In this paper, we study the boundedness, persistence, and periodicity of the positive solutions and the global asymptotic stability of the positive equilibrium points of the system of difference equations $$x_{n+1}=A+rac{x_{n-1}}{z_{n}},qquad y_{n+1}=A+ rac{y_{n-1}}{z _{n}},qquad z_{n+1}=A+rac{z_{n-1}}{y_{n}},quad n=0,1,ldots , $$ where (Ain ( 0,infty ) ) and the initial conditions (x_{i}), (y_{i}), (z_{i}in ( 0,infty ) ), (i=-1,0).KeywordsSystem of difference equations??Solution??Boundedness??Equilibrium point??Stability??Global asymptotic stability??MSC39A10??39A30??1 IntroductionDifference equation or discrete dynamical system is a diverse field which impacts almost every branch of pure and applied mathematics. Lately, there has been great interest in investigating the behavior of solutions of a system of nonlinear difference equations and discussing the asymptotic stability of their equilibrium points. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models that describe real life situations in population biology, economics, probability theory, genetics, psychology, and so forth, see [3, 5, 8, 9]. Also, similar works in two and three dimensions (limit behaviors) for more general cases, i.e., continuous and discrete cases, have been done by some authors, see [1, 11, 12, 13, 16]. There are many papers in which systems of difference equations have been studied, as in the examples given below.
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