Topological Structure of the Solutions Set of Impulsive Semilinear Differential Inclusions with Nonconvex Right-Hand Side

Abstract

In this paper, we study the topological structure of solution sets for the following first-order impulsive evolution inclusion with initial conditions: $$ \left\{ \begin{array}{rlll} y'(t)-Ay(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m\\ y(0)&=&a\in E, \end{array} \right. $$where $J:=[0,b]$ and $0 = t_0 < t_1 < \,... \,< t_m < b$, $A$ is the infinitesimal generator of a $C_0-$semigroup of linear operator $T(t)$ on a separable Banach space $E$ and $F$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, \ldots ,m$). The continuous selection of the solution set is also investigated.