CUBO A Mathematical Journal
Vol.21, No¯ 01, (79–90). April 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000100079

Positive periodic solutions of functional discrete systems
with a parameter

Youssef N. Raffoul

Department of Mathematics,

University of Dayton, Dayton, OH 45469-2316

yraffoul1@udayton.edu

Ernest Yankson

Department of Mathematics and Statistics,

University of Cape Coast, Cape Coast, Ghana.

ernestoyank@gmail.com

ABSTRACT

The existence of multiple positive periodic solutions of the system of difference equa-

tions with a parameter

x(n + 1) = A(n,x(n))x(n) + λf(n,xn),

is studied. In particular, we use the eigenvalue problems of completely continuous op-

erators to obtain our results. We apply our results to a well-known model in population

dynamics.

http://dx.doi.org/10.4067/S0719-06462019000100079


80 Youssef N. Raffoul and Ernest Yankson CUBO
21, 1 (2019)

RESUMEN

Estudiamos la existencia de soluciones periódicas múltiples del siguiente sistema de

ecuaciones diferenciales con un parámetro

x(n + 1) = A(n,x(n))x(n) + λf(n,xn).

En particular, usamos los problemas de valores propios de operadores completamente

continuos para obtener nuestros resultados. Aplicamos nuestros resultados a modelos

de dinámica poblacional bien conocidos.

Keywords and Phrases: Functional difference system, Positive periodic solution, Eigenvalue,

Population model

2010 AMS Mathematics Subject Classification: 39A10, 39A12.



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Positive periodic solutions of functional discrete systems . . . 81

1 Introduction

Let R denote the real numbers, Z the integers, Z− the negative integers, R
k
+ = {(x1,x2, ...,xk)

T ∈

R
k : xj ≥ 0, j = 1,2, ...,k}, R

+ = {x ∈ R : x > 0}, and Z+ the nonnegative integers.

Also, let BC denote the normed vector space of bounded functions φ : Z → Rk, with the norm

||φ|| =
∑k

j=1
max

n∈[0,ω−1]
|φj(n)|, where φ = (φ1,φ2, ...,φk)

T and [0,ω − 1] = {0,1, ...,ω − 1}.

Particularly for each x = (x1,x2, ...,xk)
T ∈ Rk, we define the norm |x|0 =

∑k
j=1

|xj|. Also, denote

by BCk+ = {φ ∈ BC : φ(n) ∈ R
k
+ for n ∈ Z}.

In [12], Raffoul used a Krasnoselskii’s fixed point theorem in cones to prove the existence of positive

periodic solutions of the scaler difference equation with parameter

x(n + 1) = a(n)x(n) + λh(n)f(x(n − τ(n))).

Also, in [10], Zhu and Li generalized the work in [12] by proving that the system of difference

equations with parameter

x(n + 1) = A(n)x(n) + λh(n)f(x(n − τ(n)))

where A(n) = diag[a1(n),a2(n), ...,am(n)] and h(n) = diag[h1(n),h2(n), ...,hm(n)] has positive

periodic solutions. Motivated by the above considerations we investigate the existence of multiple

positive periodic solutions of the nonautonomous system of difference equations

x(n + 1) = A(n,x(n))x(n) + λf(n,xn), (1.1)

where, λ > 0 is a parameter, A(n,x(n)) = diag[a1(n,x(n)), ...,ak(n,x(n))], aj(n+ω,.) = aj(n,.),

f(n,x) : Z × BC → Rk is continuous in x and f(n,x) is ω-periodic in n and x, whenever x is ω-

periodic, ω ≥ 1 is an integer. If x ∈ BC, then xn ∈ BC for any n ∈ Z is defined by xn(θ) = x(n+θ)

for θ ∈ Z. Throughout this paper, we denote the product of y(n) from n = a to n = b by
∏b

n=a
y(n) with the understanding that

∏b
n=a

y(n) = 1 for all a > b. Also, for two m×n matri-

ces A and B, A ≥ B (A < B) means that the inequality is satisfied entrywisely. In particular, A is

said to be a nonnegative matrix if A ≥ 0.

