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

Smaïl Djebali, Lech Górniewicz, Abdelghani Ouahab

DOI: http://dx.doi.org/10.12775/TMNA.2008.049

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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