On the solution sets of differential inclusions and the periodic problem in Banach spaces. (Q1399948)

In the first part of this interesting paper, the authors study the topological characterization of the solution set to the initial value problem for the semilinear differential inclusion \[ x'(t)\in Ax(t)+F(t,x(t)),\tag{1} \] \[ x(0)=x_0\in D,\tag{2} \] where \(A\) is the infinitesimal generator of a linear \(C_0\)-semigroup \(\{U(t)\}_{t\geq 0},\) \(F: [0,T]\times D \multimap E\) is an upper-Carathéodory set-valued map and \(D\) is a closed subset of a Banach space \(E.\) Under appropriate assumptions concerning \(D\) and some natural boundary conditions, they prove that the set of all mild solutions to (1), (2), is an \(R_{\delta}\) set in the space of continuous maps \([0,T]\to E.\) The proofs are based on some new approximation-selection techniques for set-valued maps. In the second part of the paper, applications to periodic problems and to the existence of equilibria are given.