POSITIVE PERIODIC SOLUTIONS OF FUNCTIONAL
DISCRETE SYSTEMS AND POPULATION MODELS
YOUSSEF N. RAFFOUL AND CHRISTOPHER C. TISDELL
Received 29 March 2004 and in revised form 23 August 2004
We apply a cone-theoretic fixed point theorem to study the existence of positive pe-
riodic solutions of the nonlinear system of functional difference equations x(n+1)=
A(n)x(n)+ f(n,xn).
1. Introduction
Let Rdenote the real numbers, Zthe integers, Z−the negative integers, and Z+the non-
negative integers. In this paper we explore the existence of positive periodic solutions of
the nonlinear nonautonomous system of difference equations
x(n+1)=A(n)x(n)+ fn,xn, (1.1)
where, A(n)=diag[a1(n),a2(n),...,ak(n)], ajis ω-periodic, f(n,x):Z×Rk→Rkis con-
tinuous in xand f(n,x)isω-periodic in nand x,wheneverxis ω-periodic, ω≥1isan
integer. Let ᐄbe the set of all real ω-periodic sequences φ:Z→Rk. Endowed with the
maximum norm φ=maxθ∈Zk
j=1|φj(θ)|where φ=(φ1,φ2,...,φk)t,ᐄis a Banach
space. Here tstands for the transpose. If x∈ᐄ,thenxn∈ᐄfor any n∈Zis defined by
xn(θ)=x(n+θ)forθ∈Z.
The existence of multiple positive periodic solutions of nonlinear functional differen-
tial equations has been studied extensively in recent years. Some appropriate references
are [1,14]. We are particularly motivated by the work in [8] on functional differential
equations and the work of the first author in [4,11,12] on boundary value problems
involving functional difference equations.
When working with certain boundary value problems whether in differential or dif-
ference equations, it is customary to display the desired solution in terms of a suitable
Green’s function and then apply cone theory [2,4,5,6,7,10,13]. Since our equation
(1.1) is not this type of boundary value, we obtain a variation of parameters formula and
then try to find a lower and upper estimates for the kernel inside the summation. Once
those estimates are found we use Krasnoselskii’s fixed point theorem to show the existence
of a positive periodic solution. In [11], the first author studied the existence of periodic
solutions of an equation similar to (1.1) using Schauder’s second fixed point theorem.
Copyright ©2005 Hindawi Publishing Corporation
Advances in Difference Equations 2005:3 (2005) 369–380
DOI: 10.1155/ADE.2005.369
370 Positive periodic solutions
Throughout this paper, we denote the product of y(n)fromn=ato n=bby b
n=ay(n)
with the understanding that b
n=ay(n)=1foralla>b.
In [12], the first author considered the scalar difference equation
x(n+1)=a(n)x(n)+h(n)fxn−τ(n), (1.2)
where a(n), h(n), and τ(n)areω-periodic for ωan integer with ω≥1. Under the assump-
tions that a(n), f(x), and h(n) are nonnegative with 0 <a(n)<1foralln∈[0,ω−1], it
was shown that (1.2) possesses a positive periodic solution. In this paper we generalize
(1.2) to systems with infinite delay and address the existence of positive periodic solutions
of (1.1) in the case a(n)>1.
Let R+=[0,+∞), for each x=(x1,x2,...,xn)t∈Rn,thenormofxis defined as |x|=
n
j=1|xj|.Rn
+={(x1,x2,...,xn)t∈Rn:xj≥0, j=1,2,...,n}. Also, we denote f=(f1,
f2,...,fk)t,wheretstands for transpose.
Now we list the following conditions.
(H1) a(n)= 0foralln∈[0,ω−1] with ω−1
s=0aj(s)= 1for j=1,2,...,k.
(H2) If 0 <a(n)<1foralln∈[0,ω−1] then, fj(n,φn)≥0foralln∈Zand φ:Z→Rn
+,
j=1,2,...,kwhere R+=[0,+∞).
