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Oscillation and asymptotic behavior of second order difference equations with nonlinear neutral terms. (English) Zbl 1069.39017

The authors consider second-order nonlinear neutral difference equations of the form \[ \Delta(a_n\Delta(y_n-p_ny^\alpha_{n-k})) + q_nf(y_{n+1-\ell}) = 0,\;n\geq n_0\geq 0 \] with real \(p\), \(k>0\), \(\ell\geq 0\) integers, \(\alpha\) a ratio of odd positive integers, \(\Delta y_n\) the forward difference, \(a_n>0\) such that \(\sum^\infty a_n^{-1} = \infty\), \(uf(u)>0\) and \(f\) is nondecreasing. Three theorems are proved: two on the oscillation of every solution and one on asymptotic stability. Some examples containing special cases of the above equation are discussed.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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