Comparison theorems and asymptotic behavior of solutions of discrete fractional equations
Consider the following $\nu$-th order nabla and delta fractional difference equations \begin{equation} \begin{aligned} \nabla^\nu_{\rho(a)}x(t)&=c(t)x(t),\quad \quad t\in\mathbb{N}_{a+1},\\ x(a)&>0. \end{aligned}\tag{$\ast$} \end{equation} and \begin{equation} \begin{aligned} \Delta^\nu...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University of Szeged
2015-12-01
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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=4278 |
| _version_ | 1797830618391773184 |
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| author | Baoguo Jia Lynn Erbe Allan Peterson |
| author_facet | Baoguo Jia Lynn Erbe Allan Peterson |
| author_sort | Baoguo Jia |
| collection | DOAJ |
| description | Consider the following $\nu$-th order nabla and delta fractional difference equations
\begin{equation}
\begin{aligned}
\nabla^\nu_{\rho(a)}x(t)&=c(t)x(t),\quad \quad
t\in\mathbb{N}_{a+1},\\
x(a)&>0.
\end{aligned}\tag{$\ast$}
\end{equation}
and
\begin{equation}
\begin{aligned}
\Delta^\nu_{a+\nu-1}x(t)&=c(t)x(t+\nu-1),\quad \quad
t\in\mathbb{N}_{a},\\
x(a+\nu-1)&>0.
\end{aligned}\tag{$\ast\ast$}
\end{equation}
We establish comparison theorems by which we compare the solutions $x(t)$ of ($\ast$) and ($\ast\ast$) with the solutions of the equations $\nabla^\nu_{\rho(a)}x(t)=bx(t)$ and $\Delta^\nu_{a+\nu-1}x(t)=bx(t+\nu-1),$ respectively, where $b$ is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011) 353--370].
These results show that the solutions of two fractional difference equations $\nabla^\nu_{\rho(a)}x(t)=cx(t),\ 0<\nu<1$, and $\Delta^\nu_{a+\nu-1}x(t)=cx(t+\nu-1),\ 0<\nu<1$, have similar asymptotic behavior with the solutions of the first order difference equations $\nabla x(t)=cx(t),\ |c|<1 $ and $\Delta x(t)=cx(t)$, $|c|<1 $, respectively. |
| first_indexed | 2024-04-09T13:38:58Z |
| format | Article |
| id | doaj.art-fdd4b2e36e844540a977c4d4e183eca8 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:58Z |
| publishDate | 2015-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-fdd4b2e36e844540a977c4d4e183eca82023-05-09T07:53:05ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752015-12-0120158911810.14232/ejqtde.2015.1.894278Comparison theorems and asymptotic behavior of solutions of discrete fractional equationsBaoguo Jia0Lynn Erbe1Allan Peterson2Sun Yat-Sen University, Guangzhou, ChinaUniversity of Nebraska-Lincoln, Lincoln, USAUniversity of Nebraska-Lincoln, Lincoln, USAConsider the following $\nu$-th order nabla and delta fractional difference equations \begin{equation} \begin{aligned} \nabla^\nu_{\rho(a)}x(t)&=c(t)x(t),\quad \quad t\in\mathbb{N}_{a+1},\\ x(a)&>0. \end{aligned}\tag{$\ast$} \end{equation} and \begin{equation} \begin{aligned} \Delta^\nu_{a+\nu-1}x(t)&=c(t)x(t+\nu-1),\quad \quad t\in\mathbb{N}_{a},\\ x(a+\nu-1)&>0. \end{aligned}\tag{$\ast\ast$} \end{equation} We establish comparison theorems by which we compare the solutions $x(t)$ of ($\ast$) and ($\ast\ast$) with the solutions of the equations $\nabla^\nu_{\rho(a)}x(t)=bx(t)$ and $\Delta^\nu_{a+\nu-1}x(t)=bx(t+\nu-1),$ respectively, where $b$ is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011) 353--370]. These results show that the solutions of two fractional difference equations $\nabla^\nu_{\rho(a)}x(t)=cx(t),\ 0<\nu<1$, and $\Delta^\nu_{a+\nu-1}x(t)=cx(t+\nu-1),\ 0<\nu<1$, have similar asymptotic behavior with the solutions of the first order difference equations $\nabla x(t)=cx(t),\ |c|<1 $ and $\Delta x(t)=cx(t)$, $|c|<1 $, respectively.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4278nabla and delta fractional differencediscrete mittag-leffler functionrising and falling function |
| spellingShingle | Baoguo Jia Lynn Erbe Allan Peterson Comparison theorems and asymptotic behavior of solutions of discrete fractional equations Electronic Journal of Qualitative Theory of Differential Equations nabla and delta fractional difference discrete mittag-leffler function rising and falling function |
| title | Comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| title_full | Comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| title_fullStr | Comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| title_full_unstemmed | Comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| title_short | Comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| title_sort | comparison theorems and asymptotic behavior of solutions of discrete fractional equations |
| topic | nabla and delta fractional difference discrete mittag-leffler function rising and falling function |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4278 |
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