On the Behavior of Positive Solutions of a Difference Equations System x_{n+1}={y_{n-2}+y_{n-3}}/x_{n}, y_{n+1}={x_{n-2}+x_{n-3}}/y_{n} ()
1. Introduction
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:
(1)
Motivated by the Open Problem, we introduce the difference equation system:
(2)
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:
(3)
In [3], E. Camouzis and Papaschinnopoulos studied the global asymptotic behavior of positive solution of the system of rational difference equations:
(4)
In [4], Ahmet Yasar Ozban studied the system of rational difference equations:
(5)
In [5], Abdullah Selcuk Kurbanli et al. studied the behavior of positive solutions of the system of rational difference equations:
(6)
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
(7)
We can see that (7) can be changed to
(8)
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.
(9)
where.
By condition and (9), we get:
(10)
By condition, and (9), we get:
(11)
By condition and (9), we get:
(12)
Equally, we can get:
(13)
(14)
(15)
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
(16)
(17)
in Equations (16) and (17), it implies that:
Because of, , we can get
i.e
(18)
Also, we can get
(19)
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.
(20)
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
(21)
where, which can be changed to
(22)
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
(23)
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
(24)
which can be changed into:
(25)
(26)
By the, we can get
By induction, we can get
(27)
(28)
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
(29)
Hence, we can get
i.e.
Next, we try to proof, and.
By system (2), we get
(30)
(31)
By (30) and (31), we can get
(32)
(33)
which can be changed into
(34)
(35)
By the both side of Equation (35), we can get
(36)
Assume, by Stolz Theorem we obtain that
(37)
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
(38)
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.
NOTES
#Corresponding author.