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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 \(c_i\), \(i\in\{1,2,3,4\}\).

MSC:
39A10 Additive difference equations
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References:
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