On the Symmetrical System of Rational Difference Equation 1 n n k nx A y y , 1 n n k ny A x x *

In this paper, we study the behavior of the symmetrical system of rational difference equation: 1 1 , , n k n k n n n n y x x A y A n y x           0,1,  0 where and , for . 0 A   , 0, i i x y   , 1, , i k k     


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][2][3][4][5][6][7][8][9][10].
In [1] C. Cina studied the system: In [2] A. Y. Ozban studied the difference equation system: (2) 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][2][3][4], they have the same similar character, which is the system can be reduced into a difference equation with n x or . n 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: 0 , 1 , . (4) 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: for all , thus, the system (9) reduces to the difference equation which was studied by El-owaidy in [11]. 2) 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 ,

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  x y is an arbitrary positive solution of the system (9).Then the following statements hold.
1) If k is odd, and , , 3) If k is even, we can not get some useful results.
Proof: 1) Obviously, we can have By introduction, we can get By limiting the inequality above, we can get Taking limits on the both sides of the following two equations The proof of 2) is similar, so we omit it.

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.
is a positive solution of the system (9).
Obviously, for .So we can get 1, 1, Then we can obtain Hence, we complete the proof.Theorem 3.2.Suppose A = 1, is a positive solution of the system (9).Then By system (9), we can have Obviously, , 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.m Proof: 1) As k is odd.
We set  , n n x y 1  is the solution of the system (9) with prime period two.Then there are four positive number such that , , ,  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.

The Case A > 1
Theorem 4.1.Suppose A > 1.Then every positive solution of the system (9) is bounded and persists.
Proof.Let   , n n x y be a positive solution of the system (9).Obviously, By introduction, we have , , , 1 ,2 , . ( We complete the proof.Theorem 4.2.Suppose A > 1.Then every positive solution of the system (9) converges to the equilibrium as .n   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.

On the Symmetrical System of Rational Difference Equation 
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