# On the Periodicity of Some Classes of Systems of Nonlinear Difference Equations

Authors: Stevo Stević; Mohammed A. Alghamdi; Dalal A. Maturi; Naseer Shahzad
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Some classes of systems of difference equations whose all well-defined solutions are periodic are presented in this note.


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## Body

## 1. Introduction

There has been a great recent interest in studying difference equations and systems of difference equations which do not stem from differential ones (see, e.g., [1–19] and the references therein). For some results on concrete systems of nonlinear difference equations, see, for example, [1, 3–5, 9–12, 18, 19]. Some classical results in the topic can be found, for example, in book [20].Solution(
x
n
(
1
)
,
…
,
x
n
(
l
)
)
n
≥
-
k, of the system of difference equations
(1)
x
n
(
1
)
=
f
1
(
x
n
-
1
(
1
)
,
…
,
x
n
-
k
1
(
1
)
,
…
,
x
n
-
1
(
l
)
,
…
,
x
n
-
k
l
(
l
)
)
,
x
n
(
2
)
=
f
2
(
x
n
-
1
(
1
)
,
…
,
x
n
-
k
1
(
1
)
,
…
,
x
n
-
1
(
l
)
,
…
,
x
n
-
k
l
(
l
)
)
,
⋮
x
n
(
l
)
=
f
l
(
x
n
-
1
(
1
)
,
…
,
x
n
-
k
1
(
1
)
,
…
,
x
n
-
1
(
l
)
,
…
,
x
n
-
k
l
(
l
)
)
,
where   n
∈
ℕ
0 and k
=
max
⁡
{
k
1
,
…
,
k
l
}, is called eventually periodic with period p, if there is an n
1
≥
-
k such that
(2)
x
n
+
p
(
j
)
=
x
n
(
j
)
,
for every j
=
1
,
l
¯, and n
≥
n
1. It is periodic with period p if n
1
=
-
k. Period p is prime if there is no p
^
∈
ℕ, p
^
<
p, which is a period. If all well-defined solutions of an equation or a system of difference equations are eventually periodic with the same period, then such an equation or system is called periodic. For some results on the periodicity, asymptotic periodicity and periodic equations or systems of difference equations see, for example, [1–10, 12–14, 16–19] and the related references therein.In recent paper [19], the authors formulated four results which claim that the following systems of difference equations are periodic with period ten:
(3)
x
n
+
1
=
y
n
x
n
-
1
(
1
+
y
n
)
,
y
n
+
1
=
x
n
y
n
-
1
(
1
+
x
n
)
,
n
∈
ℕ
0
;
(4)
x
n
+
1
=
y
n
x
n
-
1
(
-
1
+
y
n
)
,
y
n
+
1
=
x
n
y
n
-
1
(
-
1
+
x
n
)
,
n
∈
ℕ
0
;
(5)
x
n
+
1
=
y
n
x
n
-
1
(
1
+
y
n
)
,
y
n
+
1
=
x
n
y
n
-
1
(
-
1
+
x
n
)
,
n
∈
ℕ
0
;
(6)
x
n
+
1
=
y
n
x
n
-
1
(
-
1
+
y
n
)
,
y
n
+
1
=
x
n
y
n
-
1
(
1
+
x
n
)
,
n
∈
ℕ
0
.First, we show that all the results in [19] follow from known ones in the literature and also present some extensions of these results in the spirit of systems (3)–(6). To do this, we will use a system of difference equations related to the following, so called, Lyness difference equation:
(7)
x
n
+
1
=
1
+
x
n
x
n
-
1
,
n
∈
ℕ
0
.
It is easy to see that every well-defined solution of (7) is periodic with period five. The equation arises in frieze patterns (for the original sources, see [21–23]).Studying max-type equations and systems of difference equations is another topic of a recent interest (see, e.g, [2, 3, 5–7, 10, 11, 15–19]).Some special cases of the following max-type difference equation:(8)
x
n
=
max
⁡
{
A
n
x
n
-
s
,
x
n
-
k
}
,
n
∈
ℕ
0
,
where s, k
∈
ℕ, and (
A
n
)
n
∈
ℕ
0
⊂
ℝ, have been studied, for example, in [2, 16]. Positive solutions of (8) are periodic in many cases. However, if (
A
n
)
n
∈
ℕ
0 is not a positive sequence, it was shown in [2] that (8) can have unbounded solutions.In [5], it was shown that all solutions of the following max-type system of difference equations:
(9)
x
n
+
1
=
max
⁡
{
A
n
y
n
,
x
n
-
1
}
,
y
n
+
1
=
max
⁡
{
B
n
x
n
,
y
n
-
1
}
,
11111111111111111111111111111111
11
11
n
∈
ℕ
0
,
where x
0, x
-
1, y
0, y
-
1
∈
(
0
,
+
∞
) and (
A
n
)
n
∈
ℕ
0, (
B
n
)
n
∈
ℕ
0 are positive two-periodic sequences, are eventually periodic with, not necessarily prime, period two. This was done by direct calculation.By using direct calculation, it can be easily shown that positive solutions of the following max-type system of difference equations:(10)
x
n
+
1
=
max
⁡
{
A
n
x
n
,
y
n
-
1
}
,
y
n
+
1
=
max
⁡
{
B
n
y
n
,
x
n
-
1
}
,
11111111111111111111111111
11
11111111
n
∈
ℕ
0
,
where (
A
n
)
n
∈
ℕ
0 and (
B
n
)
n
∈
ℕ
0 are positive two-periodic sequences, are also periodic.Here, we give a noncalculatory explanation of the fact by proving that positive solutions of the following max-type system of difference equations:(11)
x
n
=
max
⁡
{
A
n
x
n
-
s
,
y
n
-
k
}
,
y
n
=
max
⁡
{
B
n
y
n
-
s
,
x
n
-
k
}
,
1111111111111111111111111111
11
111111
n
∈
ℕ
0
,
where s, k
∈
ℕ, and (
A
n
)
n
∈
ℕ
0, (
B
n
)
n
∈
ℕ
0 are positive periodic sequences of a certain period, are also periodic. We also present another extension of the result.


