In this paper, we investigate the behavior of solutions of the difference equation
where
k
∈ {1,2},
a
≥ 0, and
x_{j}
> 0,
j
= 0, 1, ...,
k
.
AMS Mathematics Subject Classification : 39A10, 39A21, 39A30.
1. Introduction
Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. More importantly, difference equations also appear in the study of discretization methods for differential equations. Several results in the theory of difference equations have been obtained as more or less natural discrete analogues of corresponding results of differential equations.
Recently there has been a lot of interest in studying the global attractivity, boundedness character, periodicity and the solution form of nonlinear difference equations. For example,
AbuSaris et al.
[1]
investigated the asymptotic stability of the difference equation
For other related results(
[2

16]
).
In this paper, we investigate the behavior of solutions of the difference equation
where
k
∈ {1,2},
a
≥ 0, and
x_{j}
> 0,
j
= 0, 1, ...,
k
.
We need the following definitions.
Definition 1.1.
Let
I
be an interval of real numbers and let
be a continuously differentiable function. Consider the difference equation
with
x_{−k}
,
x
_{−k+1}
, ...,
x
_{0}
∈
I
. Let
be the equilibrium point of Eq.(1.2). The linearized equation of Eq.(1.2) about the equilibrium point
is
where
The characteristic equation of Eq.(1.3) is
(i) The equilibrium point
of Eq.(1.2) is locally stable if for every
ϵ
> 0, there exists
δ
> 0 such that for all
x_{−k}
,
x
_{−k+1}
, ...,
x
_{−1}
,
x
_{0}
∈
I
with
we have
(ii) The equilibrium point
of Eq.(1.2) is locally asymptotically stable if
is locally stable and there exists
γ
> 0, such that for all
x_{−k}
,
x
_{−k+1}
, ...,
x
_{−1}
,
x
_{0}
∈
I
with
we have
(iii) The equilibrium point
of Eq.(1.2) is global attractor if for all
x_{−k}
,
x
_{−k+1}
, ...,
x
_{−1}
,
x
_{0}
∈
I
we have
(iv) The equilibrium point
of Eq.(1.2) is globally asymptotically stable if
is locally stable, and
is also a global attractor of Eq.(1.2).
(v) The equilibrium point
of Eq.(1.2) is unstable if
is not locally stable.
Definition 1.2.
A positive semicycle of
of Eq.(1.2) consists of a ‘string’ of terms {
x_{l}
,
x
_{l+1}
, ...,
x_{m}
}, all greater than or equal o
, with
l
≥ 
k
and
m
≤∞ and such that either
l
= 
k
or
l
> 
k
and
and either
m
= ∞ or
m
< ∞ and
A negative semicycle of
of Eq.(1.2) consists of a ‘string’ of terms {
x_{l}
,
x
_{l+1}
, ...,
x_{m}
}, all less than
, with
l
≥ 
k
and
m
≤ ∞ and such that either
l
= 
k
or
l
> 
k
and
and either
m
= ∞ or
m
< ∞ and
.
Definition 1.3.
A solution
of Eq.(1.2) is called nonoscillatory if there exists
N
≥ 
k
such that either
and it is called oscillatory if it is not nonoscillatory.
We need the following theorems.
Theorem 1.4
(
[16]
).
(i) If all roots of Eq.(1.4) have absolute value less than one, then the equilibrium point
of Eq.(1.2) is locally asymptotically stable.
(ii) If at least one of the roots of Eq.(1.4) has absolute value greater than one, then
is unstable.
The equilibrium point
of Eq.(1.2) is called a saddle point if Eq.(1.4) has roots both inside and outside the unit disk.
Theorem 1.5
(
[16]
).
Assume that
p
_{1}
,
p
_{2}
, ...,
p_{k}
∈ ℝ
and k
∈ {1, 2, ...}.
