Geodesic
Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments
Archivum mathematicum, Tome 45 (2009) no. 3, pp. 203-212 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $i=1,2,\dots , m$.
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 $i=1,2,\dots , m$.
Classification : 34K11, 39A10, 39A12, 39A21
Keywords: advanced difference equation; delay difference equation; nonlinear; oscillation 
@article{ARM_2009_45_3_a4,
     author = {\"Ocalan, \"Ozkan},
     title = {Linearized {Oscillation} of {Nonlinear} {Difference} {Equations} with {Advanced} {Arguments}},
     journal = {Archivum mathematicum},
     pages = {203--212},
     year = {2009},
     volume = {45},
     number = {3},
     mrnumber = {2591676},
     zbl = {1212.39005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a4/}
}
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Öcalan, Özkan. Linearized Oscillation of Nonlinear Difference Equations with Advanced Arguments. Archivum mathematicum, Tome 45 (2009) no. 3, pp. 203-212. http://geodesic.mathdoc.fr/item/ARM_2009_45_3_a4/