Definition 3.1. [4] Let X be a Banach space and P a closed, nonempty subset of X. P is a (convex)

cone if

(i) x,y ∈ P and α,β ∈ R+ imply αx + βy ∈ P.

(ii) x ∈ P and −x ∈ P imply x = 0.

Definition 3.2. [4] Let X be a Banach space and D ⊂ X, 0 ∈ D. The operator L : D → X is such

that L0 = 0. xλ 6= 0 is said to be an eigenvector of the eigenvalue λ of L if Lxλ = λxλ.



82 Youssef N. Raffoul and Ernest Yankson CUBO
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Lemma 3.1. [4] Suppose D is an open subset of an infinite-dimensional real Banach space X,

0 ∈ D, and P is a cone of X. If the operator Γ : P ∩ D → P is completely continuous with Γ0 = 0

and satisfies infx∈P∩∂D ||Γx|| > 0, then Γ has an eigenvector on P ∩ ∂D associated with a positive

eigenvalue. That is, there exist x0 ∈ P ∩ ∂D and µ0 > 0 such that Γx0 = µ0x0.

In this paper we make the following assumptions.

(H1) 0 < aj(n) < 1, j = 1,2, ...k, and n ∈ [0, ω − 1].

(H2) There exist B(n) = diag[b1(n),b2(n), ...,bk(n)] and C(n) = diag[c1(n),c2(n), ...,ck(n)]

where bj,cj : Z → R+ are ω-periodic with 0 < bj,cj < 1, such that

B(n) ≤ A(n,ϕ(n)) ≤ C(n)

for all (n,ϕ) ∈ Z × BCk+.

(H3) f(n,0) = 0 for all n ∈ Z.

(H4) f(n,ϕn) ≤ 0 for all (n,ϕ) ∈ Z × BC
k
+.

(H5) For any L > 0 and ǫ > 0, there exists δ > 0 such that [φ,ψ ∈ BCk+, ||φ|| ≤ L, ||ψ|| ≤

L, ||φ − ψ|| < δ, 0 ≤ s ≤ ω] imply

|f(s,φs) − f(s,ψs)| < ǫ.

To study system (1.1) we let X = {x : Z → Rk, x(n + ω) = x(n)}, then it is clear that X ⊂ BC,

endowed with the norm ||x|| =
∑k

j=1 |xj|0, where |xj|0 = maxn∈[0,ω−1] |xj(n)|.

For the next lemma we consider

xj(n + 1) = aj(n,x(n))xj(n) + fj(n,xn), j = 1,2, ...,k. (1.2)

The proof of the next lemma can be easily deduced from [12] and hence we omit it.

Lemma 3.2. Suppose that (H1) hold. If x(n) ∈ X then xj(n) is a solution of equation (1.2) if

and only if

xj(n) =

n+T−1∑

u=n

Gxj (n,u)fj(n,xn), j = 1,2, ...,k, (1.3)

where

Gxj (n,u) =

∏n+T−1
s=u+1

aj(s,x(s))

1 −
∏n+T−1

s=n
aj(s,x(s))

, u ∈ [n,n + T − 1], j = 1,2, ...,k. (1.4)



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Positive periodic solutions of functional discrete systems . . . 83

Let

σ = min 1≤j≤k

( ∏ω−1
s=0

bj(s)
)[

1 −
∏ω−1

s=0
cj(s)

]

( ∏ω−1
s=0

cj(s)
)[

1 −
∏ω−1

s=0
bj(s)

] (1.5)

It can easily be obtained from (H2) that σ < 1. We next define two cones in X as follows:

P1 =
{
y ∈ X : yj(n) ≥ σ|yj|0,n ∈ Z and j = 1, ...,k

}
,

and

P2 =
{
y ∈ X : y(n) ≥ 0, n ∈ Z

}
.