Then
is a sufficient condition for the asymptotic stability of the difference equation
2. Behavior of solutions of Eq.(1.1) when k = 1 and a = 0.
In this section we give the closed form of solutions of Eq.(1.1) when
k
= 1 and
a
= 0.
In this case the difference equation (1.1) becomes
with positive initial conditions
x
_{1}
and
x
_{0}
.
Eq. (2.1) is linear which have the solution
3. Behavior of solutions of Eq.(1.1) when k = 2 and a = 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when
k
= 2 and
a
= 0.
In this case the difference equation (1.1) becomes
with positive initial conditions
x
_{2}
,
x
_{1}
and
x
_{0}
.
Eq.(3.1) has a unique equilibrium point
Theorem 3.1.
The equilibrium point
of Eq.(3.1) is locally asymptotically
Proof
. Since the linearized equation of Eq.(3.1) about the equilibrium point
can be written in the following form
then the proof follows immediately from Theorem B.
Theorem 3.2.
The equilibrium point
of Eq.(3.1) is globally asymptotically stable.
Proof
. From Eq.(3.1) it is easy to show that
x
_{n+1}
<
x
_{n−1}
for all
n
≥ 0 and so the even terms converge to a limit (say
L
_{1}
≥ 0) and the odd terms converge to a limit (say
L
_{2}
≥ 0). Then
which implies that
L
_{1}
=
L
_{2}
= 0, and the proof is complete.
4. Behavior of solutions of Eq.(1.1) when k = 1 and a > 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when
k
= 1 and
a
> 0.
In this case the difference equation (1.1) becomes
with positive initial conditions
x
_{−1}
and
x
_{0}
.
The change of variables
reduces Eq.(4.1) to the difference equation
Eq.(4.2) has a unique positive equilibrium point
Theorem 4.1.
The equilibrium point
of Eq.(4.2) is locally asymptotically stable.
Proof
. The linearized equation of Eq.(4.2) about the equilibrium point
is
and so, the characteristic equation of Eq.(4.2) about the equilibrium point
is
which implies that 
λ
_{1}
 = 
λ
_{2}
 = 0 < 1. Hence, the proof is complete.
Theorem 4.2.
The equilibrium point
of Eq.(4.2) is globally asymptotically stable
.
Proof
. Since
for all
n
≥ 0, then we have
y_{n}
≥ 1 for all
n
≥ 1.
Furthermore
for all
n
≥ 2. So the even terms
converge to a limit (say
L
_{1}
≥ 0) and the odd terms
converge to a limit (say
L
_{2}
≥ 0). Then
which implies that
L
_{1}
=
L
_{2}
= 1. Thus, the proof is complete.
5. Behavior of solutions of Eq.(1.1) when k = 2 and a > 0.
In this section we investigate the behavior of solutions of Eq.(1.1) when
k
= 2 and
a
> 0.
In this case the difference equation (1.1) becomes
with positive initial conditions
x
_{−2}
,
x
_{−1}
and
x
_{0}
.
The change of variables
reduces Eq.(5.1) to the difference equation
Eq.(5.2) has two equilibrium points
Theorem 5.1.
The equilibrium point
of Eq.(5.2) is unstable equilibrium point.
Proof
. The linearized equation of Eq.(5.2) about the equilibrium point
is
and so, the characteristic equation of Eq.(5.2) about the equilibrium point
is
is clear that
f
(
λ
) has a root in the interval (1,∞), and so,
is an unstable equilibrium point. This completes the proof.
Theorem 5.2.
The equilibrium point
of Eq.(5.2) is locally asymptotically stable.
Proof
. The linearized equation of Eq.(5.2) about the equilibrium point
is
and so, the characteristic equation of Eq.(5.2) about the equilibrium point
is
which implies that 
λ
_{1}
 = 
λ
_{2}
 = 
λ
_{3}
 = 0 < 1, from which the proof is complete.
Lemma 5.3.
The following identities are true
Proof
. (i)
for
n
≥ 0.
