Oscillations of advanced difference equations with variable arguments
Consider the first-order advanced difference equation of the form \begin{equation*} \nabla x(n)-p(n)x(\mu (n))=0\text{, }\ n\geq 1\, [\Delta x(n)-p(n)x(\nu (n))=0, n\geq 0], \end{equation*} where $\nabla $ denotes the backward difference operator $\nabla x(n)=x(n)-x(n-1)$, $\Delta $ denotes the for...
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| Format: | Article |
| Language: | English |
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University of Szeged
2012-09-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1668 |
| _version_ | 1797830649080446976 |
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| author | George Chatzarakis Ioannis Stavroulakis |
| author_facet | George Chatzarakis Ioannis Stavroulakis |
| author_sort | George Chatzarakis |
| collection | DOAJ |
| description | Consider the first-order advanced difference equation of the form
\begin{equation*}
\nabla x(n)-p(n)x(\mu (n))=0\text{, }\ n\geq 1\, [\Delta x(n)-p(n)x(\nu (n))=0, n\geq 0],
\end{equation*}
where $\nabla $ denotes the backward difference operator $\nabla x(n)=x(n)-x(n-1)$, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$, $\left\{ p(n)\right\} $ is a sequence of nonnegative real numbers, and $\left\{ \mu (n)\right\} $ $\ \left[ \left\{ \nu (n)\right\} \right] $ is a sequence of positive integers such that
\begin{equation*}
\mu (n)\geq n+1\ \text{ for all }n\geq 1\, \left[ \nu (n)\geq n+2 \ \text{ for all }n\geq 0\right] \text{.}
\end{equation*}
Sufficient conditions which guarantee that all solutions oscillate are established. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given. |
| first_indexed | 2024-04-09T13:40:32Z |
| format | Article |
| id | doaj.art-6938bc13c5e54ca59dc214a8929b6cee |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:32Z |
| publishDate | 2012-09-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-6938bc13c5e54ca59dc214a8929b6cee2023-05-09T07:53:02ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752012-09-0120127911610.14232/ejqtde.2012.1.791668Oscillations of advanced difference equations with variable argumentsGeorge Chatzarakis0Ioannis Stavroulakis1Department of Electrical and Electronic Engineering EducatorsUniversity of Ioannina, Ioannina, GreeceConsider the first-order advanced difference equation of the form \begin{equation*} \nabla x(n)-p(n)x(\mu (n))=0\text{, }\ n\geq 1\, [\Delta x(n)-p(n)x(\nu (n))=0, n\geq 0], \end{equation*} where $\nabla $ denotes the backward difference operator $\nabla x(n)=x(n)-x(n-1)$, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$, $\left\{ p(n)\right\} $ is a sequence of nonnegative real numbers, and $\left\{ \mu (n)\right\} $ $\ \left[ \left\{ \nu (n)\right\} \right] $ is a sequence of positive integers such that \begin{equation*} \mu (n)\geq n+1\ \text{ for all }n\geq 1\, \left[ \nu (n)\geq n+2 \ \text{ for all }n\geq 0\right] \text{.} \end{equation*} Sufficient conditions which guarantee that all solutions oscillate are established. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1668advanced difference equationvariable argumentoscillatory solutionnonoscillatory solution |
| spellingShingle | George Chatzarakis Ioannis Stavroulakis Oscillations of advanced difference equations with variable arguments Electronic Journal of Qualitative Theory of Differential Equations advanced difference equation variable argument oscillatory solution nonoscillatory solution |
| title | Oscillations of advanced difference equations with variable arguments |
| title_full | Oscillations of advanced difference equations with variable arguments |
| title_fullStr | Oscillations of advanced difference equations with variable arguments |
| title_full_unstemmed | Oscillations of advanced difference equations with variable arguments |
| title_short | Oscillations of advanced difference equations with variable arguments |
| title_sort | oscillations of advanced difference equations with variable arguments |
| topic | advanced difference equation variable argument oscillatory solution nonoscillatory solution |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1668 |
| work_keys_str_mv | AT georgechatzarakis oscillationsofadvanceddifferenceequationswithvariablearguments AT ioannisstavroulakis oscillationsofadvanceddifferenceequationswithvariablearguments |