The separable difference equation \(x_{n+1}=\frac{b_{n}x_{n}}{x_{n-1}}\) (Q2922351)

The authors consider the equation NEWLINE\[NEWLINEx_{n+1} = {{b_nx_n}\over{x_{n-1}}}\;,\;n=0,1,2,\dotsNEWLINE\]NEWLINE where \(\{b_n\}\) is \(p\)-periodic (\(p>0\)), in the case \(b_i>0\), \(x_{-1}>0\), \(x_0>0\). Four theorems are proved stating respectively conditions for: a) \(\lim_{n\rightarrow\infty} x_n=1\) for every solution; b) convergence of positive solutions to a 6-periodic solution; c) \(\lim_{n\rightarrow\infty}x_n=0\) for every solution; d) \(\lim_{n\rightarrow\infty} x_n=+\infty\) for every solution.