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 $$ \begin{array}{rlll} y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2009-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=377 |
| Summary: | In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions
$$
\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. |
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| ISSN: | 1417-3875 1417-3875 |