Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions
Let $f:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ be a continuous function and let $h:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\mathbb{R}\rightarro...
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| Format: | Article |
| Language: | English |
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University of Szeged
2003-07-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=159 |
| _version_ | 1797830760289271808 |
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| author | C. Avramescu |
| author_facet | C. Avramescu |
| author_sort | C. Avramescu |
| collection | DOAJ |
| description | Let $f:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ be a continuous function and let $h:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\mathbb{R}\rightarrow \mathbb{R}^{N}$ satisfying the inequality $\left| x\left( t\right) \right| \leq k\cdot h\left( t\right) ,$ $t\in \mathbb{R},$ $k>0$, where $\left| \cdot \right| $ is the euclidean norm in $\mathbb{R}^{N},$ is given. The proof of this result is based on the use of a special function of Lyapunov type, which is often called guiding function. In the particular case $h\equiv 1$, one obtains known results regarding the existence of bounded solutions. |
| first_indexed | 2024-04-09T13:41:16Z |
| format | Article |
| id | doaj.art-1c8e18ef0f084f4b828377fa0aa3fd9f |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:16Z |
| publishDate | 2003-07-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-1c8e18ef0f084f4b828377fa0aa3fd9f2023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752003-07-012003131910.14232/ejqtde.2003.1.13159Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functionsC. Avramescu0University of Craiova, Craiova, RomaniaLet $f:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ be a continuous function and let $h:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\mathbb{R}\rightarrow \mathbb{R}^{N}$ satisfying the inequality $\left| x\left( t\right) \right| \leq k\cdot h\left( t\right) ,$ $t\in \mathbb{R},$ $k>0$, where $\left| \cdot \right| $ is the euclidean norm in $\mathbb{R}^{N},$ is given. The proof of this result is based on the use of a special function of Lyapunov type, which is often called guiding function. In the particular case $h\equiv 1$, one obtains known results regarding the existence of bounded solutions.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=159 |
| spellingShingle | C. Avramescu Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions Electronic Journal of Qualitative Theory of Differential Equations |
| title | Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| title_full | Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| title_fullStr | Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| title_full_unstemmed | Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| title_short | Asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| title_sort | asymptotic behavior of solutions of nonlinear differential equations and generalized guiding functions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=159 |
| work_keys_str_mv | AT cavramescu asymptoticbehaviorofsolutionsofnonlineardifferentialequationsandgeneralizedguidingfunctions |