On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity
We prove the existence of continuously differentiable solutions with required asymptotic properties as t → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right)...
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| Дата: | 2001 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2001
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4268 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510399141511168 |
|---|---|
| author | Zernov, A. E. Зернов, А. Е. Зернов, А. Е. |
| author_facet | Zernov, A. E. Зернов, А. Е. Зернов, А. Е. |
| author_sort | Zernov, A. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:25:45Z |
| description | We prove the existence of continuously differentiable solutions with required asymptotic properties as t → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ where α: (0, τ) → (0, +∞), g: (0, τ) → (0, +∞), and h: (0, τ) → (0, +∞) are continuous functions, 0 < g(t) ≤ t, 0 < h(t) ≤ t, t ∈ (0, τ), \(\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \\ \end{gathered}\) , and the function ϕ is continuous in a certain domain. |
| first_indexed | 2026-03-24T02:56:22Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-4268 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:56:22Z |
| publishDate | 2001 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4e/868280fc618c389994823e951d7ce54e.pdf |
| spelling | umjimathkievua-article-42682020-03-18T20:25:45Z On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity О разрешимости и асимптотике решений некоторого функционально-дифференциального уравнения с сингулярностью Zernov, A. E. Зернов, А. Е. Зернов, А. Е. We prove the existence of continuously differentiable solutions with required asymptotic properties as t → +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ where α: (0, τ) → (0, +∞), g: (0, τ) → (0, +∞), and h: (0, τ) → (0, +∞) are continuous functions, 0 < g(t) ≤ t, 0 < h(t) ≤ t, t ∈ (0, τ), \(\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \\ \end{gathered}\) , and the function ϕ is continuous in a certain domain. Доведено існування неперервно диференційовних розв'язків з потрібними асимптотичними властивостями при $t → +0$ та визначено кількість розв'язків такої задачі Коші для функціонально-диференціального рівняння: $$\alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0,$$ де $α: (0, τ) → (0, +∞),\; g: (0, τ) → (0, +∞),\; h: (0, τ) → (0, +∞)$ — неперервні функції, $0 < g(t) ≤ t, 0 < h(t) ≤ t,\; t ∈ (0, τ), $, $$\begin{gathered} \alpha \left( t \right)x\prime \left( t \right) = at + b_1 x\left( t \right) + b_2 x\left( {g\left( t \right)} \right) + \phi \left( {t,x\left( t \right),x\left( {g\left( t \right)} \right),x\prime \left( {h\left( t \right)} \right)} \right),\quad x\left( 0 \right) = 0, \hfill \\ \mathop {\lim }\limits_{t \to + 0} \alpha \left( t \right) = 0 \hfill \\ \end{gathered}$$ функція $ϕ$ неперервна в деякій області. Institute of Mathematics, NAS of Ukraine 2001-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4268 Ukrains’kyi Matematychnyi Zhurnal; Vol. 53 No. 4 (2001); 455-465 Український математичний журнал; Том 53 № 4 (2001); 455-465 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4268/5240 https://umj.imath.kiev.ua/index.php/umj/article/view/4268/5241 Copyright (c) 2001 Zernov A. E. |
| spellingShingle | Zernov, A. E. Зернов, А. Е. Зернов, А. Е. On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title | On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title_alt | О разрешимости и асимптотике решений некоторого функционально-дифференциального уравнения с сингулярностью |
| title_full | On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title_fullStr | On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title_full_unstemmed | On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title_short | On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity |
| title_sort | on the solvability and asymptotics of solutions of one functional differential equation with singularity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4268 |
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