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

Smaïl Djebali, Lech Górniewicz, Abdelghani Ouahab

Abstract


In this paper, we first present an impulsive version of Filippov's
Theorem for first-order semilinear functional differential
inclusions of the form:
$$
\cases
(y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\},
\\
y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m,
\\
y(t)=\phi(t) &\text{for } t\in[-r,0],
\endcases
$$
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-Wa{\plr ż}ewski 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.

Keywords


Impulsive functional differential inclusions; mild solution; Filippov's theorem; relaxation; solution set; compactness; AR; $R_\delta$; contractibility; acyclicity

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