| Abstract: |
| We study the following problem
$$
x``(t)\in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds,\quad x(0)=x_0,\quad x`(0)=y_0,\eqno (1)
$$
where $F:[0,T]\times X\to \mathcal{P}(X)$ is a set-valued map, $X$ is a Banach space, $\{A(t)\}_{t\geq 0}$ is a family of linear closed operators from $X$ into $X$ that genearates an evolution system of operators $\{{\cal U}(t,s)\}_{t,s\in [0,T]}$, $\Delta =\{(t,s)\in [0,T]\times [0,T];t\geq s\}$, $K(.,.):\Delta \to \mathbf{R}$ is continuous and $x_0,y_0\in X$.
The general framework of evolution operators $\{A(t)\}_{t\geq 0}$ that define problem (1) has been developed by Kozak ([2]) and improved by Henriquez ([1]).
We consider this problem in the case when the set-valued map is not convex valued, but is Lipschitz in the second variable.
We obtain several existence results for mild solutions of this problem using fixed point techniques and using classical selection results as Kuratowsky and Ryll-Nardzewski, Bressan and Colombo, De Blasi andi Pianigiani.
\begin{thebibliography}{9}
\bibitem{1} H.R. Henriquez, Existence of solutions of nonautonomous second order functional differential equations with infinite delay, {\it Nonlinear Anal.}, {\bf 74} (2011), 3333-3352.
\bibitem{2} M. Kozak, A fundamental solution of a second-order differential equation in a Banach space, {\it Univ.
Iagel. Acta. Math.}, \textbf{32} (1995), 275-289.
\end{thebibliography} |
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