Nonconvex perturbations of second order maximal monotone differential inclusions (Q1958993)

Consider differential inclusions of the form \[ -x''(t)\in A(t)x(t)+F(t,x(t),x'(t))\quad a.e.\, ([0,1]),\quad x(0)=x(1)=0,\tag{1} \] where \(A:E\to {\mathcal P}(E)\) is a maximal monotone operator, \(E\) is a finite dimensional space and \(F:[0,1]\times E\times E\to {\mathcal P}(E)\) is a nonempty closed valued multifunction. Under certain technical hypotheses, the existence of solutions to problem (1) is proved in the case when \(F\) has \textit{convex values} and \(F(t,.,.)\) is upper semicontinuous for all \(t\in [0,1]\) and in the case when the values of \(F\) are not necessarily convex but \(F(t,.,.)\) is lower semicontinuous for all \(t\in [0,1]\).