Periodic solutions for time-dependent subdifferential evolution inclusions (Q523953)

In a separable Hilbert space \(H\), the problem \[ \begin{cases} -u'(t)\in \partial \varphi (t,u(t))+ F(t,u(t)) & \text{for almost all } t\in [0,b],\\ u(0)=u(b) \end{cases} \] is considered, where \(\varphi:[0,b]\times H\to \bar{ \mathbb{R}}=\mathbb{R}\cup \{+\infty\}\) is a proper, convex and lower semicontinuous map; \(\partial\varphi(t,x)\) is the subdifferential in the sense of convex analysis; \(F:[0,b]\times H\to 2^H\setminus\{\emptyset\}\) is a multifunction. A \textit{strong} solution for the problem is a function \(u\in W^{1,2}((0,b),H)\) such that \(u(t)\in \text{dom } \varphi(t,\cdot)\) for all \(t\in [0,T]\), \(u(0)=u(b)\) and there exists \(f\in S^2_{F(\cdot, u(\cdot))}\) such that \[ -u'(t)\in \partial\varphi(t,u(t)) + f(t)\;\text{for almost all } t\in [0,b]. \] The existence of strong solutions for the convex case, i.e. when the multimap \(F\) is convex valued, and for the nonconvex case is proved. Further, the existence of extremal periodic solutions is show. Then, a strong relaxation theorem is provided showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally, some examples illustrate the results of the paper.