Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations
The two-dimensional system of nonlinear differential equations $$ x'' = p(t)y^{\alpha}, \quad y'' = q(t)x^{\beta}\qquad {\textrm{(A)}} $$ with positive exponents $\alpha$ and $\beta$ satisfying $\alpha\beta<1$ is analyzed in the framework of regular variation. Under the assump...
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| Format: | Article |
| Language: | English |
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University of Szeged
2013-04-01
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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=1892 |
| _version_ | 1797830668002000896 |
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| author | Jaroslav Jaroš Takaŝi Kusano |
| author_facet | Jaroslav Jaroš Takaŝi Kusano |
| author_sort | Jaroslav Jaroš |
| collection | DOAJ |
| description | The two-dimensional system of nonlinear differential equations
$$
x'' = p(t)y^{\alpha}, \quad y'' = q(t)x^{\beta}\qquad {\textrm{(A)}}
$$
with positive exponents $\alpha$ and $\beta$ satisfying $\alpha\beta<1$ is analyzed in the framework of regular variation. Under the assumption that $p(t)$ and $q(t)$ are nearly regularly varying it is shown that system (A) may possess three types of positive solutions $(x(t),y(t))$ which are strongly monotone in the sense that (i) both components are strongly decreasing, (ii) both components are strongly increasing, and (iii) one of the components is strongly decreasing, while the other is strongly increasing. The solutions in question are sought in the three classes of nearly regularly varying functions of positive or negative indices. It is also shown that if we make a stronger assumption that $p(t)$ and $q(t)$ are regularly varying, then the solutions from the above three classes are fully regularly varying functions, too. |
| first_indexed | 2024-04-09T13:39:44Z |
| format | Article |
| id | doaj.art-af0d35e817714e31a58b40c6705bb139 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:44Z |
| publishDate | 2013-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-af0d35e817714e31a58b40c6705bb1392023-05-09T07:53:03ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752013-04-0120132312310.14232/ejqtde.2013.1.231892Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equationsJaroslav Jaroš0Takaŝi Kusano1Comenius University, Bratislava, SlovakiaHiroshima University, Higashi-Hiroshima, JapanThe two-dimensional system of nonlinear differential equations $$ x'' = p(t)y^{\alpha}, \quad y'' = q(t)x^{\beta}\qquad {\textrm{(A)}} $$ with positive exponents $\alpha$ and $\beta$ satisfying $\alpha\beta<1$ is analyzed in the framework of regular variation. Under the assumption that $p(t)$ and $q(t)$ are nearly regularly varying it is shown that system (A) may possess three types of positive solutions $(x(t),y(t))$ which are strongly monotone in the sense that (i) both components are strongly decreasing, (ii) both components are strongly increasing, and (iii) one of the components is strongly decreasing, while the other is strongly increasing. The solutions in question are sought in the three classes of nearly regularly varying functions of positive or negative indices. It is also shown that if we make a stronger assumption that $p(t)$ and $q(t)$ are regularly varying, then the solutions from the above three classes are fully regularly varying functions, too.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1892systems of nonlinear differential equationsstrongly increasing (decreasing) solutionsregularly varying functions |
| spellingShingle | Jaroslav Jaroš Takaŝi Kusano Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations Electronic Journal of Qualitative Theory of Differential Equations systems of nonlinear differential equations strongly increasing (decreasing) solutions regularly varying functions |
| title | Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| title_full | Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| title_fullStr | Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| title_full_unstemmed | Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| title_short | Asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| title_sort | asymptotic behavior of positive solutions of a class of systems of second order nonlinear differential equations |
| topic | systems of nonlinear differential equations strongly increasing (decreasing) solutions regularly varying functions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=1892 |
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