Define an operator T on X by T : X → X by

(Tx) = (T1x,T2x,...,Tkx)
T. (1.6)

where

(Tjx)(n) =

n+ω−1∑

u=n

Gxj (n,u)fj(u,xu), j = 1, ...,k.

It is not very difficult to see that Gxj (n+ω,u+ω) = G
x
j (n,u). Also, it can easily be verified that

x∗(n) = (x∗1(n), ...,x
∗
k(n)) ≥ 0 is a positive ω-periodic solution of system (1.1) associated with λ

∗

if and only if x∗ ∈ P2 is an eigenvector of the operator T associated with the eigenvalue
1
λ∗
> 0,

that is Tx∗ = 1
λ∗
x∗.

Lemma 3.2. Suppose that (H1) and (H2) hold. Then the mapping T maps P1 into P1, i.e.,

TP1 ⊂ P1.

Proof. In view of (H1) and (H2), we have that, for j = 1,2, ...,k, and 0 ≤ u ≤ ω − 1,

∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

≤ Gxj (n,u) ≤

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

(1.7)



84 Youssef N. Raffoul and Ernest Yankson CUBO
21, 1 (2019)

|(Tjx)(n)| ≤

n+ω−1∑

u=n

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

|fj(u,xu)|

≤

∏ω−1
s=0 cj(s)

1 −
∏ω−1

s=0
cj(s)

ω−1∑

u=0

|fj(u,xu)|

It follows that

|(Tjx)|0 ≤

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

ω−1∑

u=0

|fj(u,xu)|

or

ω−1∑

u=0

|fj(u,xu)| ≥
1 −

∏ω−1
s=0

cj(s)
∏ω−1

s=0
cj(s)

|(Tjx)|0.

Therefore,

(Tjx)(n) ≥

∏ω−1
s=0 bj(s)

1 −
∏ω−1

s=0
bj(s)

ω−1∑

u=0

|fj(u,xu)|

≥

( ∏ω−1
s=0 bj(s)

)[

1 −
∏ω−1

s=0 cj(s)
]

( ∏ω−1
s=0

cj(s)
)[

1 −
∏ω−1

s=0
bj(s)

] |(Tjx)|o

≥ σ|(Tjx)|o,

which means that Tx ∈ P1. This completes the proof.

Lemma 3.3. Suppose (H5) hold. Then the operator T : P2 → X is completely continuous.

Proof. In view of (H5) and the assumption that f(n,x) is continuous in x, we have that the

operator T is continuous. We will show that T is compact.

Let U ⊆ P2 be any bounded set. Then, by the (H5), there exists a constant M > 0 such that

|fj(n,xn)| ≤ M, for (n,x) ∈ [0,ω − 1] × U, j = 1,2, ...,k.

Thus we have,

|(Tjx)| ≤

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

Mω.

It follows that,



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Positive periodic solutions of functional discrete systems . . . 85

||(Tx)|| =

k∑

j=1

|Tjx|0

≤ Mω

k∑

j=1

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

≤ Mkωγ,

where

γ = max 1≤j≤k

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

.

Next, we show that T maps bounded subsets into compact sets. Let J > 0 be given, and define

ρ = {ϕ ∈ P2 :‖ ϕ ‖≤ J} and Q = {(Tϕ)(n) : ϕ ∈ ρ}, then ρ is a subset of R
ωk which is closed and

bounded thus compact. As T is continuous in ϕ it maps compact sets into compact sets. Therefore

Q = T(ρ) is compact.

This completes the proof of lemma 3.3.

2 Main Results

In this section we state and prove our main results. For our main results we let

f0 = lim
φ∈P1, ||φ||→0

∑ω−1
u=0

|f(u,xu)|

||φ||
, and f∞ = lim

φ∈P1, ||φ||→∞

∑ω−1
u=0

|f(u,xu)|

||φ||
.

Also, define, for r a positive number, Ωr, by

Ωr = {x ∈ X : ||x|| < r }.

