Global behavior of a generalized Lyness difference equation under linear perturbation

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We study the generalized Lyness difference equation under linear perturbation un+2=αun+1+βun(γun+1+δ)+ηun,n=0,1,2,…,with initial values u0,u1>0 where α,β,γ,δ≥0, α+β>0, γ+δ>0, 0≤η<1. It is proved that the solutions are globally asymptotically stable for 0<η<1. Therefore, it is concluded that the generalized Lyness difference equation under linear perturbation holds the dichotomy property as follows: for 0<η<1, all of its solutions are globally asymptotically stable; for η=0, almost all of its solutions are diverge and strictly oscillatory.