On the recursive sequence \(y_{n+1}=(p+y_{n-1})/(qy_n+y_{n-1})\). (English) Zbl 0967.39004

Kosmala, W. A.; Kulenović, M. R. S.; Ladas, G.; Teixeira, C. T.

J. Math. Anal. Appl. 251, No. 2, 571-586 (2000).

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 \(I\).

Reviewer: Ahmed Hegazi (Mansoura)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

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

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