In this paper, we study the behavior of the symmetrical system of rational difference equation:
where A > o and xi, yi ∈(0, ∞), for i= -k,-k+1,…,0.
Symmetrical System; Difference Equation; Boundedness; Period-Two Solution1. Introduction
Recently there has been a great interest in studying difference equations and systems, and quite a lot of papers about the behavior of positive solutions of system of difference equation. We can read references [1-10].
In [1] C. Cina studied the system:
In [2] A. Y. Ozban studied the difference equation system:
In [3] A. Y. Ozban studied the behavior of positive solutions of the difference equation system:
In [4] X. Yang, Y. Liu, S. Bai studied the difference equation system:
We can see in [1-4], they have the same similar character, which is the system can be reduced into a difference equation with or.
In [5] G. Papaschinopoulos, C. J. Schinas studied the behavior of positive solutions of the difference equation system:
In [6] G. Papaschinopoulos, Basil K. Papadopoulos studied the behavior of positive solutions of the difference equation system:
In [7] E. Camouzis, G. Papaschinopoulos studied the behavior of positive solutions of the difference equation system:
In [8] Yu Zhang, Xiaofan Yang, David J. Evans, Ce Zhu studied the behavior of positive solutions of the difference equation system:
Motivated by systems above, we introduce the symmetrical system:
with parameter, the initial conditions, for, and is a positive integer. We can easily get the system (9) has the unique positive equilibrium.
There are two cases we need to consider:
1) If the initial conditions in the system (9) for, then for all, thus, the system (9) reduces to the difference equation
which was studied by El-owaidy in [11].
2) If for, then the system (9) is similar to the system in [8]. We study the system (9) basing on this condition in this paper.
In this paper, we try to give some results of the system (9) by using the methods in [8]. We consider the following cases of, and.
2. The Case 0 < A < 1
In this section, we give the asymptotic behavior of positive solution to the system (9).
Theorem 2.1. Suppose and is an arbitrary positive solution of the system (9). Then the following statements hold.
1) If k is odd, and, , , for, then
2) If is odd, and, , , for, then
3) If k is even, we can not get some useful results.
Proof: 1) Obviously, we can have
By introduction, we can get
So for,
By limiting the inequality above, we can get
. Similarly, we can also get.
Taking limits on the both sides of the following two equations
we can obtain,.
The proof of 2) is similar, so we omit it.
3. The Case A = 1
In this section, we try to get the boundedness, persistence, and periodicity of positive solutions of the system (9).
Theorem 3.1. Suppose A = 1. Then every positive solution of the system (9) is bounded and persists.
Proof. is a positive solution of the system (9).
Obviously, for. So we can get
where, ,
, for.
Then we can obtain
By introduction, we have
Hence, we complete the proof.
Theorem 3.2. Suppose A = 1, is a positive solution of the system (9). Then
Proof: By (10), we can get
By system (9), we can have
which implies
Hence, we can obtain
which can be changed into
Obviously, , we complete the proof.
Theorem 3.3. Suppose.
1) If is odd, then every positive solution of the system (9) with prime period two takes the form
or
with.
2) If is even, there do not exist positive nontrival solution of the system (9) with prime period two.
Proof: 1) As k is odd.
We set is the solution of the system (9) with prime period two. Then there are four positive number such that
If, by the system (9) we can get, which is contradiction with the condition, hence. Similarly, we can get. Then we obtain
From Theorem 3.2, we can get
Next, we consider the following possibilities:
Case 1: Either(I) A < C and B < D or (II) A > C and B > D. Then A = B, C = D.
Case 2: Either(I) A < C and B > D or (II) A > C and B < D. Then A = D, B = C.
Therefore by the system (9), we can get 1) holds.
2) Obviously, if k is even, the system (9) just has trival solution with prime period two.
We complete the proof.
4. The Case A > 1
Theorem 4.1. Suppose A > 1. Then every positive solution of the system (9) is bounded and persists.
Proof. Let be a positive solution of the system (9). Obviously, , , for. So we can get
where, ,
, for. Then we can obtain
By introduction, we have
We complete the proof.
Theorem 4.2. Suppose A > 1. Then every positive solution of the system (9) converges to the equilibrium as.
Proof: By (13), we can get
By system (9), we can have
which imply
By the condition, we can get
Besides, and, so we can get
and
i.e.
we complete the proof.