## 2. Some Extensions of Systems (3)–(6)

In this section, we present some periodic systems of difference equations in the spirit of systems (3)–(6).Theorem 1.
Consider the following system of difference equations(12)
x
n
+
1
(
1
)
=
f
1
-
1
(
1
+
f
2
(
x
n
(
2
)
)
f
3
(
x
n
-
1
(
3
)
)
)
,
⋮
x
n
+
1
(
k
-
1
)
=
f
k
-
1
-
1
(
1
+
f
k
(
x
n
(
k
)
)
f
1
(
x
n
-
1
(
1
)
)
)
,
x
n
+
1
(
k
)
=
f
k
-
1
(
1
+
f
1
(
x
n
(
1
)
)
f
2
(
x
n
-
1
(
2
)
)
)
,
n
∈
ℕ
0
,
where k
∈
ℕ
∖
{
1
}, and functions f
j, j
=
1
,
k
¯, are continuous on their domains; map the set ℝ
∖
{
0
} onto itself and, for each fixed j
∈
{
1
,
…
,
k
}, f
j is simultaneously increasing or decreasing on the intervals (
-
∞
,
0
) and (
0
,
+
∞
).
Then the following statements hold.(a)
Ifk
≢
0
(
mod
⁡
5
), then every well-defined solution of system (12) is periodic with period 5
k.
(b)
Ifk
≡
0
(
mod
⁡
5
), then every well-defined solution of system (12) is periodic with period k.Proof.
From the conditions of the theorem, it follows that for eachj
∈
{
1
,
…
,
k
}, there is f
j
-
1 which continuously map the set ℝ
∖
{
0
} onto itself. Using the change of variables
(13)
y
n
(
j
)
=
f
j
(
x
n
(
j
)
)
,
j
=
1
,
k
¯
,
system (12) is easily transformed into the next one
(14)
y
n
+
1
(
1
)
=
1
+
y
n
(
2
)
y
n
-
1
(
1
)
,
y
n
+
1
(
2
)
=
1
+
y
n
(
1
)
y
n
-
1
(
2
)
,
for k
=
2,
(15)
y
n
+
1
(
1
)
=
1
+
y
n
(
2
)
y
n
-
1
(
3
)
,
y
n
+
1
(
2
)
=
1
+
y
n
(
3
)
y
n
-
1
(
1
)
,
y
n
+
1
(
3
)
=
1
+
y
n
(
1
)
y
n
-
1
(
2
)
,
for k
=
3, and
(16)
y
n
+
1
(
1
)
=
1
+
y
n
(
2
)
y
n
-
1
(
3
)
,
y
n
+
1
(
2
)
=
1
+
y
n
(
3
)
y
n
-
1
(
4
)
,
…
,
y
n
+
1
(
k
)
=
1
+
y
n
(
1
)
y
n
-
1
(
2
)
,
for k
≥
4. In [4], it was proved that, if k
≢  (
mod
⁡
5
), then every well-defined solution of systems (14)–(16) is periodic with period 5
k, and, if k
≡
0
(
mod
⁡
5
), then every well-defined solution of systems (14)–(16) is periodic with period k. Using this along with the fact
(17)
x
n
(
j
)
=
f
j
-
1
(
y
n
(
j
)
)
,
j
=
1
,
k
¯
,
following from (13), the results in (a) and (b) follow.The following theorem is proved in a similar way. Therefore, the proof will be omitted.Theorem 2.
Consider the following system of difference equations(18)
x
n
+
1
(
1
)
=
f
1
-
1
(
1
+
f
k
(
x
n
(
k
)
)
f
k
-
1
(
x
n
-
1
(
k
-
1
)
)
)
,
x
n
+
1
(
2
)
=
f
2
-
1
(
1
+
f
1
(
x
n
(
1
)
)
f
k
(
x
n
-
1
(
k
)
)
)
,
⋮
x
n
+
1
(
k
)
=
f
k
-
1
(
1
+
f
k
-
1
(
x
n
(
k
-
1
)
)
f
k
-
2
(
x
n
-
1
(
k
-
2
)
)
)
,
n
∈
ℕ
0
,
where k
∈
ℕ
∖
{
1
}, and functions f
j, j
=
1
,
k
¯, are continuous on their domains; map the set ℝ
∖
{
0
} onto itself and, for each fixed j
∈
{
1
,
…
,
k
}, f
j is simultaneously increasing or decreasing on the intervals (
-
∞
,
0
) and (
0
,
+
∞
).
Then the following statements hold.(a)
Ifk
≢
0
(
mod
⁡
5
), then every well-defined solution of system (18) is periodic with period 5
k.
(b)
Ifk
≡
0
(
mod
⁡
5
