii.
Taking limits in the first equation in SystemF, we set that
0 lim
n→ ∞
⎛
⎝bin−aiinxin− l
j1,j /i
aijn l
s−∞
Kijn−sxjs
⎞ ⎠
≥bi−Aiixi− l
j1,j /i
Aijαj>0,
3.16
which is a contradiction. It follows that3.10holds. This proof is complete.
By the same arguments of Lemmas2.2,2.3andProposition 2.5, we obtain Lemmas3.2,
3.3andProposition 3.4. So, we will omit to these proofs.
Lemma 3.2. LetKbe the closed bounded set inRlsuch that
Kx1, x2, . . . , xl∈Rl;βi ≤xi ≤αi for eachi1,2, . . . , l
. 3.17
ThenK is invariant for SystemH, that is, one can see that for anyn0 ∈ Zand anyϕsuch that
ϕs∈K,s≤0, every solution of Hthroughn0, ϕremains inKfor alln≥n0andi1,2, . . . , l.
Lemma 3.3. If a compact setKinRlofLemma 3.2is invariant for SystemH, thenKis invariant also for every limiting equation of SystemH.
Proposition 3.4. Under the assumption (iv), (v), and (vi), if the solution un of System H is
K, ρ-WUAS inΩH, then the solutionunof SystemHisK, ρ-TS.
By making changes of the variablesxin exp{yin}and defining the Liapunov functional
V by
Vvn, ynl i1
vin−yin ∞
s0
Kijs n−1
ln−s
cislexp{vil} −exp yil
,
3.18
where yn and vn are solutions of changing equation H! for H which remains in K, the arguments similar to theTheorem 2.6lead to the following results.
Theorem 3.5. one assumes conditions (iv), (v), and (vi). Then System H has a unique almost periodic solutionpnin compact setK.
Corollary 3.6. Under the assumption (iv), (v), and (vi), suppose thatbinandaijnare positiveω -periodic sequencesω∈Zfor alli, j1,2, . . . , l. Then System (H) has a uniqueω-periodic solution inK.
4. Examples
For simplicity, we consider the following prey-predator system with finite delay:
x1n1 x1nexp
It is easy to verify that SystemE1satisfies all the assumptions in ourTheorem 2.6. Thus, SystemE1has an almost periodic solution.
We next consider the following competitive system with finite delay:
for SystemE2. Thus,
α1≈4.077, α2≈7.389. 4.4
It is easy to verify that SystemE2satisfies all the assumptions inTheorem 3.5. Thus, System
E2has an almost periodic solution.
Acknowledgment
The authors would like to express their gratitude to the referees for their many helpful comments.
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