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Periodically forced Pielou’s equation. (English) Zbl 1120.39003
The authors consider the parametrically excited Pielou equation \[ x_{n+1}={{\beta_nx_n}\over{1+x_{n-1}}} \] with \(\{\beta_n\}_n\) a \(k\)-periodic sequence. Several results on the convergence of positive solutions are given. If \(\{\beta_n\}_n\) is constant and equal to \(\beta>1\), every positive solution converges to 0. If \({\prod_0^{k-1}\beta_i\leq 1}\) then every nonnegative solution converges to 0. If \(\{\beta_n\}_n\) is positive periodic with period \(2p\) (\(p\) prime) and \({\prod_0^{k-1}\beta_i>1}\) then any positive solution converges to a solution that is periodic with period \(2p\); the same is true for the case of period \(2p+1\).

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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