Filippov-Ważewski theorems and structure of solution sets for first order impulsive semilinear functional differential inclusions

Abstract

In this paper, we first present an impulsive version of Filippov's Theorem for first-order semilinear functional differential inclusions of the form: $$ \begin{cases} (y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\}, \\ y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m, \\ y(t)=\phi(t) &\text{for } t\in[-r,0], \end{cases} $$ where $J=[0,b]$, $A$ is the infinitesimal generator of a $C_0$-semigroup 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 $t_k$ ($k=1,\ldots,m$). Then the convexified problem is considered and a Filippov-Ważewski result is proved. Further to several existence results, the topological structure of solution sets - closeness and compactness - is also investigated. Some results from topological fixed point theory together with notions of measure on noncompactness are used. Finally, some geometric properties of solution sets, AR, $R_\delta$-contractibility and acyclicity, corresponding to Aronszajn-Browder-Gupta type results, are obtained.