Extremal selections of multifunctions generating a continuous flow

Alberto Bressan; Graziano Crasta

Annales Polonici Mathematici, Tome 60 (1994), p. 101-117 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $F : [0, T] \times ℝ^{n} \to 2^{ℝ^{n}}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property: (LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε, then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem ẋ(t) = f(t,x(t)), x(0) = x₀, has a unique Carathéodory solution, depending continuously on x₀. We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class for which (LSP) holds consists of those continuous multifunctions F whose values are compact and have convex closure with nonempty interior.

Publié le : 1994-01-01

Zbl 0816.34009

EUDML-ID : urn:eudml:doc:262467

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     author = {Alberto Bressan and Graziano Crasta},
     title = {Extremal selections of multifunctions generating a continuous flow},
     journal = {Annales Polonici Mathematici},
     volume = {60},
     year = {1994},
     pages = {101-117},
     zbl = {0816.34009},
     language = {en},
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Alberto Bressan; Graziano Crasta. Extremal selections of multifunctions generating a continuous flow. Annales Polonici Mathematici, Tome 60 (1994) pp. 101-117. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv60z2p101bwm/

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