Abstract
This paper is devoted to investigating the asymptotic behavior of the recursive sequence
x
n
+
1
=
α
−
β
x
n
−
k
g
(
x
n
,
x
n
−
1
,
…
,
x
n
−
k
+
1
)
,
n
=
0
,
1
,
…
where
α
≥
0
and
β
>
0
and
g
is continuous on
R
k
. We show that under certain conditions this equation has a unique positive (negative) equilibrium point which is a global attractor with some basin
S
⊂
R
k
+
1
. Also we establish the oscillation of all solutions with initial conditions
{
x
−
i
}
i
=
0
k
such that
(
x
0
,
x
−
1
,
…
,
x
−
k
)
∈
S
. We apply these results to the recursive sequence
x
n
+
1
=
α
−
β
x
n
−
k
γ
+
∑
k
−
1
i
=
0
(
a
i
x
n
−
i
±
b
i
x
n
−
i
2
)
,
n
=
0
,
1
,
…
where
α
,
γ
,
a
i
,
b
i
≥
0
,
i
=
0
,
…
,
k
−
1
, and
β
>
0
.