Selections of Lipschitz multifunctions generating a continuous flow
Let F: \({\mathbb{R}}\times {\mathbb{R}}^ n\mapsto 2^{{\mathbb{R}}^ n}\) be a Lipschitz continuous multifunction with compact, not necessarily convex values. The theorem proved in this paper states the existence of a directionally continuous selection f, with f(t,x)\(\in F(t,x)\forall t,x\), such that the Cauchy problem \(\dot x=f(t,x)\), \(x(t_ 0)=x_ 0\) has a unique solution, depending continuously on the initial data \((t_ 0,x_ 0)\). The construction of f given in the paper is a refinement of the directionally continuous selection theorems for lower semi-continuous multifunctions, previously developed by the author. In particular, the result implies the existence of a (time dependent) continuous flow on \({\mathbb{R}}^ n\), whose trajectories are all solutions to the differential inclusion \(\dot x=F(t,x)\).
scientific article; zbMATH DE number 4190164