Asymptotic Periodicity of a Higher-Order Difference Equation

• Published in: Discrete Dynamics in Nature and Society Vol. 2007; pp. 44 - 52
• Main Author: Stevic, Stevo
• Format: Journal Article
• Language: English
• Published: Hindawi Limiteds 01-01-2007; Hindawi Limited
• ISSN: 1026-0226; 1607-887X
• DOI: 10.1155/2007/13737

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation:
x
n
=
f
(
x
n
−
p
1
,
…
,
x
n
−
p
k
,
x
n
−
q
1
,
…
,
x
n
−
q
m
)
,
n
∈
ℕ
0
, where
p
i
,

i
∈
{
1
,
…
,
k
}
, and
q
j
,

j
∈
{
1
,
…
,
m
}
, are natural numbers such that
p
1
<
p
2
<
⋯
<
p
k
,
q
1
<
q
2
<
⋯
<
q
m
and
gcd
(
p
1
,
…
,
p
k
,
q
1
,
…
,
q
m
)
=
1
, the function
f
∈
C
[
(
0
,
∞
)
k
+
m
,
(
α
,
∞
)
]
,

α
>
0
, is increasing in the first
k
arguments and decreasing in other
m
arguments, there is a decreasing function
g
∈
C
[
(
α
,
∞
)
,
(
α
,
∞
)
]
such that
g
(
g
(
x
)
)
=
x
,

x
∈
(
α
,
∞
)
,
x
=
f
(
x
,
…
,
x
︸
k
,
g
(
x
)
,
…
,
g
(
x
)
︸
m
)
,
x
∈
(
α
,
∞
)
,
lim
x
→
α
+
g
(
x
)
=
+
∞
, and
lim
x
→
+
∞
g
(
x
)
=
α
. It is proved that if all
p
i
,

i
∈
{
1
,
…
,
k
}
, are even and all
q
j
,

j
∈
{
1
,
…
,
m
}
are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.