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Oscillation criteria for certain even order quasilinear difference equations. (English) Zbl 1186.39011

By assuming that \(\sigma(n) \leq n\) is a real increasing sequence and \(f(n,u) \, \text{sgn} \,u \geq q_n\, |u|^{\alpha-1}u\), with \(q_n\) a positive sequence, the author gives a new oscillation criterion for the high order quasilinear difference equation
\[ \Delta(|\Delta^{k-1}x_n|^{\alpha-1} \Delta^{k-1}x_n)+f(n,x_{\sigma(n)})=0. \]
Here \(n \in \{0,1,2,\dots\}\), \(\Delta x_n=x_{n+1}-x_n\) and \(\Delta^dx_n=\Delta(\Delta^{d-1}x_n)\). \(\alpha \geq 1\) is a constant and \(k\) is an even integer.

MSC:

39A21 Oscillation theory for difference equations
39A10 Additive difference equations
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References:

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