), then every well-defined solution of system (18) is periodic with period k.Now, we show that all the results on the periodicity of the solutions of systems (3)–(6) in [19] follow from Theorems 1 and 2.Corollary 3.
Systems of difference equations (3)–(6) are all periodic with period ten.Proof.
For the systems of difference equations (3)–(6), we use the following changes of variables, respectively:
(19)
x
n
=
1
u
n
,
y
n
=
1
v
n
,
n
∈
ℕ
0
;
x
n
=
-
1
u
n
,
y
n
=
-
1
v
n
,
n
∈
ℕ
0
;
x
n
=
-
1
u
n
,
y
n
=
1
v
n
,
n
∈
ℕ
0
;
x
n
=
1
u
n
,
y
n
=
-
1
v
n
,
n
∈
ℕ
0
.
By using them, systems (3)–(6) are transformed into system (14). By applying Theorem 1(a), ten-periodicity of all well-defined solutions of system (14) follows, from which ten-periodicity of all well-defined solutions of systems (3)–(6) follows.The following four examples are natural extensions of systems (3)–(6).Example 4.
If we use the following functions:(20)
f
1
(
x
)
=
1
x
l
,
f
2
(
x
)
=
1
x
m
,
where l and m are odd integers, we see that all the conditions in Theorem 1 are applied with k
=
2, so by using the theorem we obtain that the system of difference equations
(21)
x
n
+
1
=
1
x
n
-
1
(
y
n
m
1
+
y
n
m
)
1
/
l
,
y
n
+
1
=
1
y
n
-
1
(
x
n
l
1
+
x
n
l
)
1
/
m
,
n
∈
ℕ
0, is also periodic with period ten.Example 5.
For(22)
f
1
(
x
)
=
-
1
x
l
,
f
2
(
x
)
=
-
1
x
m
,
where l and m are also odd integers, all the conditions of Theorem 1 are again satisfied with k
=
2. Using the theorem, it follows that the system
(23)
x
n
+
1
=
1
x
n
-
1
(
y
n
m
-
1
+
y
n
m
)
1
/
l
,
y
n
+
1
=
1
y
n
-
1
(
x
n
l
-
1
+
x
n
l
)
1
/
m
,
n
∈
ℕ
0, is ten-periodic.Example 6.
For(24)
f
1
(
x
)
=
1
x
l
,
f
2
(
x
)
=
-
1
x
m
,
where l and m are odd integers, all the conditions of Theorem 1 are also satisfied with k
=
2. Applying the theorem, it follows that the system
(25)
x
n
+
1
=
1
x
n
-
1
(
y
n
m
-
1
+
y
n
m
)
1
/
l
,
y
n
+
1
=
1
y
n
-
1
(
x
n
l
1
+
x
n
l
)
1
/
m
,
n
∈
ℕ
0, is ten-periodic.Example 7.
Finally, for(26)
f
1
(
x
)
=
-
1
x
l
,
f
2
(
x
)
=
1
x
m
,
where l and m are odd integers and applying Theorem 1 with k
=
2, we get that the system
(27)
x
n
+
1
=
1
x
n
-
1
(
y
n
m
1
+
y
n
m
)
1
/
l
,
y
n
+
1
=
1
y
n
-
1
(
x
n
l
-
1
+
x
n
l
)
1
/
m
,
n
∈
ℕ
0, is ten-periodic.The main results in [4] can be relatively easily extended to a very general situation, which have been noticed by Iričanin and Stević soon after publishing [4], and later also proved by several other authors. Namely, the following result holds (see, e.g., [1]).Theorem 8.
Assume that the following difference equation(28)
x
n
=
f
(
x
n
-
1
,
…
,
x
n
-
k
)
,
n
∈
ℕ
0
,
is periodic with period p.
Then the following system of difference equations(29)
x
n
(
i
)
=
f
(
x
n
-
1
(
σ
(
i
)
)
,
x
n
-
2
(
σ
[
2
]
(
i
)
)
,
…
,
x
n
-
k
(
σ
[
k
]
(
i
)
)
)
,
1111111111111
i
=
1
,
l
¯
,
n
∈
ℕ
0
,
where σ
(
i
)
=
i
+
1, for 1
≤
i
≤
l
-
1, σ
(
l
)
=
1 and σ
[
j
]
(
i
)
=
σ
(
σ
[
j
-
1
]
(
i
)
), j
=
1
,
k
¯, and σ
[
0
]
(
i
)
=
i, i
=
1
,
l
¯, is periodic with period lcm

