**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. 31. (2008), pp. 1-40.

Communicated by S. K. Ntouyas. | Received on 2008-07-11 Appeared on 2008-10-15 |

**Abstract: **In this paper, we present an impulsive version of Filippov's Theorem for the first-order nonresonance impulsive differential inclusion

$$

\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,\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 $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion

$$

\begin{array}{rlll}

y'(t) &\in& \varphi(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,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $\varphi: J\times \R^n\to{\cal P}(\R^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator.

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