Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides
Sufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation \[x'(t) = f(t, x_t), \qquad (x_t(s):= x(t+s),\ -h\le s\le 0)\] are given. In the stability theory of non-autonomous differential equations a result is of M...
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| Format: | Article |
| Language: | English |
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University of Szeged
2015-10-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4198 |
| _version_ | 1797830573611286528 |
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| author | László Hatvani |
| author_facet | László Hatvani |
| author_sort | László Hatvani |
| collection | DOAJ |
| description | Sufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation
\[x'(t) = f(t, x_t), \qquad (x_t(s):= x(t+s),\ -h\le s\le 0)\]
are given. In the stability theory of non-autonomous differential equations a result is of Marachkov type if it contains some kind of boundedness or growth condition on the right-hand side of the equation with respect to $t$. Using Lyapunov's direct method and the annulus argument we prove theorems for equations whose right-hand sides may be unbounded with respect to $t$. The derivative of the Lyapunov function is not supposed to be negative definite, it may be negative semi-definite. The results are applied to the retarded
scalar differential equation with distributed delay
\[x'(t) = -a(t) x(t) + b(t) \int^t_{t-h} x(s) \,{\rm{d}}s,\qquad (a(t)>0),\]
where $a$ and $b$ may be unbounded on $[0,\infty)$. The growth conditions do not concern function $a$, they contain only function $b$. In addition, the function $t\mapsto a(t) - \int^{t+h}_t |b(u)| \,{\rm{d}}u$, measuring the dominance of the negative instantaneous feedback over the delayed feedback, is not supposed to remain above a positive constant, even it may vanish on long intervals. |
| first_indexed | 2024-04-09T13:39:21Z |
| format | Article |
| id | doaj.art-4f6a56e4b67a454aa48651c6213b0652 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:39:21Z |
| publishDate | 2015-10-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-4f6a56e4b67a454aa48651c6213b06522023-05-09T07:53:05ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752015-10-0120156411110.14232/ejqtde.2015.1.644198Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sidesLászló Hatvani0Bolyai Institute, University of Szeged, Szeged, HungarySufficient conditions for uniform equi-asymptotic stability and uniform asymptotic stability of the zero solution of the retarded equation \[x'(t) = f(t, x_t), \qquad (x_t(s):= x(t+s),\ -h\le s\le 0)\] are given. In the stability theory of non-autonomous differential equations a result is of Marachkov type if it contains some kind of boundedness or growth condition on the right-hand side of the equation with respect to $t$. Using Lyapunov's direct method and the annulus argument we prove theorems for equations whose right-hand sides may be unbounded with respect to $t$. The derivative of the Lyapunov function is not supposed to be negative definite, it may be negative semi-definite. The results are applied to the retarded scalar differential equation with distributed delay \[x'(t) = -a(t) x(t) + b(t) \int^t_{t-h} x(s) \,{\rm{d}}s,\qquad (a(t)>0),\] where $a$ and $b$ may be unbounded on $[0,\infty)$. The growth conditions do not concern function $a$, they contain only function $b$. In addition, the function $t\mapsto a(t) - \int^{t+h}_t |b(u)| \,{\rm{d}}u$, measuring the dominance of the negative instantaneous feedback over the delayed feedback, is not supposed to remain above a positive constant, even it may vanish on long intervals.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4198lyapunov functional with negative semi-definite derivative; annulus argument; uniform asymptotic stability; uniform equi-asymptotic sability. |
| spellingShingle | László Hatvani Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides Electronic Journal of Qualitative Theory of Differential Equations lyapunov functional with negative semi-definite derivative; annulus argument; uniform asymptotic stability; uniform equi-asymptotic sability. |
| title | Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides |
| title_full | Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides |
| title_fullStr | Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides |
| title_full_unstemmed | Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides |
| title_short | Marachkov type stability conditions for non-autonomous functional differential equations with unbounded right-hand sides |
| title_sort | marachkov type stability conditions for non autonomous functional differential equations with unbounded right hand sides |
| topic | lyapunov functional with negative semi-definite derivative; annulus argument; uniform asymptotic stability; uniform equi-asymptotic sability. |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4198 |
| work_keys_str_mv | AT laszlohatvani marachkovtypestabilityconditionsfornonautonomousfunctionaldifferentialequationswithunboundedrighthandsides |