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On global periodicity of a class of difference equations. (English) Zbl 1180.39005
Summary: We show that the difference equation $x_n= f_3(x_{n-1})f_2(x_{n-2})f_1(x_{n-3})$, $n\in\Bbb N_0$, 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 $$x_n= f_4(x_{n-1})f_3(x_{n-2})f_2(x_{n-3})f_1(x_{n-4}), \quad n\in\Bbb N_0,$$ 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 $c_i$, $i\in\{1,2,3,4\}$.

39A10Additive difference equations
Full Text: DOI EuDML
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