Extremal Selections of Multifunctions Generating a Continuous Flow

n be a continuous multifunction with compact, not necessarily convex values. If F is Lipschitz continuous, it was shown in [4] that there exists a measurable selection f of F such that, for every x 0 , the Cauchy problem ˙ x(t) = f (t, x(t)), x(0) = x 0 has a unique Caratheodory solution, depending continuously on x 0. In this paper, we prove that the above selection f can be chosen so that f (t, x) ∈ extF (t, x) for all t, x. More generally, the result remains valid if F satisfies the following Lipschitz Selection Property: (LSP) For every t, x, every y ∈ coF (t, x) and ε > 0, there exists a Lipschitz selection φ of coF , defined on a neighborhood of (t, x), with |φ(t, x) − y| < ε. (LSP). Another interesting class, for which (LSP) holds, consists of those continuous mul-tifunctions F whose values are compact and have convex closure with nonempty interior.