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

Öcalan, Ö.

Arch. Math., Brno 45, No. 3, 203-212 (2009).

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

Keywords:

advanced difference equation; delay difference equation; nonlinear; oscillation