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

Volume 16, Number 2 (2014), 72 - 91

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

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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 $t_k$ ($k=1, ... ,m$). The continuous selection of the solution set is also investigated.