(H3) If a(n)>1foralln∈[0,ω−1] then, fj(n,φn)≤0foralln∈Zand φ:Z→Rn
+,
j=1,2,...,kwhere R+=[0,+∞).
(H4) For any L>0andε>0, there exists δ>0suchthat[φ,ψ∈ᐄ,φ≤L,ψ≤
L,φ−ψ<δ,0≤s≤ω]imply
fs,φs−fs,ψs
<ε. (1.3)
2. Preliminaries
In this section we state some preliminaries in the form of definitions and lemmas that are
essential to the proofs of our main results. We start with the following definition.
Definition 2.1. Let XbeaBanachspaceandKbe a closed, nonempty subset of X. The set
Kis a cone if
(i) αu +βv ∈Kfor all u,v∈Kand all α,β≥0
(ii) u,−u∈Kimply u=0.
We now state the Krasnosel’skii fixed point theorem [9].
Theorem 2.2 (Krasnosel’skii). Let Ꮾbe a Banach space, and let ᏼbe a cone in Ꮾ.Suppose
Ω1and Ω2are open subsets of Ꮾsuch that 0∈Ω1⊂Ω1⊂Ω2and suppose that
T:ᏼ∩Ω2\Ω1−→ ᏼ(2.1)
is a completely continuous operator such that
(i) Tu≤u,u∈ᏼ∩∂Ω1,andTu≥u,u∈ᏼ∩∂Ω2;or
(ii) Tu≥u,u∈ᏼ∩∂Ω1,andTu≤u,u∈ᏼ∩∂Ω2.
Then Thas a fixed point in ᏼ∩(Ω2\Ω1).
Y. N. Ra ffoul and C. C. Tisdell 371
For the next lemma we consider
xj(n+1)=ajxj(n)+ fjn,xn,j=1,2,...,k. (2.2)
The proof of the next lemma can be easily deduced from [11] and hence we omit it.
Lemma 2.3. Suppose (H1) holds. Then xj(n)∈ᐄis a solution of (2.2)ifandonlyif
xj(n)=
n+ω−1
u=n
Gj(n,u)fju,xu,j=1,2,...,k, (2.3)
where
Gj(n,u)=n+ω−1
s=u+1 aj(s)
1−n+ω−1
s=naj(s),u∈[n,n+ω−1], j=1,2,...,k. (2.4)
Set
G(n,u)=diagG1(n,u),G2(n,u),...,Gk(n,u).(2.5)
It is clear that G(n,u)=G(n+ω,u+ω)forall(n,u)∈Z2. Also, if either (H2) or (H3)
holds, then (2.4) implies that
Gj(n,u)fju,φu≥0 (2.6)
for (n,u)∈Z2and u∈Z,φ:Z→Rn
+. To define the desired cone, we observe that if (H2)
holds, then
ω−1
s=0aj(s)
1−n+ω−1
s=naj(s)≤
Gj(n,u)
≤ω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s)(2.7)
for all u∈[n,n+ω−1]. Also, if (H3) holds then
ω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s)
≤
Gj(n,u)
≤ω−1
s=0aj(s)
1−n+ω−1
s=naj(s)
(2.8)
for all u∈[n,n+ω−1]. For all (n,s)∈Z2,j=1,2, ...,k,wedefine
σ2:=minω−1
s=0
aj(s)2
,j=1,2,...,n,
σ3:=minω−1
s=0
a−1
j(s)2
,j=1,2,...,n.
(2.9)
We note that if 0 <a(n)<1foralln∈[0,ω−1], then σ2∈(0,1). Also, if a(n)>1for
all n∈[0,ω−1], then σ3∈(0,1). Conditions (H2) and (H3) will have to be handled
372 Positive periodic solutions
separately. That is, we define two cones; namely, ᏼ2andᏼ3. Thus, for each y∈ᐄset
ᏼ2=y∈ᐄ:y(n)≥0, n∈Z,andy(n)≥σ2y,
ᏼ3=y∈ᐄ:y(n)≥0, n∈Z,andy(n)≥σ3y.(2.10)
Define a mapping T:ᐄ→ᐄby
(Tx)(n)=
n+ω−1
u=n
G(n,u)fu,xu, (2.11)
where G(n,u) is defined following (2.4). We denote
(Tx)=T1x,T2x,...,Tnxt.(2.12)
It is clear that (Tx)(n+ω)=(Tx)(n).
