Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach
In this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system \begin{equation*} x'-p(t)\varphi_{1/\alpha}(y)=0,\qquad y'+q(t)\varphi_{\alpha}(x)=0, \tag{A} \end{equation*} where $\alpha>...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2018-10-01
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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=7030 |
| _version_ | 1797830512521248768 |
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| author | Jaroslav Jaroš Takaŝi Kusano T. Tanigawa |
| author_facet | Jaroslav Jaroš Takaŝi Kusano T. Tanigawa |
| author_sort | Jaroslav Jaroš |
| collection | DOAJ |
| description | In this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system
\begin{equation*}
x'-p(t)\varphi_{1/\alpha}(y)=0,\qquad y'+q(t)\varphi_{\alpha}(x)=0,
\tag{A}
\end{equation*}
where $\alpha>0$ is a constant, $p(t)$ and $q(t)$ are positive continuous functions on $[0,\infty)$, and $\varphi_{\gamma}(u)=|u|^{\gamma}\textrm{sgn}\;u, \;u\in {\mathbb R},\;\gamma>0$. A systematic analysis of the existence and asymptotic behavior of solutions of (A) is proposed for this purpose. A special mention should be made of the fact that all possible types of nonoscillatory solutions of (A) can be constructed by solving the Riccati type differential equations associated with (A). Worthy of attention is that all the results for (A) can be applied to the second order half-linear differential equation
\begin{equation*}
(p(t)\varphi_{\alpha}(x'))'+q(t)\varphi_{\alpha}(x)=0,
\tag{E}
\end{equation*}
to build automatically a nonoscillation theory for (E). |
| first_indexed | 2024-04-09T13:38:24Z |
| format | Article |
| id | doaj.art-6e4274bad9324299b6440b1d5b30c3ae |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:24Z |
| publishDate | 2018-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-6e4274bad9324299b6440b1d5b30c3ae2023-05-09T07:53:08ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752018-10-0120189212810.14232/ejqtde.2018.1.927030Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approachJaroslav Jaroš0Takaŝi Kusano1T. Tanigawa2Comenius University, Bratislava, SlovakiaHiroshima University, Higashi-Hiroshima, JapanDepartment of Mathematical Sciences, Osaka Prefecture University, Osaka, JapanIn this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system \begin{equation*} x'-p(t)\varphi_{1/\alpha}(y)=0,\qquad y'+q(t)\varphi_{\alpha}(x)=0, \tag{A} \end{equation*} where $\alpha>0$ is a constant, $p(t)$ and $q(t)$ are positive continuous functions on $[0,\infty)$, and $\varphi_{\gamma}(u)=|u|^{\gamma}\textrm{sgn}\;u, \;u\in {\mathbb R},\;\gamma>0$. A systematic analysis of the existence and asymptotic behavior of solutions of (A) is proposed for this purpose. A special mention should be made of the fact that all possible types of nonoscillatory solutions of (A) can be constructed by solving the Riccati type differential equations associated with (A). Worthy of attention is that all the results for (A) can be applied to the second order half-linear differential equation \begin{equation*} (p(t)\varphi_{\alpha}(x'))'+q(t)\varphi_{\alpha}(x)=0, \tag{E} \end{equation*} to build automatically a nonoscillation theory for (E).http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7030half-linear differential systemsnonoscillatory solutionsriccati equation |
| spellingShingle | Jaroslav Jaroš Takaŝi Kusano T. Tanigawa Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach Electronic Journal of Qualitative Theory of Differential Equations half-linear differential systems nonoscillatory solutions riccati equation |
| title | Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach |
| title_full | Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach |
| title_fullStr | Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach |
| title_full_unstemmed | Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach |
| title_short | Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach |
| title_sort | nonoscillatory solutions of planar half linear differential systems a riccati equation approach |
| topic | half-linear differential systems nonoscillatory solutions riccati equation |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7030 |
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