Periodic viable trajectories of differential inclusions (Q1818146)

Let \(X\) be a Banach space, \( I=[0,T] \) \((T>0)\subset\mathbb{R}\), \( F: I\times X\to 2^X\setminus\varPhi \) be a set-valued map. Consider the following initial value problem for the differential inclusion \[ x'(t)\in F(t,x(t)),\quad \text{a.e.}\quad x(0)=x_0\in X.\tag{1} \] An existence theorem of periodic viable trajectories to differential inclusions (1) in a finite-dimensional space is proved. With the above results and Galerkin's approximation, an existence theorem of periodic viable trajectories to partial differential inclusions in a Hilbert space is proved.