In this paper, solutions of difference equation for are investigated, where , are both arbitrary nonzero real numbers. The results are applied to the following difference equation for n=0,1,... .
Amleh, Grove and Ladas [1] studied the global stability boundedness character and periodic nature of positive solutions of difference equation
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
Let. Then equation (2) can be rewritten as follows
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
and
Lemma 2.1 (Kocic and Ladas [3]) Consider the difference equation
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
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
Then by induction, we have
where, and
Change equation (8) into
or
where.
From equation (4), we get
where
Equation (11) can be changed into
Let and. Then we obtain that
and
By induction, we have
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
b) If and, then the solution
of equation (3) is periodic with period-3 as follows
c) If and, then the solution of equation (3) is periodic with period-3 as follows
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
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.
REFERENCESReferencesA. M. Amleh, E. A. Grove and G. Ladas, “On the Recursive Sequence ,” Journal of Mathematical Analysis and Applications, Vol. 233, No. 2, 1999, pp. 790-798. doi:10.1006/jmaa.1999.6346M. R. S Kulenovic and L. Glass, “Dynamics of Second Order Rational Difference Equations,” Chapman Hall/ CRC, USA, 2002.V. L. Kocic and G. Ladas, “Global Asymptotic Behavior of Nonlinear Difference Equation of Higher Order with Applications,” Kluwer Academic, Dordrecht, 1993.
and initial conditions 
and 
are both arbitrary positive real numbers.
and 
be a solution of equation (1)
and
.
.
and
, then
. So, the theorem A does not hold for
.
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
. Then equation (2) can be rewritten as follows
and 
are arbitrary nonzero real numbers. To this end, we study equation (3) and use the results of equation (3) to equation (2).
is a 
function and 
is an equilibrium of equation (5).
is
, then the equilibrium 
of equation (5) is asymptotically stable;
of equation (5) is unstable.
and
.
has two roots and the proof is complete.
and
. Then the following statements are true.
, where
, where
is the solution of equation (3) with the initial
,
.
,
. Then by equation (3) we have
, 
and 
.
and
. Then we obtain that
,
. Then the following statements are true.
, then the positive solution 
of equation (3) converges to 1, i.e,
.
, then the positive solution 
of equation (3) has the properties
, then the positive solution 
of equation (3) has the properties
,
. Then the following statements are true.
and
, then the solution 
and
, then the solution 
and
, then the solution 
of equation (3) is periodic with period-3 as follows
. Then the following statements are true.
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).
and at least one of p and q is less than 0, then 
of equation (3) has the following properties
and at least one of p and q is less than 0, then 
of equation (3) has the following properties
.
and at least one of p and q is less than 0, then every solution of equation (3) strictly oscillates about the equilibrium
.
of equation (3) is unstable.
is
. 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.
and
. Then the following statements are true.
of equation (2) satisfies 
for 
, then the solution 
, then the solution 
, then the solution 
. Then the following statements are true.
and at least one of 
is less than 0, then 
of equation (2) converges to a period-3 solution of equation (2) as one of the following:
and at least one of 
is less than 0, then every solution 
of equation (2) has the following properties:
and at least one of 
is less than 0, then every solution 
of equation (2) has the following properties:
and at least one of 
is less than 0, then every solution 
of equation (2) strictly oscillates about the equilibrium 
of equation (2).