Motivated by an open problem in the literature “Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture”, we introduce a difference equation system: xn+1={yn-2+yn-3}/xn, yn+1={xn-2+xn-3}/yn , n=o, 1 ...* Where xi, yi ∈（0, ∞）， i=-3, -2, -1, 0. We try to find out some conditions such that the solution of system converges to periodic solution. This model can be applied to the two species competition and population biology.
In the monograph of Dynamics of Second Order Difference Equation [1], M. R. S. Kulenović and G. Ladas gave an open problem (see [1], p. 199) as following:
Open problem 11.4.8:
Determine whether every positive solutions of the following equation converges to a periodic solution of the corresponding equation:
Motivated by the Open Problem, we introduce the difference equation system:
where the initial points
Recently, there has been great interest in studying difference equation systems. One of the reasons for this is the necessity for some techniques that can be used in investigating equations arising in mathematical models describing real life situations in population biology, economics, probability theory, etc. There are many papers related to the difference equations system for example, such as [2-9].
In [2], Cinar studied the solutions of the system of difference equations:
In [3], E. Camouzis and Papaschinnopoulos studied the global asymptotic behavior of positive solution of the system of rational difference equations:
In [4], Ahmet Yasar Ozban studied the system of rational difference equations:
In [5], Abdullah Selcuk Kurbanli et al. studied the behavior of positive solutions of the system of rational difference equations:
In this paper, we try to find out some conditions such that the solution of system (2) converges to periodic solution. At the same time, we can get the oscillatory of system (2).
Before giving some results of the system (2), we need some definitions as follows [6]:
Definition 1.1 A pair of sequences of positive real numbers that satisfies system (2) is a positive solution of system (2). If a positive solution of system (2) is a pair of positive constants, that solution is the equilibrium solution.
Definition 1.2 A “string” of consecutive terms (resp.), (,) is said to be a positive semicycle if (resp.), , (resp.), and (resp.). Otherwise, that is said to be a negative semicycle.
A “string” of consecutive terms is said to be a positive(resp.negative) semicycle if, are positive (resp.negative) semicycle.
A solution (resp.) oscillates about (resp.) if for every, there exist, , , such that (resp.). We say that a solution
of system oscillates about if
oscillates about or oscillates about.
2. Some Lemmas
Lemma 2.1 The system (2) has a unique positive equilibrium.
The proof of lemma 2.1 is very easy, so we omit it.
Lemma 2.2 If, , , Then ever positive solution of system (2) with prime period two takes the forms
or
is a period-two solution of system (2).
Proof: Let be a period-two solution of system (2).
Then, by system (2) we get
We can see that (7) can be changed to
Form (8), we can obtain
or
and
Therefore, we complete the proof.
Lemma 2.3 Assume that the initial points , and is a positive solution of system (2). Then the following cases are true:
(a) If, ,
;, , , then and
are both increasing.
(b) If, ,
;, , , then and
are both decreasing.
Proof: (a) By system (2), we can get
i.e.
where.
By condition and (9), we get:
By condition, and (9), we get:
By condition and (9), we get:
Equally, we can get:
Hence, by induction and (10)-(15), we proof that
and are both increasing.
Using the same method, we can prove that case (b) holds.
Therefore, we complete the proof.
Lemma 2.4 Assume that
. Then there does not exist a positive solution of system (2) such that and are both increasing or both decreasing.
Proof: By Equation (9), we can get and have the same monotonous.
Firstly, we proof that there does not exist positive solution such that
and are both increasing.
Assume, for the sake of contradiction, that we have the following results:
(i) is increasing;
(ii) is also increasing.
By system (2), we obtain
in Equations (16) and (17), it implies that:
Because of, , we can get
i.e
Also, we can get
Because of the assumptions (i) and (ii), it is easy to see that (18) and (19) do not hold.
This is a contradiction and we proof the case of increasing does not hold.
Next, we proof there does not exist positive solution of system (2) such that and
are both decreasing.
Assume, for the sake of contradiction, that we have the following results:
(i) is decreasing;
(ii) is also decreasing.
By the Limiting Theorem we know that
and are both decreasing into a pair of constants.
We set, , ,
, and, , ,.
By system (2), we know that these constants satisfy the system (2)i.e.
However, if, , , , Equation (20) do not holds, which is contradiction.
Hence, we complete the proof of lemma 10.
Lemma 2.5 Assume that
. Then there does not exist a positive solution of system (2) such that and are both decreasing or both increasing.
Proof: First, we proof there does not exist positive solution of system (2) such that and are both decreasing, the proof of increasing is similar, so we omit it.
