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On the recursive sequence $y_{n+1}=(p+y_{n-1})/(qy_n+y_{n-1})$. (English) Zbl 0967.39004
The difference equation $$y(n+1)=(p+y(n-1))/(qy(n)+y(n-1)) \tag 1$$ with positive $p,q$ and initial conditions is studied. It is shown that this system has a unique equilibrium point which is locally asymptotically stable if $q<1+4p$. If $q>1+4p$ then it is a saddle point and a cycle with prime period-two exists. It is proved that the interval $I$ with end points 1 and $p/q$ is an invariant interval of the system (1). Hence the authors proved that if $q< 1+4p$ then the equilibrium point is a global attractor of (1). If $q>1+4p$ then every solution of (1) eventually enters and remains inside $I$.

39A11Stability of difference equations (MSC2000)
Full Text: DOI
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