Nonlinear multivalued periodic systems (Q2424946)

The paper studies the following differential inclusion with periodic conditions \[ \begin{gathered} -u'(t)\in A(t,u(t))+\partial \varphi (u(t))+F(t,u(t))\quad \text{for a.a. } t\in I,\\ u(0)=u(b), \end{gathered}\tag{1} \] where $I=[0,b]$, $A(.,.):I\times {\mathbb{R}}^n\to \mathcal{P}({\mathbb{R}}^n)$ is a set-valued map which is maximal monotone in the second variable, $\partial \varphi $ denotes the subdifferential in the sense of convex analysis of the lower semicontinuous convex and proper function $\varphi (.):{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $F(.,.):I\times {\mathbb{R}}^n\to \mathcal{P}({\mathbb{R}}^n)$ is a set-valued map. Under certain hypothesis on the map $A$ the existence of solutions for problem (1) is proved in the case when $F(.,.)$ has nonempty compact convex values and in the case when $F(.,.)$ has nonempty closed values and is lower semicontinuous in the second variable. Afterwards, when $F(.,.)$ has nonempty compact convex values and is continuous in the second variable the existence of solutions for the problem \[ \begin{gathered} -u'(t)\in A(t,u(t))+\partial \varphi (u(t))+\operatorname{ext} F(t,u(t))\quad \text{for a.a. } t\in I,\\ u(0)=u(b) \end{gathered} \] is obtained, where $\operatorname{ext} C$ denotes the set of the extreme points of the set $C\subset {\mathbb{R}}^n$. Finally, it is proved that any solution of problem (1) in which $F(.,.)$ is assumed to have nonempty compact convex values can be obtained as the limit in $C(I,{\mathbb{R}}^n)$-norm of certain extremal trajectories.