Theorem 4.1 Suppose that (H1)-(H5) hold and 0 < f∞ < ∞. Then there exist positive constants

R0, λ1, and λ2 with λ1 < λ2 such that, for any r > R0, system (1.1) has a positive ω-periodic

solution xr(n) associated with some λr ∈ [λ1,λ2] and ||x
r|| = r.

Proof. Since 0 < f∞ < +∞, there exist ǫ2 > ǫ1 > 0 and R0 > 0 such that

ǫ1||φ|| <

ω−1∑

u=0

|f(u,φu)| < ǫ2||φ|| for ||φ|| ≥ R0, φ ∈ P1.



86 Youssef N. Raffoul and Ernest Yankson CUBO
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Suppose r > R0, then Ωr is a bounded open subset of X and 0 ∈ Ωr. For x ∈ P1 ∩ ∂Ωr, we have

||Tx|| =

k∑

j=1

max
n∈[0,ω−1]

|(Tjx)(n)|

≥

k∑

j=1

|(Tjx)(n)|

=

k∑

j=1

ω−1∑

u=0

Gxj (n,u)fj(u,xu)

≥

k∑

j=1

∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

ω−1∑

u=0

fj(u,xu)

≥ min
1≤j≤k

∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

ω−1∑

u=0

k∑

j=1

|fj(u,xu)|

≥ min
1≤j≤k

∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

ǫ1r > 0.

It follows that

inf
x∈P1∩∂Ωr

||Tx|| ≥ min
1≤j≤k

{ ∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

}
ǫ1r > 0.

Since, T is completely continuous with T(0) = 0, it follows from Lemma 3.1 that the operator T

has an eigenvector xr ∈ P1 associated with the eigenvalue µr > 0 such that ||x
r|| = r. Set λr =

1
µr
.

Then xr is a positive ω-periodic solution of system (1.1).

We next determine λ1 and λ2 as follows. From

(xr)j(n) = λr

n+ω−1∑

u=n

Gx
r

j (n,u)fj(u,x
r
u)

≤ λr

ω−1∑

u=0

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

|fj(u,x
r
u)|

≤ λr

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

ω−1∑

u=0

|fj(u,x
r
u)|

≤ λr

∏ω−1
s=0

cj(s)

1 −
∏ω−1

s=0
cj(s)

ǫ2r, j = 1,2, ...,k,

and ||xr|| = r we can get

λr ≥
1

ǫ2
∑k

j=1

∏
ω−1
s=0

cj(s)

1−
∏

ω−1
s=0

cj(s)

=: λ1



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Positive periodic solutions of functional discrete systems . . . 87

On the other hand,

(xr)j(n) ≥ λr

∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

ω−1∑

u=0

|fj(u,x
r
u)|, j = 1, ...,k.

It follows from

||xr|| = r ≥ λr min
1≤j≤k

{ ∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

} ω−1∑

u=0

|f(u,xru)|

≥ λr min
1≤j≤k

{ ∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

}
ǫ1r

that

λr ≤ λr max
1≤j≤k

{1 −
∏ω−1

s=0
bj(s)

ǫ1
∏ω−1

s=0
bj(s)

}
:= λ2.

Therefore, λr ∈ [λ1,λ2] and this completes the proof.

Theorem 4.2. Suppose that (H1)-(H5) hold and 0 < f0 < ∞. Then there exist positive constants

r0 > 0, λ̃1 and λ̃2 with λ̃1 < λ̃2 such that, for any 0 < r < r0, system (1.1) has a positive

ω-periodic solution x̃r(n) associated with some λ̃r ∈ [λ̃1, λ̃2] and ||x̃r|| = r.

Proof. Since 0 < f0 < ∞, there exist 0 < l1 < l2 and r0 > 0 such that

l1||φ|| <

ω−1∑

u=0

|f(u,φu)| < l2||φ|| for 0 < ||φ|| < r0, φ ∈ P1.

