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Comparison and oscillation theorem for second-order nonlinear neutral difference equations of mixed type. (English) Zbl 1280.39010

Summary: We establish some comparison theorems for the oscillation of second-order neutral difference equations of mixed type \[ \Delta(a_n \Delta(x_n+b_nx_{n-\sigma_1} + c_nx_{n+\sigma_2})^\alpha) + q_nx_{n-\tau_1}^\beta + p_nx_{n+\tau_2}^\beta = 0, \] where \(\alpha\) and \(\beta\) are ratio of odd positive integers, \(\sigma_1\), \(\sigma_2\), \(\tau_1\), and \(\tau_2\) are positive integers. Our results are new even if \(p_n=c_n=0\). Examples are provided to illustrate the results.

MSC:

39A21 Oscillation theory for difference equations
39A12 Discrete version of topics in analysis
34K40 Neutral functional-differential equations
39A10 Additive difference equations
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