Structure of solutions sets and a continuous version of Filippov's theorem for first order impulsive differential inclusions with periodic conditions

J. R. Graef, The University of Tennessee at Chattanooga
A. Ouahab, Université de Sidi Bel Abbés, Sidi Bel Abbés, Algérie

E. J. Qualitative Theory of Diff. Equ., No. 24. (2009), pp. 1-23.

Abstract: 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 $F: J\times \R^n\to{\cal P}(\R^n)$ 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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