Extremal selections of multifunctions generating a continuous flow

Alberto Bressan, and Graziano Crasta. "Extremal selections of multifunctions generating a continuous flow." Annales Polonici Mathematici 60.2 (1994): 101-117. <http://eudml.org/doc/262467>.

@article{AlbertoBressan1994,
abstract = {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.},
author = {Alberto Bressan, Graziano Crasta},
journal = {Annales Polonici Mathematici},
keywords = {differential inclusion; extremal selection; Cauchy problem; unique Carathéodory solution},
language = {eng},
number = {2},
pages = {101-117},
title = {Extremal selections of multifunctions generating a continuous flow},
url = {http://eudml.org/doc/262467},
volume = {60},
year = {1994},
}

TY - JOUR
AU - Alberto Bressan
AU - Graziano Crasta
TI - Extremal selections of multifunctions generating a continuous flow
JO - Annales Polonici Mathematici
PY - 1994
VL - 60
IS - 2
SP - 101
EP - 117
AB - 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.
LA - eng
KW - differential inclusion; extremal selection; Cauchy problem; unique Carathéodory solution
UR - http://eudml.org/doc/262467
ER -