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Stability of the recursive sequence x n+1 =(α-βx n )/(γ+x n-1 ). (English) Zbl 0990.39009

Consider the recursive sequence

x n+1 =α+βx n γ+x n-1 ,n=0,1,(*)

where α,β and γ are nonnegative and the initial conditions x 1 and x 0 are arbitrary. Equation (*) has two equilibrium points positive and negative.

If there exists k2 such that γkα/β and αkβ 2 , then the positive equilibrium point is a global attractor with some given basin. The asymptotic properties in the case α=0, β<0, γ>0 are investigated in details.


MSC:
39A11Stability of difference equations (MSC2000)
39B05General theory of functional equations