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Extremal selections of multifunctions generating a continuous flow

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Bressan A., Crasta G.

Annales Polonici Mathematici

1994-1995

tom 60

nr 2

101-117

Let $F:[0,T] × ℝ^n → 2^{ℝ^n}$ 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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Bressan A.

1 Crasta G.

1 1994

Wyniki wyszukiwania

Extremal selections of multifunctions generating a continuous flow