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On the global periodicity of some difference equations of third order. (English) Zbl 1130.39003
This paper is restricted to the equation \[ x_{n+3}=x_i f(x_j,x_k), \] where \(i,j,k\in \{n,n+1,n+2\}\) are pairwise distinct, \(f: (0,\infty)^2\to (0,\infty)^2\) is continuous, and the initial conditions are positive real numbers. The authors are interested in the periodic character of its solutions. The equation is said to be globally periodic if all sequences \(\{x_n\}\) generated by the equation are periodic. The equation is called a \(q\)-cycle if additionally \(q\) is the least common multiple of periods of those periodic sequences generated by the equation.
In this paper the authors want to detect the map \(f\) such that the equation is a \(q\)-cycle. They prove the following results: 1. The unique \(3\)-cycle is given by \(x_{n+3}=x_n\). 2. The unique \(4\)-cycle is given by \(x_{n+3}=x_n(x_{n+2}/x_{n+1})\). 3. There are two \(5\)-cycles, which are given by \(x_{n+3}=x_n(x_{n+2}/x_{n+1})^u\) and \(x_{n+3}=x_n(x_{n+2}/x_{n+1})^v\), where \(u,v\) are two roots of the quadratic equation \(z^2-z-1=0\).

MSC:
39A11 Stability of difference equations (MSC2000)
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