First order impulsive differential inclusions with periodic conditions

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 $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). 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 $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ 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.