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.

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.

You can download the full text of this paper in DVI, PostScript or PDF format.