Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations
We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown tha...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2023-03-01
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| Series: | Opuscula Mathematica |
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| Online Access: | https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4313.pdf |
| _version_ | 1797859127043555328 |
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| author | Manabu Naito |
| author_facet | Manabu Naito |
| author_sort | Manabu Naito |
| collection | DOAJ |
| description | We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition
\[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t\to\infty\). |
| first_indexed | 2024-04-09T21:24:28Z |
| format | Article |
| id | doaj.art-0ba28de52de749e1bc249fbe50d2c84c |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-04-09T21:24:28Z |
| publishDate | 2023-03-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-0ba28de52de749e1bc249fbe50d2c84c2023-03-27T18:04:20ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742023-03-01432221246https://doi.org/10.7494/OpMath.2023.43.2.2214313Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equationsManabu Naito0Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, JapanWe consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t\to\infty\).https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4313.pdfasymptotic behaviornonoscillatory solutionhalf-linear differential equation |
| spellingShingle | Manabu Naito Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations Opuscula Mathematica asymptotic behavior nonoscillatory solution half-linear differential equation |
| title | Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations |
| title_full | Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations |
| title_fullStr | Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations |
| title_full_unstemmed | Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations |
| title_short | Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations |
| title_sort | existence and asymptotic behavior of nonoscillatory solutions of half linear ordinary differential equations |
| topic | asymptotic behavior nonoscillatory solution half-linear differential equation |
| url | https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4313.pdf |
| work_keys_str_mv | AT manabunaito existenceandasymptoticbehaviorofnonoscillatorysolutionsofhalflinearordinarydifferentialequations |