For r ∈ (0,r0), Ωr is a bounded subset of X and 0 ∈ Ωr. Moreover, for x ∈ P1 ∩ ∂Ωr,

||Tx|| ≥

k∑

j=1

|(Tjx)(n)|

=

k∑

j=1

n+ω−1∑

u=n

Gxj (n,u)fj(u,xu)

≥ min
1≤j≤k

{ ∏ω−1
s=0

bj(s)

1 −
∏ω−1

s=0
bj(s)

}
l1r > 0.

This implies that infx∈P1∩∂ωr ||Tx|| > 0. The remaining part of the proof is similar to that of The-

orem 4.1 and so we omit it. This completes the proof.



88 Youssef N. Raffoul and Ernest Yankson CUBO
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Using arguments similar to that of Theorem 4.1 and Theorem 4.2, the following results can be

established respectively.

Theorem 4.3. Suppose that (H1)-(H5) hold and f∞ = ∞. Then there exist positive constants R̆0

and λ̆ such that, for any r > R̆0, system (1.1) has a positive ω-periodic solution x̆
r(n) associated

with some λ̆r ≤ λ̆ and ||x̆
r|| = r.

Theorem 4.4. Suppose that (H1)-(H5) hold and f0 = ∞. Then there exist positive constants

r̄0 and λ̄ such that, for any 0 < r < r̄0, system (1.1) has a positive ω-periodic solution x̄
r(n)

associated with some λ̄r ≤ λ̄ and ||x̄
r|| = r.

3 An application

In this section, we apply our results from the previous section to the Volterra discrete system

xj(n + 1) = xj(n)
[

aj(n) − λ

k∑

i=1

(

bji(n)xi(n) +

n∑

s=−∞

Cji(n,s)gji(xi(s))
)]

,

j = 1,2, ...,k,

(3.1)

where xj(n) is the population of the jth species, aj,bji : Z → R+ are ω-periodic and Cji(n,s) ≥ 0

and Cji(n + ω,s + ω) = Cji(n,s) for all (n,s) ∈ Z
2; gji : R+ → R+, i, j = 1, ...,k.

Theorem 5.1. Suppose that maxn∈Z
∑n

s=−∞ |Cji(n,s)| < +∞. Then there exist positive con-

stants R0 and λ0 such that, for any r > R0, system (3.1) has a positive ω-periodic solution x
r(n)

associated with λr ≤ λ0 and ||x
r|| = r.

Proof. Note that A(n,x(n)) = diag[a1(n),a2(n), ...,ak(n)] and f = (f1,f2, ...,fk) where

fj(n,xn) = −xj(n)

k∑

i=1

(

bji(n)xi(n) +

n∑

s=−∞

Cji(n,s)gji(xi(s))
)

for j = 1,2, ...,k and (H1)-(H5) are satisfied.



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Positive periodic solutions of functional discrete systems . . . 89

For x ∈ P1 and j = 1, ...,k we have

ω−1∑

u=0

|fj(u,xu)| =

k∑

i=1

ω−1∑

u=o

xj(u)
(

xi(u)bji(u) +

u∑

s=−∞

Cji(u,s)gji(xi(s))
)

≥

k∑

i=1

ω−1∑

u=o

xj(u)xi(u)bji(u)

≥

ω−1∑

u=o

x2j (u)bjj(u)

≥ σ2|xj|
2
0

ω−1∑

u=o

bjj(u).

Thus,

ω−1∑

u=0

|f(u,xu)| =

k∑

j=1

ω−1∑

u=0

|fj(u,xu)|

≥

k∑

j=1

σ2|xj|
2
0

ω−1∑

u=o

bjj(u)

≥ σ2 min
1≤j≤k

ω−1∑

u=o

bjj(u)

k∑

j=1

|xj|
2
0

≥
σ2

k
||x||2 min

1≤j≤k

ω−1∑

u=o

bjj(u).

It follows that

∑ω−1
u=0

|f(u,xu)|

||x||
→ as ||x|| → ∞.

The conclusion follows directly from Theorem 4.3 and this completes the proof.



90 Youssef N. Raffoul and Ernest Yankson CUBO
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	Introduction
	Main Results
	An application