Periodic solutions of a system of nonlinear difference equations with periodic coefficients

In this paper it is dealt with the following system of difference equations x_{n+1}=((a_{n})/(x_{n}))+((b_{n})/(y_{n})), y_{n+1}=((c_{n})/(x_{n}))+((d_{n})/(y_{n})), n in N_0, where the initial values x_0,y_0 are positive real numbers and the coefficients (a_{n})_{n>=0}, (b_{n})_{n>=0}, (c_{n})_{n>=0}, (d_{n})_{n>=0} are two-periodic sequences with positive terms. The system is an extention of a system that every positive solution is two-periodic or converges to its a two-periodic solution. Here, the long-term behavior of posistive solutions of the system is examined by using a new method to solve the system.


Introduction
Studying concrete nonlinear difference equations and systems have attracted a great recent interest. Particularly, there have been a renewed interest in solvable nonlinear difference equations and systems for fifteen years (see, e.g., [2]- [4], [6], [7], [10]- [13], [15]- [19] and the related references therein). Solvable difference equations are not interesting for themselves only, but they can be also applied in other areas of mathematics, as well as other areas of science (see, e.g., [5], [9]).
One of the first examples of solvable nonlinear difference equations is presented in note [1] where Brand solves the nonlinear difference equation where the initial value x 0 is a real number and the parameters a, b, c, d are real numbers with the restrictions c = 0, ad − bc = 0, and studies long-term behavior of solutions the equation. The note presents a transformation which transforms the nonlinear equation into a linear one. The idea has been used many times in showing solvability of some difference equations, as well as of some systems of difference equations (see, e.g., [6], [7], [10], [11], [15]- [17], [19]). Another example of solvable nonlinear difference equations is the following system of nonlinear difference equation where the initial value x 0 , y 0 are positive real numbers and the parameters a, b, c, d are positive real numbers. System (2) can be transformed into an equation of form (1) dividing the first equation of (2) by its second one. So, the results on Eq. (1) can be used to obtained the results on system (2). System (2) was studied for the first time in [4] by using the method described above. Also, in [4], it is shown that every positive solution of system (2) is two-periodic or converges to its a two-periodic solution. For more results on system (2), see [5], [12], [13]. System (2) can be extended by interchanging the constant coefficients a, b, c, d with two-periodic ones. More concretely, another extension with two-periodic coefficients of (2) is the following system of difference equations where the initial values x 0 , y 0 are positive real numbers and (a n ) n≥0 , (b n ) n≥0 , (c n ) n≥0 , (d n ) n≥0 are two-periodic sequences of positive real numbers. For extensions with periodic coefficients of some difference equations and systems, see [2], [14], [18]. Our main purpose in this paper is to determine the long-term behavior of posistive solutions of system (3). We also use a new method to solve the system without needing some other nonlinear difference equations such as (1). Throughout this paper we assume that a 2n = a 0 , a 2n+1 = a 1 , We also adopt the assumptions x n+1 = f (x n , y n ) , y n+1 = g (x n , y n ) , n ∈ N 0 , is eventually periodic with period p, if there is an n 1 > 0 such that (x n+p , y n+p ) = (x n , y n ), then for n ≥ n 1 . If n 1 = 0, then the solution is periodic with period p.
The folowing result is extracted from [8].
(1 + α k ) with positive terms α k is convergent if and only if ∞ k=0 α k converges.

Main Results
In this section we formulate and prove our main results.
Theorem 3. Assume that x 0 , y 0 > 0 and (a n ) n≥0 , (b n ) n≥0 , (c n ) n≥0 , (d n ) n≥0 are two-periodic sequences of positive real numbers. Then, system of difference equations (3) can be solved in closed form.
Proof. First, it is easy to show by induction that x n , y n > 0, for all n ∈ N 0 . Multiplying both equations in (3) by the following positive product for all n ∈ N 0 . Note that the equalities (4)-(5) constitute a linear system with respect to the following products Therefore, we can write this system in the vector form where u 0 = x 0 , v 0 = y 0 , which is simplier, for all n ∈ N 0 . Let Then, since the sequences (a n ) n≥0 , (b n ) n≥0 , (c n ) n≥0 , (d n ) n≥0 are two-periodic, the matrix A n becomes Now we decompose (7) with respect to even-subscript and odd-subscript terms as follows: for all n ∈ N 0 . From which (8) and (9) follows that Let A 0 A 1 = A. Then, we consider two cases of the matrix A as the following: Case 1: rank (A) = 1. In this case the first row in the matrix A are linearly dependent on the second one. Without loss of generality we may assume that where K is a positive constant such that Using (11) in system (10) we have for all n ∈ N 0 . By the last three relations we have from which it follows that for all n ∈ N. Using (12) and (13) in (8) we obtain for all n ∈ N. Also, the changes of variables in (6) yield for all n ∈ N. Hence, from (16) and (17), we obtain and respectively. By employing (8) in (18)-(21), we have the following closed formulas which is valid for all n ∈ N 0 , respectively. Consequently, in the case rank (A) = 1, by using the formulas (12)- (15) in (22)-(25), we have the general solution of (3) as follows: where for all n ∈ N 0 , respectively. Case 2: rank (A) = 2. In this case both rows in the matrix A are linearly independent of each other. This case also implies that A has two different eigenvalues given by and (32) Since these eigenvalues will correspond to two linear independent eigenvectors, we may write the matrix A as follows: Therefore we may write system (10) as the following where Z 2n = u 2n v 2n , for all n ∈ N 0 . From (33) we have for all n ∈ N 0 . Multiplying both sides of (34) by the matrix P , we have or after some computations for all n ∈ N 0 . From the last vectorial equality, we have for all n ∈ N 0 . From (8), (40) and (41), we have the formulas and for all n ∈ N 0 . Also, we can write the formulas (22)-(25) as the following: for all n ∈ N 0 . Finally, by employing (40)-(43) in (44)-(47), we have the general solution of (3) as the following The following theorem determines and characterizes the long-term behavior of positive solutions of (3) according to the parameters in the case rank (A) = 1.
Proof. The proof follows directly from formulas (26)-(29). That is to say, it is clearly seen from these formulas that if K (a 1 d 0 then every solution of (3) is two-periodic such that x 2n = x 0 , x 2n+1 = x 1 , y 2n = y 0 , y 2n+1 = y 1 .
The following theorem determines and characterizes the long-term behavior of positive solutions of (3) according to the parameters in the case rank (A) = 2.