Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions
In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions $$ \begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2009-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=377 |
| _version_ | 1827951411425443840 |
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| author | John Graef Abdelghani Ouahab |
| author_facet | John Graef Abdelghani Ouahab |
| author_sort | John Graef |
| collection | DOAJ |
| description | In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions
$$
\begin{array}{rlll}
y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in
J\backslash \{t_{1},\ldots,t_{m}\},\\
y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\
y(0)&=&y(b),
\end{array}
$$
where $J=[0,b]$ and $F: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved. |
| first_indexed | 2024-04-09T13:41:20Z |
| format | Article |
| id | doaj.art-2d9b8effedd24558bc9ea54445166882 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:41:20Z |
| publishDate | 2009-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-2d9b8effedd24558bc9ea544451668822023-05-09T07:52:59ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752009-04-0120092412310.14232/ejqtde.2009.1.24377Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditionsJohn Graef0Abdelghani Ouahab1University of Tennessee at Chattanooga, Chattanooga, TN, U.S.A.Université de Sidi Bel Abbés, Sidi Bel Abbés, AlgérieIn this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions $$ \begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\ y(0)&=&y(b), \end{array} $$ where $J=[0,b]$ and $F: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=377 |
| spellingShingle | John Graef Abdelghani Ouahab Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions Electronic Journal of Qualitative Theory of Differential Equations |
| title | Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions |
| title_full | Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions |
| title_fullStr | Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions |
| title_full_unstemmed | Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions |
| title_short | Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions |
| title_sort | structure of solutions sets and a continuous version of filippov s theorem for first order impulsive differential inclusions with periodic conditions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=377 |
| work_keys_str_mv | AT johngraef structureofsolutionssetsandacontinuousversionoffilippovstheoremforfirstorderimpulsivedifferentialinclusionswithperiodicconditions AT abdelghaniouahab structureofsolutionssetsandacontinuousversionoffilippovstheoremforfirstorderimpulsivedifferentialinclusionswithperiodicconditions |