REFERENCESNOTESReferencesC. Cinar, “On the Positive Solutions of the Difference Equation Systemx<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, No. 2, 2004, pp. 303-305. doi:10.1016/j.amc.2003.08.073A. Y. Ozban, “On the Positive Solutions of the System of Rational Difference Equations x<sub>n+1</sub>=1/y<sub>n-k</sub>,y<sub>n+1</sub>=y<sub>n</sub>/x<sub>n-m</sub>y<sub>n-m-k</sub>,” Journal of Mathematical Analysis and Applications, Vol. 323, No. 1, 2006, pp. 26-32.
doi:10.1016/j.jmaa.2005.10.031A. 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, No. 1, 2007, pp. 833-837. doi:10.1016/j.amc.2006.10.034X. Yang, Y. Liu and S. Bai, “On the System of High Order Rational Difference Equations x<sub>n+1</sub>=a/y<sub>n-p</sub> ,y<sub>n+1</sub>=by<sub>n-p</sub>/x<sub>n-q</sub>y<sub>n-q</sub>,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 853-856.
doi:10.1016/j.amc.2005.01.092G. Papaschinopoulos and C. J. Schinas, “On a System of Two Nonlinear Difference Equations,” Journal of Mathematical Analysis and Applications, Vol. 219, No. 2, 1998, pp. 415-426. doi:10.1006/jmaa.1997.5829G. Papaschinopoulos and B. K. Papadopoulos, “On the Fuzzy Difference Equation x<sub>n+1</sub>=A+x<sub>n</sub>/x<sub>n-m</sub>,” Fuzzy Sets and Systems, Vol. 129, No. 1, 2002, pp. 73-81.
doi:10.1016/S0165-0114(01)00198-1E. Camouzis and G. Papaschinopoulos, “Global Asymptotic Behavior of Positive Solutions on the System of Rational Difference Equation 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>m-n</sub>,” Applied Mathematics and Letters, Vol. 17, No. 6, 2004, pp. 733-737.
doi:10.1016/S0893-9659(04)90113-9Y. Zhang, X. F. Yang, D. J. Evans and C. Zhu, “On the Nonlinear Difference Equation System x<sub>n+1</sub>=A+y<sub>n-m</sub>/x<sub>n</sub>,y<sub>n+1</sub>=A+x<sub>n-m</sub>/ y<sub>n</sub>” Computers and Mathematics with Application, Vol. 53, No. 10, 2007, pp. 1561-1566.
doi:10.1016/j.camwa.2006.04.030E. 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, No. 7, 2012, pp. 1205-1252.
doi:10.1080/10236198.2011.555406E. 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></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, No. 1, 2012, pp. 89-110.
doi:10.1080/10236198.2010.515591H. M. El-Owaidy, A. M. Ahmed and M. S. Mousa, “On Asymptotic Behavior of the Difference x<sub>n+1</sub>=α+x<sub>n-k</sub>/x<sub>n</sub>,” Applied Mathematics and Computation, Vol. 147, No. 1, 2004, pp. 163-167.
doi:10.1016/S0096-3003(02)00659-8
where A > o and xi, yi ∈(0, ∞), for i= -k,-k+1,…,0.
or
.
, the initial conditions
, for
, and 
is a positive integer. We can easily get the system (9) has the unique positive equilibrium
.
in the system (9) for
, then 
for all
, thus, the system (9) reduces to the difference equation
for
, then the system (9) is similar to the system in [
, 
and
.
and 
is an arbitrary positive solution of the system (9). Then the following statements hold.
, 
, 
, 
for
, then
is odd, and
, 
, 
, 
for
, then
,
. Similarly, we can also get
.
,
.
is a positive solution of the system (9).
for
. So we can get
, 
,
, for
.
is a positive solution of the system (9). Then
, we complete the proof.
.
is odd, then every positive solution of the system (9) with prime period two takes the form
.
is even, there do not exist positive nontrival solution of the system (9) with prime period two.
is the solution of the system (9) with prime period two. Then there are four positive number 
such that
, by the system (9) we can get
, which is contradiction with the condition
, hence
. Similarly, we can get
. Then we obtain
be a positive solution of the system (9). Obviously, 
, 
, for
. So we can get
, 
,
, for
. Then we can obtain
.
, we can get
and
, so we can get
and 
