Periodicities of a System of Difference Equations

We study the periodicities of a system of difference equations xn+1 = max{1/xn, An/yn−k}, yn+1 = max{1/yn, Bn/xn−k}, where initial values (x−k, y−k), . . . , (x0, y0) ∈ (0, +∞) × (0, +∞). We show that if An, Bn ∈ (0, 1) are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If k is even, there must be one in {xn} and {yn} converges to period two solution.


Introduction
Difference equations are powerful tool that describe the law of nature.Recently, the theory of difference equations has received extensive attention because it can be applied to many areas of science and technology, such as the fields of information and e-science.Moreover, people pay more attention to the dynamics of max-type difference equations; they are concerned about the research of the period character of solutions and the convergence of positive solutions.Please refer to [1][2][3][4][5][6][7].
Furthermore, every solution of this equation is periodic with period 4. Yang [10] studied the max-type difference equation where 0 <  < 1,  > 0; then every solution of this equation tends to be  = 1 or is eventually periodic with period 4. Sun et al. [11] studied the dynamics of max-type difference equation where The initial values  0 ,  −1 are positive real numbers.Let lim sup →∞   = ; if there exist infinitely many  such that   ≥  and  +1 ≥ , then {  } is eventually equal to 1. every positive solution of this equation is eventually periodic with period 2 provided   ∈ (0, 1).

Main Results
Before proceeding with the proof of our main results in this section, we shall need the following lemmas.
Lemma 2. Let {(  ,   )} +∞ =− be a solution of system (5).If  > , then Proof.By Lemma 1 and   < 1, we obtain that, for any  > , Similarly, we also obtain that, for any  > , This completes the proof of the lemma.
for all  ≥ .