Existence and solution sets of impulsive functional differential inclusions with multiple delay

Abstract

In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: $\begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} \\ y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, \\ y(t)=\phi(t), & t \in [-r,0],\end{cases}$ where $J:=[0,b]$ and $0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b$ ($m \in \mathbb{N}^*$), $F$ is a set-valued map and g is single map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,...,m$). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuous dependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.