Global Behavior of a Nonlinear Difference Equation with Applications ()
1. Introduction
Amleh, Grove and Ladas [1] studied the global stability boundedness character and periodic nature of positive solutions of difference equation
(1)
where 
and initial conditions 
and 
are both arbitrary positive real numbers.
Amleh, Grove and Ladas [1] obtain the following theorem.
Theorem A (Amleh, Grove and Ladas [1]) Let
and 
be a solution of equation (1)
with initial conditions 
and
.
Then the following statements are true.
1) 
2)
.
Now, we can see that if 
and
, then
. So, the theorem A does not hold for
.
Kulenovic and Glass in their monograph [2] give an open problem as follows.
Open Problem 6.10.7. For the following difference equation determine the “good” set 
of the initial conditions 
throng with the equation is well defined for all
. Then for every
, investigate the long-term behavior of the solution 
of
(2)
Let
. Then equation (2) can be rewritten as follows
(3)
where 
and 
are arbitrary nonzero real numbers. To this end, we study equation (3) and use the results of equation (3) to equation (2).
2. Some Lemmas
It is easy for one to see that if
then we have
(4)
and
Lemma 2.1 (Kocic and Ladas [3]) Consider the difference equation
(5)
Assume that 
is a 
function and 
is an equilibrium of equation (5).
Then the linearized equation associated with equation (5) about the equilibrium 
is
and the following statements are true.
a) If all roots of the polynomial equation
(6)
lie in the open unit disk
, then the equilibrium 
of equation (5) is asymptotically stable;
b) If at least one root of equation (6) has absolute value greater than one, then equilibrium 
of equation (5) is unstable.
One can refer to Kocic and Ladas [3, Corallary 1.3.2, p14 ].
Lemma 2.2 Equation (3) has two equilibriums 
and
.
It is easy to see that 
has two roots and the proof is complete.
3. Main Results
Theorem 3.1 Let 
and
. Then the following statements are true.
a)
, where
and
b)
, where
and
where 
is the solution of equation (3) with the initial
,
.
Proof: Part a).
Let
,
. Then by equation (3) we have
we assume that
(7)
Then by induction, we have
(8)
where
, 
and 
Change equation (8) into
(9)
or
(10)
where
.
From equation (4), we get
(11)
where
Equation (11) can be changed into
Let 
and
. Then we obtain that
(12)
and 
By induction, we have
(13)
Therefore,
Hence, the proof of part (a) is complete.
The proof of part (b) can be similarly given, so we omit it. This can complete the proof of theorem 3.1.
By theorem 3.1, we get the following corollary.
Corollary 3.1 Assume that
,
. Then the following statements are true.
a) If
, then the positive solution 
of equation (3) converges to 1, i.e,
.
b) If
, then the positive solution 
of equation (3) has the properties
c) If
, then the positive solution 
of equation (3) has the properties
Theorem 3.2 Assume that
,
. Then the following statements are true.
a) If 
and
, then the solution 
of equation (3) is periodic with period-3 as follows
(14)
b) If 
and
, then the solution 
of equation (3) is periodic with period-3 as follows
(15)
c) If 
and
, then the solution 
of equation (3) is periodic with period-3 as follows
(16)
The proof of theorem 3.2 is very easy, so we will omit it.
By theorems 3.1 and 3.2, we can obtain the following corollary.
Corollary 3.2 Assume that
. Then the following statements are true.
a) If 
and at least one of p and q is less than 0, then 
of equation (3) converges to a period-3 solution of equation (3) as one of (10)-(12).
b) If 
and at least one of p and q is less than 0, then 
of equation (3) has the following properties
c) If 
and at least one of p and q is less than 0, then 
of equation (3) has the following properties
d) If at least one of p and q is less than 0, then every solution of equation (3) strictly oscillates about the equilibrium
.
e) If 
and at least one of p and q is less than 0, then every solution of equation (3) strictly oscillates about the equilibrium
.
Theorem 3.3 The equilibrium 
of equation (3) is unstable.
Proof: The linearize equation associated with equation (3) about the equilibrium 
is
(17)
The characteristic equation of (17) is
Thus, we obtain two roots
. Noting that
. Therefore, by lemma 3.1, we know that the equilibrium 
of equation (3) is unstable. The proof of theorem 3.3 is complete.
4. Application
By theorem 3.1, we have the following theorem.
Theorem 4.1 Assume that 
and
. Then the following statements are true.
a) Every solution 
of equation (2) satisfies 
for 
b) If
, then the solution 
of equation (2) converges to 0.
c) If
, then the solution 
of equation (2) has the following properties
d) If
, then the solution 
of equation (2) has the following properties
By corollary 3.2, we get the following theorem.
Theorem 4.2 Assume that
. Then the following statements are true.
a) If 
and at least one of 
and 
is less than 0, then 
of equation (2) converges to a period-3 solution of equation (2) as one of the following:
i) 
ii) 
iii) 
b) If 
and at least one of 
and 
is less than 0, then every solution 
of equation (2) has the following properties:
c) If 
and at least one of 
and 
is less than 0, then every solution 
of equation (2) has the following properties:
d) If 
and at least one of 
and 
is less than 0, then every solution 
of equation (2) strictly oscillates about the equilibrium 
of equation (2).
5. Acknowledgements
Research supported by Distinguished Expert Foundation and Youth Science Foundation of Naval Aeronautical and Astronautical University.