Spectral characterizations for Hyers-Ulam stability
First we prove that an $n\times n$ complex linear system is Hyers-Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imaginary axis $i\mathbb{R}$). Also we show that the scalar differential equation of order $n,$ \[\begin{split} x^{(n)}(t)=a_1x^{(n-1)}...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2014-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2686 |
| _version_ | 1797830674307088384 |
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| author | Constantin Buse Olivia Saierli Afshan Tabassum |
| author_facet | Constantin Buse Olivia Saierli Afshan Tabassum |
| author_sort | Constantin Buse |
| collection | DOAJ |
| description | First we prove that an $n\times n$ complex linear system is Hyers-Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imaginary axis $i\mathbb{R}$). Also we show that the scalar differential equation of order $n,$
\[\begin{split}
x^{(n)}(t)=a_1x^{(n-1)}(t)+\ldots+a_{n-1}{x}'(t)+a_nx(t),\quad t\in\mathbb{R}_+:=[0, \infty),
\end{split}\]
is Hyers-Ulam stable if and only if the algebraic equation
\[
\begin{split}
z^n=a_1z^{n-1}+\cdots +a_{n-1}z+a_n,
\end{split}
\]
has no roots on the imaginary axis. |
| first_indexed | 2024-04-09T13:39:49Z |
| format | Article |
| id | doaj.art-c55f1d4e8d554809b027e9dc6f3fd397 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:49Z |
| publishDate | 2014-06-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-c55f1d4e8d554809b027e9dc6f3fd3972023-05-09T07:53:04ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752014-06-0120143011410.14232/ejqtde.2014.1.302686Spectral characterizations for Hyers-Ulam stabilityConstantin Buse0Olivia Saierli1Afshan Tabassum2West University of Timisoara, Timisoara, RomaniaTibiscus University of Timisoara, Department of Computer Sciences,Government College University, Abdus Salam School of Mathematical Sciences, (ASSMS), Lahore, PakistanFirst we prove that an $n\times n$ complex linear system is Hyers-Ulam stable if and only if it is dichotomic (i.e. its associated matrix has no eigenvalues on the imaginary axis $i\mathbb{R}$). Also we show that the scalar differential equation of order $n,$ \[\begin{split} x^{(n)}(t)=a_1x^{(n-1)}(t)+\ldots+a_{n-1}{x}'(t)+a_nx(t),\quad t\in\mathbb{R}_+:=[0, \infty), \end{split}\] is Hyers-Ulam stable if and only if the algebraic equation \[ \begin{split} z^n=a_1z^{n-1}+\cdots +a_{n-1}z+a_n, \end{split} \] has no roots on the imaginary axis.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2686diferential equationsdichotomyhyers-ulam stability |
| spellingShingle | Constantin Buse Olivia Saierli Afshan Tabassum Spectral characterizations for Hyers-Ulam stability Electronic Journal of Qualitative Theory of Differential Equations diferential equations dichotomy hyers-ulam stability |
| title | Spectral characterizations for Hyers-Ulam stability |
| title_full | Spectral characterizations for Hyers-Ulam stability |
| title_fullStr | Spectral characterizations for Hyers-Ulam stability |
| title_full_unstemmed | Spectral characterizations for Hyers-Ulam stability |
| title_short | Spectral characterizations for Hyers-Ulam stability |
| title_sort | spectral characterizations for hyers ulam stability |
| topic | diferential equations dichotomy hyers-ulam stability |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=2686 |
| work_keys_str_mv | AT constantinbuse spectralcharacterizationsforhyersulamstability AT oliviasaierli spectralcharacterizationsforhyersulamstability AT afshantabassum spectralcharacterizationsforhyersulamstability |