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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:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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References:
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