Linearized oscillation of nonlinear difference equations with advanced arguments

Özkan Öcalan

Address: Afyon Kocatepe University, Faculty of Science and Arts, Department of Mathematics, ANS Campus, 03200, Afyonkarahisar, Turkey


Abstract: 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 \lbrace \dots ,-2,-1\rbrace $ for $i=1,2,\dots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with 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 $p_{in}\le 0$ and $k_{i}\in \lbrace \dots ,-2,-1\rbrace $ for $i=1,2,\dots , m$.

AMSclassification: primary 39A10.

Keywords: advanced difference equation, delay difference equation, nonlinear, oscillation.