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Zbl 1157.39004
Cheng, Jinfa; Chu, Yuming
Oscillation theorem for second-order difference equations.
(English)
[J] Taiwanese J. Math. 12, No. 3, 632-633 (2008). ISSN 1027-5487

The authors study the following second-order difference equation, $$\Delta (r_{n-1}\Delta x_{n-1})+p_nx^{\gamma}_n=0, \qquad n=1,2,\ldots, \tag*$$ where $\Delta x_n=x_{n+1}-x_n$, $\gamma$ is the quotient of odd positive integers and $p_n$, $r_n\in (0,\infty)$ for $n=1$, $2$, $\ldots$ with $p_n$ not eventually zero. The obtained results include sufficient condition for the existence of bounded nonoscillatory solution to (*) and respective sufficient and necessary conditions for every bounded solution of (*) to oscillate for the cases where $\gamma>1$, $\gamma=1$, and $\gamma\in (0,1)$. These results not only generalize those for the case where $r_n\equiv 1$ but also improve some of them.
[Yuming Chen (Waterloo)]
MSC 2000:
*39A11 Stability of difference equations
39A10 Difference equations

Keywords: oscillation; nonoscillatory solution; contraction principle; second-order difference equation; bounded solution

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