Extensions of Lipschitz selections and an application to differential inclusions

WE ESTABLISH in one section of this paper, the existence of a Lipschitz extension of a Lipschitz selection of a Lipschitz convex-compact set-valued map into R”. That is, given a Lipschitz selection on a subdomain, it can be extended to a Lipschitz selection on the entire domain of definition of the set-valued map. The extension result is applied in the closing section. We show that under the minimal natural conditions, a Lipschitz solution of an autonomous Lipschitz differential inclusion, is a solution of an autonomous Lipschitz ordinary differential equation within the inclusion.