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Linearized oscillation of nonlinear difference equations with advanced arguments. (English) Zbl 1212.39005

Summary: This paper is concerned with the nonlinear advanced difference equation with constant coefficients \[ x_{n+1}-x_{n}+\sum _{i=1}^{m}p_{i}f_{i}(x_{n-k_{i}})=0,\quad n=0,1,\dots, \] where \(p_{i}\in (-\infty ,0)\) and \(k_{i}\in \{\dots ,-2,-1\}\) for \(i=1,2,\dots ,m\). We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the above difference equation by comparing to the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients \[ x_{n+1}-x_{n}+\sum _{i=1}^{m}p_{in}f_{i}(x_{n-k_{i}})=0, \quad n=0,1,\dots, \] where \(p_{in}\leq 0\) and \(k_{i}\in \{\dots ,-2,-1\}\) for \(i=1,2,\dots ,m.\)

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34K11 Oscillation theory of functional-differential equations
39A21 Oscillation theory for difference equations
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