Lemma 2.4. If (H1) and (H2) hold, then the operator Tᏼ2⊂ᏼ2. If (H1) and (H3) hold,
then Tᏼ3⊂ᏼ3.
Proof. Suppose (H1) and (H2) hold. Then for any x∈ᏼ2wehave
Tjx(n)≥0, j=1,2,...,k. (2.13)
Also, for x∈ᏼ2 by using (2.4), (2.7), and (2.11)wehavethat
Tjx(n)≤ω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s)
n+ω−1
u=n
fju,xu
,
Tjx
=max
n∈[0,ω−1]
Tjx(n)
≤ω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s)
n+ω−1
u=n
fju,xu
.
(2.14)
Therefore,
Tjx(n)=
n+ω−1
u=n
Gj(n,u)fju,xu
≥ω−1
s=0aj(s)
1−n+ω−1
s=naj(s)
n+ω−1
u=n
fju,xu
≥ω−1
s=0
aj(s)2
Tjx
≥σ2
Tjx
.
(2.15)
That is, Tᏼ2 is contained in ᏼ2. The proof of the other part follows in the same manner
by simply using (2.8), and hence we omit it. This completes the proof.
Y. N. Ra ffoul and C. C. Tisdell 373
To simplify notation, we state the following notation:
A2=min
1≤j≤kω−1
s=0aj(s)
1−n+ω−1
s=naj(s), (2.16)
B2=max
1≤j≤kω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s), (2.17)
A3=min
1≤j≤kω−1
s=0a−1
j(s)
1−n+ω−1
s=naj(s)
, (2.18)
B3=max
1≤j≤kω−1
s=0aj(s)
1−n+ω−1
s=naj(s)
, (2.19)
where kis defined in the introduction.
Lemma 2.5. If (H1), (H2), and (H4) hold, then the operator T:ᏼ2→ᏼ2is completely
continuous. Similarly, if (H1), (H3), and (H4) hold, then the operator T:ᏼ3→ᏼ3is com-
pletely continuous.
Proof. Suppose (H1), (H2), and (H4) hold. First show that Tis continuous. By (H4), for
any L>0andε>0, there exists a δ>0suchthat[φ,ψ∈ᐄ,φ≤L,ψ≤L,φ−ψ<
δ]imply
max
0≤s≤ω−1
fs,φs−fs,ψs
<ε
B2ω, (2.20)
where B2is given by (2.17). If x,y∈ᏼ2withx≤L,y≤L,andx−y<δ,then
(Tx)(n)−(Ty)(n)
≤
n+ω−1
u=n
G(n,u)
fu,xu−fu,yu
≤B2
ω−1
u=0
fu,xu−fu,yu
<ε
(2.21)
for all n∈[0,ω−1], where |G(n,u)|=max1≤j≤n|Gj(n,u)|,j=1,2,...,k. This yields
(Tx)−(Ty)<ε.Thus,Tis continuous. Next we show that Tmaps bounded sub-
sets into compact subsets. Let ε=1. By (H4), for any µ>0 there exists δ>0suchthat
[x,y∈ᐄ,x≤µ,y≤µ,x−y<δ]imply
fs,xs−fs,ys
<1.(2.22)
We choose a positive integer Nso that δ>µ/N.Forx∈ᐄ,definexi(n)=ix(n)/N,for
i=0,1,2,...,N.Forx≤µ,
xi−xi−1
=max
n∈Z
ix(n)
N−(i−1)x(n)
N
≤x
N≤µ
N<δ.
(2.23)