(
p
,
l
) (the least common multiple of numbers p and l).Theorem8 can be used in constructing numerous periodic cyclic systems of difference equations based on scalar periodic difference equations, which, with some changes of variables, give some other periodic cyclic systems of difference equations.


## 3. Periodicity of Positive Solutions of System (11)

In this section, we study positive solutions of system (11). By gcd
(
s
,
k
), we denote the greatest common divisor of natural numbers s and k.Theorem 9.
Consider system (11). Assume that s, k
∈
ℕ, and (
A
n
)
n
∈
ℕ
0 and  (
B
n
)
n
∈
ℕ
0 are positive k
gcd

(
s
,
k
)-periodic sequences. Then every positive solution of system (11) is periodic with, not necessarily prime, period
(30)
p
=
2
k
gcd


(
s
,
k
)
.Proof.
Letr
=
gcd
(
s
,
k
). Then we have that s
=
r
s
1 and k
=
r
k
1 for some s
1, k
1
∈
ℕ such that
(31)
gcd
(
s
1
,
k
1
)
=
1
.
Since everyn
∈
ℕ
0 can be written as n
=
m
r
+
i, for some m
∈
ℕ
0 and i
=
0
,
r
-
1
¯, system (11) becomes
(32)
x
m
r
+
i
=
max
⁡
{
A
m
r
+
i
x
r
(
m
-
s
1
)
+
i
,
y
r
(
m
-
k
1
)
+
i
}
,
y
m
r
+
i
=
max
⁡
{
B
m
r
+
i
y
r
(
m
-
s
1
)
+
i
,
x
r
(
m
-
k
1
)
+
i
}
,
for every m
∈
ℕ
0 and i
=
0
,
r
-
1
¯.
Using the next change of variables(33)
x
t
(
i
)
=
x
t
r
+
i
,
y
t
(
i
)
=
y
t
r
+
i
,
where  t
≥
-
max
⁡
{
s
1
,
k
1
},  i
=
0
,
r
-
1
¯, in (32), we have that (
x
t
(
i
)
)
t
≥
-
max
⁡
{
s
1
,
k
1
}, (
y
t
(
i
)
)
t
≥
-
max
⁡
{
s
1
,
k
1
},  i
=
0
,
r
-
1
¯, are r independent solutions of the next systems
(34)
x
t
=
max
⁡
{
A
t
r
+
i
x
t
-
s
1
,
y
t
-
k
1
}
,
y
t
=
max
⁡
{
B
t
r
+
i
y
t
-
s
1
,
x
t
-
k
1
}
,
which are systems of the form in (11) with s
1 and k
1 instead of s and k, and where the sequences (
A
t
r
+
i
)
t
∈
ℕ
0 and  (
B
t
r
+
i
)
t
∈
ℕ
0, i
=
1
,
r
¯, are k-periodic.
Hence, it is enough to prove the theorem whengcd
(
s
,
k
)
=
1 and the sequences (
A
n
)
n
∈
ℕ
0, and  (
B
n
)
n
∈
ℕ
0 are positive k-periodic.
Now note that from the equations in (11), we have that
(35)
x
n
≥
y
n
-
k
,
y
n
≥
x
n
-
k
,
for
n
∈
ℕ
0
