Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0
We prove the existence of continuously differentiable solutions \(x:(0,{\rho ]} \to \mathbb{R}^n\) such that $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \...
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| Date: | 2002 |
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| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2002
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4209 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510339634823168 |
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| author | Zernov, A. E. Kuzina, Yu. V. Зєрнов, А. Є. Кузіна, Ю. В. |
| author_facet | Zernov, A. E. Kuzina, Yu. V. Зєрнов, А. Є. Кузіна, Ю. В. |
| author_sort | Zernov, A. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:24:06Z |
| description | We prove the existence of continuously differentiable solutions \(x:(0,{\rho ]} \to \mathbb{R}^n\) such that $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0$$ or $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$ where $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$ |
| first_indexed | 2026-03-24T02:55:26Z |
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| id | umjimathkievua-article-4209 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:55:26Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/07/80b10df25397d590ad929e46bec2e107.pdf |
| spelling | umjimathkievua-article-42092020-03-18T20:24:06Z Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 Асимптотическое поведение решений задачи Коши x′ = f(t, x, x′), x(0) = 0 Zernov, A. E. Kuzina, Yu. V. Зєрнов, А. Є. Кузіна, Ю. В. We prove the existence of continuously differentiable solutions \(x:(0,{\rho ]} \to \mathbb{R}^n\) such that $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0$$ or $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$ where $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },{ \rho is sufficiently small}{.}$$ Доводиться існування неперервно диференційовних розв'язків $x:(0,{\rho ]} \to \mathbb{R}^n$ таких, що $$\left\| {x\left( t \right) - {\xi }\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {\xi '}\left( t \right)} \right\| = O\left( {{\eta }\left( t \right)/t} \right),{ }t \to + 0,$$ або $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t \to + 0,$$ де $${\xi }:\left( {0,{\tau }} \right) \to \mathbb{R}^n ,{ \eta }:\left( {0,{\tau }} \right) \to \left( {0, + \infty } \right),{ }\left\| {{\xi }\left( t \right)} \right\| = o\left( 1 \right),$$ $${\eta }\left( t \right) = o\left( t \right),{ \eta }\left( t \right) = o\left( {\left\| {{\xi }\left( t \right)} \right\|} \right),{ }t \to + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {\rho } < {\tau },\rho$ — достаньо мале. Institute of Mathematics, NAS of Ukraine 2002-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4209 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 12 (2002); 1698-1703 Український математичний журнал; Том 54 № 12 (2002); 1698-1703 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4209/5125 https://umj.imath.kiev.ua/index.php/umj/article/view/4209/5126 Copyright (c) 2002 Zernov A. E.; Kuzina Yu. V. |
| spellingShingle | Zernov, A. E. Kuzina, Yu. V. Зєрнов, А. Є. Кузіна, Ю. В. Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title | Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title_alt | Асимптотическое поведение решений задачи Коши x′ = f(t, x, x′), x(0) = 0 |
| title_full | Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title_fullStr | Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title_full_unstemmed | Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title_short | Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 |
| title_sort | asymptotic behavior of solutions of the cauchy problem x′ = f(t, x, x′), x(0) = 0 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4209 |
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