A critical case of stability of one quasilinear difference equation of the second order
We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where \(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L...
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| Дата: | 1999 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4764 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510932919123968 |
|---|---|
| author | Vitrychenko, I. E. Витриченко, И. Е. Витриченко, И. Е. |
| author_facet | Vitrychenko, I. E. Витриченко, И. Е. Витриченко, И. Е. |
| author_sort | Vitrychenko, I. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:13:33Z |
| description | We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where \(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0\) and \(\alpha \in \left] {0, + \infty } \right[\) . The resuits obtained are valid for the case where \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) . |
| first_indexed | 2026-03-24T03:04:51Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-4764 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:51Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d9/ed39b3ec6f679b2129d4e691ea5716d9.pdf |
| spelling | umjimathkievua-article-47642020-03-18T21:13:33Z A critical case of stability of one quasilinear difference equation of the second order Критический случай устойчивости одного квазилинейного разностного уравнения второго порядка Vitrychenko, I. E. Витриченко, И. Е. Витриченко, И. Е. We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where \(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0\) and \(\alpha \in \left] {0, + \infty } \right[\) . The resuits obtained are valid for the case where \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) . Отримано достатні умови стійкості за Перроном тривіального розв'язку дійсного різницевого рівняння типу $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ),\; n \in N$$ де $y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0$, $\alpha \in \left] {0, + \infty } \right[$. Результати охоплюють випадки, коли $\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty$. Institute of Mathematics, NAS of Ukraine 1999-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4764 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 12 (1999); 1593–1603 Український математичний журнал; Том 51 № 12 (1999); 1593–1603 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4764/6229 https://umj.imath.kiev.ua/index.php/umj/article/view/4764/6230 Copyright (c) 1999 Vitrychenko I. E. |
| spellingShingle | Vitrychenko, I. E. Витриченко, И. Е. Витриченко, И. Е. A critical case of stability of one quasilinear difference equation of the second order |
| title | A critical case of stability of one quasilinear difference equation of the second order |
| title_alt | Критический случай устойчивости одного квазилинейного разностного уравнения второго порядка |
| title_full | A critical case of stability of one quasilinear difference equation of the second order |
| title_fullStr | A critical case of stability of one quasilinear difference equation of the second order |
| title_full_unstemmed | A critical case of stability of one quasilinear difference equation of the second order |
| title_short | A critical case of stability of one quasilinear difference equation of the second order |
| title_sort | critical case of stability of one quasilinear difference equation of the second order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4764 |
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