# Article

 Title: Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments (English) Author: Öcalan, Özkan Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 45 Issue: 3 Year: 2009 Pages: 203-212 Summary lang: English . Category: math . 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 $k_{i}\in \lbrace \dots ,-2,-1\rbrace$ 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 $i=1,2,\dots , m$. (English) Keyword: advanced difference equation Keyword: delay difference equation Keyword: nonlinear Keyword: oscillation MSC: 34K11 MSC: 39A10 MSC: 39A12 MSC: 39A21 idZBL: Zbl 1212.39005 idMR: MR2591676 . Date available: 2009-09-18T11:25:30Z Last updated: 2013-09-19 Stable URL: http://hdl.handle.net/10338.dmlcz/134234 . Reference: [1] Agarwal, R. P.: Difference Equations and Inequalities.Marcel Dekker, New York, 2000. Zbl 0952.39001, MR 1740241 Reference: [2] Elaydi, S.: An Introduction to Difference Equation.Springer-Verlag, New York, 1999. MR 1711587 Reference: [3] Erbe, L. H., Zhang, B. G.: Oscillation of discrete analogues of delay equations.Differential Integral Equations 2 (3) (1989), 300–309. Zbl 0723.39004, MR 0983682 Reference: [4] Györi, I., Ladas, G.: Linearized oscillations for equations with piecewise constant arguments.Differential Integral Equations 2 (1989), 123–131. MR 0984181 Reference: [5] Györi, I., Ladas, G.: Oscillation theory of delay differential equations with applications.Clarendon Press, Oxford, 1991. MR 1168471 Reference: [6] Ladas, G.: Oscillations of equations with piecewise constant mixed arguments.International Conference on Differential Equations and Population Biology, Ohio University, March 21-25, New York, 1988. Zbl 0711.34083, MR 1026200 Reference: [7] Ladas, G.: Explicit conditions for the oscillation of difference equations.J. Math. Anal. Appl. 153 (1990), 276–287. Zbl 0718.39002, MR 1080131, 10.1016/0022-247X(90)90278-N Reference: [8] Öcalan, Ö.: Oscillation of nonlinear difference equations with several coefficients.Commun. Math. Anal. 4 (1) (2008), 35–44. Zbl 1163.39004, MR 2365921 Reference: [9] Öcalan, Ö., Akin, Ö.: Oscillation properties for advanced difference equations.Novi Sad J. Math. 37 (1) (2007), 39–47. Zbl 1224.39017, MR 2402049 .

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