Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments

Özkan Öcalan

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 3, page 203-212
  • ISSN: 0044-8753

Abstract

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This paper is concerned with the nonlinear advanced difference equation with constant coefficients x n + 1 - x n + ∑ i = 1 m p i f i ( x n - k i ) = 0 , n = 0 , 1 , ⋯ where p i ∈ ( - ∞ , 0 ) and k i ∈ { ⋯ , - 2 , - 1 } for i = 1 , 2 , ⋯ , 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 + ∑ i = 1 m p i n f i ( x n - k i ) = 0 , n = 0 , 1 , ⋯ where p i n ≤ 0 and k i ∈ { ⋯ , - 2 , - 1 } for i = 1 , 2 , ⋯ , m .

How to cite

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Öcalan, Özkan. "Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments." Archivum Mathematicum 045.3 (2009): 203-212. <http://eudml.org/doc/250688>.

@article{Öcalan2009,
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 $i=1,2,\dots , m$.},
author = {Öcalan, Özkan},
journal = {Archivum Mathematicum},
keywords = {advanced difference equation; delay difference equation; nonlinear; oscillation; advanced difference equation; delay difference equation; nonlinear; oscillation},
language = {eng},
number = {3},
pages = {203-212},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments},
url = {http://eudml.org/doc/250688},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Öcalan, Özkan
TI - Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 3
SP - 203
EP - 212
AB - 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 $i=1,2,\dots , m$.
LA - eng
KW - advanced difference equation; delay difference equation; nonlinear; oscillation; advanced difference equation; delay difference equation; nonlinear; oscillation
UR - http://eudml.org/doc/250688
ER -

References

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  3. Erbe, L. H., Zhang, B. G., Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (3) (1989), 300–309. (1989) Zbl0723.39004MR0983682
  4. Györi, I., Ladas, G., Linearized oscillations for equations with piecewise constant arguments, Differential Integral Equations 2 (1989), 123–131. (1989) MR0984181
  5. Györi, I., Ladas, G., Oscillation theory of delay differential equations with applications, Clarendon Press, Oxford, 1991. (1991) MR1168471
  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. Zbl0711.34083MR1026200
  7. Ladas, G., 10.1016/0022-247X(90)90278-N, J. Math. Anal. Appl. 153 (1990), 276–287. (1990) Zbl0718.39002MR1080131DOI10.1016/0022-247X(90)90278-N
  8. Öcalan, Ö., Oscillation of nonlinear difference equations with several coefficients, Commun. Math. Anal. 4 (1) (2008), 35–44. (2008) Zbl1163.39004MR2365921
  9. Öcalan, Ö., Akin, Ö., Oscillation properties for advanced difference equations, Novi Sad J. Math. 37 (1) (2007), 39–47. (2007) Zbl1224.39017MR2402049

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