| 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 |
| . |