(ii)
for
n
≥ 0.
(iii)
for
n
≥ 0.
(
iv
)
(
v
)
Then, the proof is complete.
Theorem 5.4.
Let
be a solution of Eq.(5.2), then the following statements are true
(i) If
for some
n
_{0}
∈ {1, 0, 1, 2, ...},
then
for all
n
≥
n
_{0}
+ 2.
Also if
then
for all n
≥ 3.
(ii) If
for some
n
_{0}
∈ {2,1, 0, 1, 2, ...},
then
for all n
∈
n
_{0}
.
(iii) If (i) and (ii) are not satisfied, then
oscillates about
,
with positive semicycles of length at most three, and negative semicycles of length at most two.
Proof
. (i) Let
for some
n
_{0}
∈ {1, 0, 1, 2, ...},
then from Eq.(5.3) we have
for all n
≥
n
_{0}
+ 2.
If
then from Eq.(5.3) we have
which implies that
for all
n
≥ 3.
(ii) Let
for some
n
_{0}
∈ {2,1, 0, 1, 2, ...}, then from Eq.(5.3) we have
for all
n
∈
n
_{0}
.
(iii) Suppose without loss of generality that there exists
n
_{0}
∈ {2,1, 0, 1, 2, ...}, such that
Then from Eq.(5.3) we have
The proofs of the other possibilities are similar, and will be omitted.
Theorem 5.5.
The equilibrium point
of Eq.(5.2) is globally asymptotically stable.
Proof
. We proved that
of Eq.(5.2) is locally asymptotically stable, and so it suffices to show that lim
_{n→∞}
z_{n}
= 1. If there exists
n
_{0}
∈ {2,1, 0, 1, 2, ...}g, such that
then from Theorem 5.4 we have lim
_{n→∞}
z_{n}
= 1. Also, if
then by Theorem 5.4 we have
for all
n
≥ 2, and from Eq.(5.4), we have
z
_{n+1}
>
z
_{n1}
, for
n
≥ 0. So the sequences
are increasing and bounded, which implies that the even terms
converge to a limit (say
M
_{1}
> 0) and the odd terms
converge to a limit (say
M
_{2}
> 0). Then
which implies that
M
_{1}
=
M
_{2}
= 1.
Now, Suppose that
then from Eqs.(5.4)  (5.7) we have the following results
The sequence
is decreasing and bounded, and so converges to a limit (say
L
_{0}
> 0).
The sequence
is increasing and bounded, and so converges to a limit (say
L
_{1}
> 0).
The sequence
is increasing and bounded, and so converges to a limit (say
L
_{2}
> 0).
The sequence
is decreasing and bounded, and so converges to a limit (say
L
_{3}
> 0).
The sequence
is increasing and bounded, and so converges to a limit (say
L
_{4}
> 0).
The sequence
is decreasing and bounded, and so converges to a limit (say
L
_{5}
> 0).
The sequence
is decreasing and bounded, and so converges to a limit (say
L
_{6}
> 0).
So we have from Eq.(5.2) that
The solution of this system is either
L_{i}
= 1,
i
= 0, 1, ...6, or
L_{i}
= 0,
i
= 0, 1, ...6, or
L_{i}
= 1,
i
= 0, 1, ...6. Since
L_{i}
> 0,
i
= 0, 1, ...6 , we have lim
_{n→∞}
z_{n}
= 1.
The proofs for the other cases are as follows.
or
are similar to the proof of the last case, and will be omitted. Therefore the proof is complete.
BIO
A.M. Ahmed obtained his B.Sc. (1997), M.Sc. (2000) and Ph. D. (2004) from AlAzhar University. He obtained the rank associate professor from AlAzhar University in 2012. His research interests in difference equations and its applications.
Mathematics Department, Faculty of Science, AlAzhar University , Nasr City(11884), Cairo, Egypt.
email: ahmedelkb@yahoo.com
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