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Zbl 0904.39009
Zhang, Yuzhu; Yan, Jurang; Zhao, Aimin
Oscillation criteria for a difference equation.
(English)
[J] Indian J. Pure Appl. Math. 28, No.9, 1241-1249 (1997). ISSN 0019-5588; ISSN 0975-7465/e

Consider the difference inequality $$y(t)- y(t-\tau)+ p\cdot y(t- \sigma)\le 0,\tag i$$ where $\tau$, $\sigma$, $p\in\bbfR^+$ and $\sigma>\tau$. The inequality (i) has an eventually positive solution if and only if the difference equation $$y(t)- y(t-\tau)+ p\cdot y(t-\sigma)= 0\tag ii$$ has an eventually positive solution. If $$\liminf_{t\to\infty} \Biggl[\limsup_{t\to\infty}\Biggr]p(t)> [<]\ {(\sigma/\tau- 1)^{\sigma/\tau- 1}\over (\sigma/\tau)^{\sigma/\tau}}$$ then all solutions of (ii) are oscillatory [(ii) has a nonoscillatory solution] ($p:=p(t)\in C(\bbfR^+,\bbfR^+)$).
[D.Bobrowski (Poznań)]
MSC 2000:
*39A12 Discrete version of topics in analysis
39A10 Difference equations

Keywords: oscillatory solution; difference inequality; eventually positive solution; difference equation

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