.
Further, by using the equations in (11), we also get
(36)
x
n
=
max
⁡
{
A
n
x
n
-
s
,
y
n
-
k
}
=
max
⁡
{
A
n
x
n
-
s
,
B
n
-
k
y
n
-
k
-
s
,
x
n
-
2
k
}
,
y
n
=
max
⁡
{
B
n
y
n
-
s
,
x
n
-
k
}
=
max
⁡
{
B
n
y
n
-
s
,
A
n
-
k
x
n
-
k
-
s
,
y
n
-
2
k
}
,
for n
≥
k.
Using relations (36), we get
(37)
x
n
=
max
⁡
{
A
n
x
n
-
s
,
B
n
-
k
y
n
-
k
-
s
,
x
n
-
2
k
}
=
max
⁡
{
A
n
x
n
-
s
,
B
n
-
k
y
n
-
k
-
s
,
A
n
-
2
k
x
n
-
2
k
-
s
,
B
n
-
3
k
y
n
-
3
k
-
s
,
x
n
-
4
k
}
,
y
n
=
max
⁡
{
B
n
y
n
-
s
,
A
n
-
k
x
n
-
k
-
s
,
y
n
-
2
k
}
=
max
⁡
{
B
n
y
n
-
s
,
A
n
-
k
x
n
-
k
-
s
,
B
n
-
2
k
y
n
-
2
k
-
s
,
A
n
-
3
k
x
n
-
3
k
-
s
,
y
n
-
4
k
}
,
for n
≥
3
k.
Now, note that, from the inequalities in (35), we have that
(38)
x
n
≥
x
n
-
2
k
,
y
n
≥
y
n
-
2
k
,
for
n
≥
k
.
Using (38) and k-periodicity of the sequences A
n and B
n, we obtain
(39)
A
n
x
n
-
s
=
A
n
-
2
k
x
n
-
s
≤
A
n
-
2
k
x
n
-
2
k
-
s
,
B
n
-
k
y
n
-
k
-
s
=
B
n
-
3
k
y
n
-
k
-
s
≤
B
n
-
3
k
y
n
-
3
k
-
s
,
B
n
y
n
-
s
=
B
n
-
2
k
y
n
-
s
≤
B
n
-
2
k
y
n
-
2
k
-
s
,
A
n
-
k
x
n
-
k
-
s
=
A
n
-
3
k
x
n
-
k
-
s
≤
A
n
-
3
k
x
n
-
3
k
-
s
.
Employing (39) into (37), we get
(40)
x
n
=
max
⁡
{
A
n
-
2
k
x
n
-
2
k
-
s
,
B
n
-
3
k
y
n
-
3
k
-
s
,
x
n
-
4
k
}
=
x
n
-
2
k
,
y
n
=
max
⁡
{
B
n
-
2
k
y
n
-
2
k
-
s
,
A
n
-
3
k
x
n
-
3
k
-
s
,
y
n
-
4
k
}
=
y
n
-
2
k
,
from which it follows that in this case the solutions of system (11) are 2
k-periodic. From all the above, the theorem follows.By a slight modification of the proof of Theorem9, the next result can be proved. We omit the proof.Theorem 10.
Consider the following system of difference equations(41)
x
n
=
max
⁡
{
A
n
x
n
-
s
,
y
n
-
k
}
,
y
n
=
max
⁡
{
B
n
y
n
-
s
,
z
n
-
k
}
,
z
n
=
max
⁡
{
C
n
z
n
-
s
,
x
n
-
k
}
,
n
∈
ℕ
0
,
where s, k
∈
ℕ, and (
A
n
)
n
∈
ℕ
0,  (
B
n
)
n
∈
ℕ
0, and (
C
n
)
n
∈
ℕ
0 are positive k
gcd

(
s
,
k
)-periodic sequences. Then, every positive solution of system (41) is periodic with, not necessarily prime, period
(42)
p
=
3
k
gcd


(
s
,
k
)
.



---

*Source: 982378-2014-01-23.xml*