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On global periodicity of a class of difference equations. (English) Zbl 1180.39005
Summary: We show that the difference equation $$x_n= f_3(x_{n-1})f_2(x_{n-2})f_1(x_{n-3})$$, $$n\in\mathbb N_0$$, where $$f_i\in C[(0,\infty),(0,\infty)]$$, $$\in\{1,2,3\}$$, is periodic with period 4 if and only if $$f_i(x)=c_i/x$$ for some positive constants $$c_i$$, $$i\in\{1,2,3\}$$ or if $$f_i(x)= c_i/x$$ when $$i=2$$ and $$f_i(x)=c_ix$$ if $$i\in\{1,3\}$$, with $$c_1c_2c_3=1$$. Also, we prove that the difference equation
$x_n= f_4(x_{n-1})f_3(x_{n-2})f_2(x_{n-3})f_1(x_{n-4}), \quad n\in\mathbb N_0,$ where $$f_i\in C[(0,\infty),(0,\infty)]$$, $$i\in\{1,2,3,4\}$$, is periodic with period 5 if and only if $$f_i(x)=c_i/x$$, for some positive constants $$c_i$$, $$i\in\{1,2,3,4\}$$.