Assume, for the sake of contradiction, that we have the following results:
(i) is decreasing;
(ii) is also decreasing.
We set, ,.
By Limit Theorem,we know that and are both decreasing into a pair of constants.
Obviously, the limits of
can not decrease into zero.
By system (2), we can get
where, which can be changed to
However, if, Equation (22) can not hold.
This is a contradiction and we complete the proof.
The the proof of the case of increasing is similar with the proof of the the case of decreasing, so we omit it.
In addition to the method above, we can proof the Lemma 2.5 by the method of Lemma 2.4. Here, we omit it.
3. Main Results
Theorem 3.1 Assume that, ,
;, , , and is a positive solution of system (2). Then and
are both decreasing; and converges to a period-two solution as following
where satisfy,
.
Proof: By lemma 2.3(a), we can obtain that
and are both decreasing.
Then by the Limit Theorem, we can get, , , and, all exist and are positive.
We can set
By lemmas 2.4 and 2.5, we know that there does not exist a positive solution or
such that and
are both decreasing.
Hence, there is at least one of satisfy and at least one of satisfy
By system (2), we get
It is to see that is a period-two solution of system (2), and satisfy,.
We complete the proof.
Corollary 3.1 Suppose that is a positive solution of system (2). Then the following statement is true:
If
the solution of system (2)
eventually oscillates about equilibrium
.
Theorem 3.2 Assume that, ,
, and is a positive solution of system (2). Then and
are both increasing; and
converges to a period-two solution as following
where satisfy,
.
Proof: By lemma 2.3(a), we obtain that and are both increasing.
We set, , ,
By Equation (9), we can get
which can be changed into:
By the, we can get
By induction, we can get
From Lemma 10, we know that there at least one. Then by Limiting Theorem, we can get at least one of the limiting of must exist. With no loss generality, we set the limit of exist,we can know.
By limiting Equation (27), we can get
Hence, we can get
i.e.
Next, we try to proof, and.
By system (2), we get
By (30) and (31), we can get
which can be changed into
By the both side of Equation (35), we can get
Assume, by Stolz Theorem we obtain that
Because, then we can get the limit of
However, there exist, such that, which is conduction.
Hence, the assume does not hold. We obtain
.
Use the same method, we can also get
.
By system (2), we get
It is to see that is a period-two solution of system (2), and satisfy,.
Therefore, we complete the proof.
Corollary 3.2 Suppose that is a positive solution of system (2). Then the following statement is true:
If, ,;, , then the solution of system (2) oscillates about equilibrium.
Theorem 3.3 Assume that, ,
;, ,
, and is a positive solution of system (2). Then the system (2) has prime period two solutions, and for
Proof: By the lemma 8, we can complete the proof. Here, we omit it.
REFERENCESNOTESReferencesM. R. S. Kulenovi? and G. Ladas, “Dynamics of Second Order Rational Difference Equation with Open Problems and Conjectures,” CRC Press, Chapman Hall, 2002.C. Cinar, “On the Positive Solutions of the Difference Equation System x<sub>n+1</sub>=1/y<sub>n</sub>, y<sub>n+1</sub>=y<sub>n</sub>/x<sub>n-1</sub>y<sub>n-1</sub> ,” Applied Mathematics and Computation, Vol. 158, 2004, pp. 303-305.E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equations x<sub>n+1</sub>=1+x<sub>n</sub>/y<sub>n-m</sub>, y<sub>n+1</sub>=1+y<sub>n</sub>/x<sub>n-m</sub>,” Applied Mathematics and Letters, Vol. 17, 2004, pp. 733-737.A. Y. Ozban, “On the System of Rational Difference Equations
