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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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##### References:
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