| Abstract: |
| We consider the existence problem for the following second order periodic system:
$\begin{equation}
\begin{cases} m(u^\prime(t))^\prime \in F(u(t))+ G(t,u(t),u`(t)) & \mbox{for a.a. } t \in [0,t_{\max}],\ u(0)=u(t_{\max})=0, \, u^\prime(0)=u^\prime(t_{\max})=0, &\end{cases}
\end{equation}$
where $m: \mathbb{R}^N \to \mathbb{R}^N$ is a monotone-type map, including as special case the $p$-Laplacian operator $m(y):=|y|^{p-2}y$ with $p \in (1,+\infty)$. In the reaction, we have the combined effects of a maximal monotone multivalued map $F:D(F) \subseteq \mathbb{R}^N \to 2^{\mathbb{R}^N}$ and a graph measurable multivalued map $G: [0,t_{\max}] \times \mathbb{R}^N \times \mathbb{R}^N \to 2^{\mathbb{R}^N} \setminus \{\emptyset\}$.
We develop a topological approach based on the theory of monotone-type nonlinear operators (see [1]) and multivalued analysis (see [2]). The starting point of the study is a joint work with N. S. Papageorgiou (see [3]). We discuss the cases when $G$ has convex values and non-convex values, respectively, by imposing different hypotheses on the data.
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[1] L. Gasi\`nski and N. S. Papageorgiou, \textit{Nonlinear Analysi}s. Ser. Math. Anal. Appl., vol. 9. CRC Press Boca Raton, 2006.
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[2] S. Hu and N. S. Papageorgiou, \textit{Handbook of Multivalued Analysis}. Vol. I: Theory. Kluwer Academic, Dordrecht, 1997.
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[3] N. S. Papageorgiou and C. Vetro, Existence and relaxation results for second order
multivalued systems, \textit{Acta Appl. Math.}, 173 (2021), Paper No. 5, 36 pp. |
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