Global behavior of a generalized Lyness difference equation under linear perturbation

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 concl...

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Bibliographic Details
Published inApplied mathematics letters Vol. 99; p. 106009
Main Authors Deng, Guifeng, Lu, Qiuying, Qian, Lili
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.01.2020
Subjects
Asymptotic stability
Dichotomy
Generalized Lyness equation
Lyapunov function
Dichotomy
Asymptotic stability
Generalized Lyness equation
Lyapunov function
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Summary: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.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2019.106009