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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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