x<sub>n</sub>=a/y<sub>n-3</sub>, y<sub>n</sub>=by<sub>n-3</sub>/x<sub>n-q</sub>y<sub>n-q</sub>,” Applied Mathematics and Computation, Vol. 188, 2007, pp. 833-837.A. S. Kurbanli, C. Cinar and I. Yalcinkaya, “On the Behavior of Positive Solutions of the System of Rational Difference Equations, x<sub>n</sub>=x<sub>n-1</sub>/y<sub>n</sub>x<sub>n-1</sub>+1, y<sub>n</sub>=y<sub>n-1</sub>/x<sub>n</sub>y<sub>n-1</sub>+1 ,” Mathematical and Computer Modelling, Vol. 53, 2011, pp. 1261-1267.G. Papaschinopoulos, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematicsal Analysis and Applications, Vol. 219, 1998, pp. 415-426.G. Papaschinopoulos, M. Radin and C. J. Schinas, “Study of the Asymptotic Behavior of the Solutions of Three Systems of Difference Equations of Exponential Form,” Applied Mathematics and Computation, Vol. 218, 2012, pp. 5310-5318.[8] E. Camouzis, C. M. Kent, G. Ladas and C. D. Lynd, “On the Global Character of Solutions of the System: x<sub>n+1</sub>=α<sub>1</sub>+y<sub>n</sub>/x<sub>n</sub>, y<sub>n+1</sub>=α<sub>2</sub>+β<sub>2</sub>x<sub>n</sub>+γ<sub>2</sub>y<sub>n</sub>/A<sub>2</sub>+B<sub>2</sub>x<sub>n</sub>+C<sub>2</sub>y<sub>n</sub>,” Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 1205-1252.[9] E. Camouzis, E. Drymonis, G. Ladas and W. Tikjha, “Patterns of Boundedness of the Rational System x<sub>n+1</sub>=α<sub>1</sub>/A<sub>1</sub>+B<sub>1</sub>x<sub>n</sub>+C<sub>1</sub>y<sub>n</sub>, y<sub>n+1</sub>=α<sub>2</sub>+β<sub>2</sub>x<sub>n</sub>+γ<sub>2</sub>y<sub>n</sub>/A<sub>2</sub>+B<sub>2</sub>x<sub>n</sub>+C<sub>2</sub>y<sub>n</sub>,” Journal of Difference Equations and Applications, Vol. 18, 2012, pp. 89-110.
that satisfies system (2) is a positive solution of system (2). If a positive solution of system (2) is a pair of positive constants
, that solution is the equilibrium solution.
(resp.
), (
,
) is said to be a positive semicycle if 
(resp.
), 
, 
(resp.
), and 
(resp.
). Otherwise, that is said to be a negative semicycle.
is said to be a positive(resp.negative) semicycle if
, 
are positive (resp.negative) semicycle.
(resp.
) oscillates about 
(resp.
) if for every
, there exist
, 
, 
, such that 
(resp.
). We say that a solution
of system oscillates about 
if 
or 
oscillates about
.
.
, 
, 
, Then ever positive solution of system (2) with prime period two takes the forms
be a period-two solution of system (2).
, and 
is a positive solution of system (2). Then the following cases are true:
, 
,
;
, 
, 
, then 
and
are both increasing.
, 
,
;
, 
, 
, then 
and
are both decreasing.
.
and (9), we get:
, and (9), we get:
and (9), we get:
and 
are both increasing.
. Then there does not exist a positive solution 
of system (2) such that 
and 
are both increasing or both decreasing.
and 
have the same monotonous.
such that
and 
are both increasing.
is increasing;
is also increasing.
, 
, we can get
and
are both decreasing.
is decreasing;
is also decreasing.
are both decreasing into a pair of constants.
, 
, 
,
, and
, 
, 
,
.
, 
, 
, 
, Equation (20) do not holds, which is contradiction.
. Then there does not exist a positive solution 
of system (2) such that 
and 
are both decreasing or both increasing.
and 
are both decreasing, the proof of increasing is similar, so we omit it.
is decreasing;
is also decreasing.
, 
,
.
and 
are both decreasing into a pair of constants.
can not decrease into zero.
, which can be changed to
, Equation (22) can not hold.
, 
,
;
, 
, 
, and 
is a positive solution of system (2). Then 
and
are both decreasing; and 
converges to a period-two solution as following
satisfy
,
.
and 
are both decreasing.
, 
, 
, and
, all exist and are positive.
or
such that 
and
are both decreasing.
satisfy 
and at least one of 
satisfy 
is a period-two solution of system (2), and 
satisfy
,
.
is a positive solution of system (2). Then the following statement is true:
the solution of system (2)
eventually oscillates about equilibrium
.
, 
,
, and 
is a positive solution of system (2). Then 
and
are both increasing; and 
satisfy
,
.
and 
are both increasing.
, 
, 
, 
, we can get
. Then by Limiting Theorem, we can get at least one of the limiting of 
must exist. With no loss generality, we set the limit of 
exist,we can know
.
, and
.
, by Stolz Theorem we obtain that
, then we can get the limit of 
, 
such that
, which is conduction.
.
.
is a period-two solution of system (2), and 
satisfy
,
.
is a positive solution of system (2). Then the following statement is true:
, 
,
;
, 
, 
then the solution of system (2) oscillates about equilibrium
.
, 
,
;
, 
,
, and 
is a positive solution of system (2). Then the system (2) has prime period two solutions, and 
for 