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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 $$I$$.

##### MSC:
 39A11 Stability of difference equations (MSC2000)
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