| Abstract: |
| In recent joint papers with Tiziana Cardinali we investigate in Banach spaces the existence of impulsive mild solutions for a problem driven by the following semilinear second order differential inclusion
$\begin{equation*}
x''(t) \in Ax(t)+F(t,x(t),x'(t)),\ \text{a.e.}\ t \in [0,\infty) \setminus \lbrace t_k \rbrace_k
\end{equation*}$
and the existence of strong solutions for a Dirichlet problem governed by the following Duffing differential inclusion
$\begin{equation*}
-x''(t)-r(t)x(t) \in F(t,x(t)),\ t \in [0,a]
\end{equation*}$
To establish the first goal, we show the existence of mild solutions on a bounded interval. Then, by using a glueing method, we achieve the existence of impulsive mild solutions on $[0,\infty)$.
While to study the Duffing Dirichlet problem, we apply a fixed point result to an appropriate solution operator.
All results are proved without strong compactness assumptions.
Finally, thanks to these findings, the controllability for problems driven by ordinary/partial differential equations is obtained. |
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