Multivalued differential equations on closed convex sets in Banach spaces (Q1321594)

The authors consider the problem of proving the existence of absolutely continuous solutions for the problem \(\dot x(t)\in F(t,x(t))\) a.e. \(t\in [0,T]\), \(x(0)= a\in D\), \(x(t)\in D\), \(\forall t\in [0,T]\), where \(F\) is a multifunction from \([0,T]\times E\) to the set of nonempty convex compact subsets of a separable Banach space \(E\), globally measurable on \([0,T]\times E\), and upper semicontinuous on \(E\) and \(D\) is a nonempty closed convex subset of \(E\). The existence result is deduced from abstract results concerning Lipschitzian approximations and differential properties for upper semi- continuous integrands and upper semi-continuous multifunctions which allow the authors to obtain a new version of Scorza-Dragoni theorem for \(F\) and also upper semicontinuous properties and differentiable properties of the approximates \(1/h\int_ t^{t+ h} F(s,x)ds\) for \(h> 0\). The results are compared with the existing literature.