Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities
We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alph...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2015-01-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3504.pdf |
| _version_ | 1811307106695905280 |
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| author | Jaroslav Jaroš Kusano Takaŝi |
| author_facet | Jaroslav Jaroš Kusano Takaŝi |
| author_sort | Jaroslav Jaroš |
| collection | DOAJ |
| description | We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions. |
| first_indexed | 2024-04-13T08:58:36Z |
| format | Article |
| id | doaj.art-467057f075284e2c82a55b347464c3a9 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-04-13T08:58:36Z |
| publishDate | 2015-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-467057f075284e2c82a55b347464c3a92022-12-22T02:53:12ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742015-01-013514769http://dx.doi.org/10.7494/OpMath.2015.35.1.473504Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearitiesJaroslav Jaroš0Kusano Takaŝi1Comenius University, Faculty of Mathematics, Physics and Informatics, Department of Mathematical Analysis and Numerical Mathematics, 842 48 Bratislava, SlovakiaHiroshima University, Faculty of Science, Department of Mathematics, Higashi-Hiroshima 739-8526, JapanWe consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.http://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3504.pdfsystems of differential equationspositive solutionsasymptotic behaviorregularly varying functions |
| spellingShingle | Jaroslav Jaroš Kusano Takaŝi Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities Opuscula Mathematica systems of differential equations positive solutions asymptotic behavior regularly varying functions |
| title | Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities |
| title_full | Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities |
| title_fullStr | Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities |
| title_full_unstemmed | Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities |
| title_short | Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities |
| title_sort | strongly increasing solutions of cyclic systems of second order differential equations with power type nonlinearities |
| topic | systems of differential equations positive solutions asymptotic behavior regularly varying functions |
| url | http://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3504.pdf |
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