Olech's correction to the proof of his theorem on existence of solutions to differential inclusions (Q1987777)

The paper studies the Cauchy problem \[ x'\in F(t,x),\quad x(0)=\xi \] where \(\xi \in \mathbb{R}^d\), \(I=[0,1]\), \(F(.,.):I\times \mathbb{R}^d\to \mathcal{P}(\mathbb{R}^d)\) is a set-valued map such that \(F(.,x)\) is measurable \(\forall x\in \mathbb{R}^d\), \(F(t,.)\) is upper semicontinuous with compact values \(\forall t\in I\), \(F(.,.)\) is integrably bounded and if the set \(F(t,x)\) is not convex then \(F(t,.)\) is continuous at \(x\). A new proof of the existence of solutions for this problem is provided.