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Zbl 0967.39004
Kosmala, W.A.; Kulenović, M.R.S.; Ladas, G.; Teixeira, C.T.
On the recursive sequence $y_{n+1}=(p+y_{n-1})/(qy_n+y_{n-1})$. (English)
[J] J. Math. Anal. Appl. 251, No.2, 571-586 (2000). ISSN 0022-247X

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 $I$.
[Ahmed Hegazi (Mansoura)]

MSC 2000:: *39A11 Stability of difference equations

Keywords: recursive sequence; stability; asymptotics; difference equation; saddle point; cycle; global attractor

Cited in: Zbl 1004.39010

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