# Extremal selections of multifunctions generating a continuous flow

• Volume: 60, Issue: 2, page 101-117
• ISSN: 0066-2216

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## Abstract

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Let $F:\left[0,T\right]×{ℝ}^{n}\to {2}^{{ℝ}^{n}}$ 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.

## How to cite

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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 -

## References

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1. [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
2. [2] A. Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), 459-470. Zbl0676.34014
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5. [5] A. Bressan, Selections of Lipschitz multifunctions generating a continuous flow, Differential Integral Equations 4 (1991), 483-490. Zbl0722.34009
6. [6] A. Bressan and G. Colombo, Boundary value problems for lower semicontinuous differential inclusions, Funkcial. Ekvac. 36 (1993), 359-373. Zbl0788.34007
7. [7] A. Cellina, On the set of solutions to Lipschitzean differential inclusions, Differential Integral Equations 1 (1988), 495-500. Zbl0723.34009
8. [8] F. S. De Blasi and G. Pianigiani, On the solution set of nonconvex differential inclusions, J. Differential Equations, to appear. Zbl0853.34013
9. [9] F. S. De Blasi and G. Pianigiani, Topological properties of nonconvex differential inclusions, Nonlinear Anal. 20 (1993), 871-894. Zbl0774.34010
10. [10] A. LeDonne and M. V. Marchi, Representation of Lipschitz compact convex valued mappings, Atti Accad. Naz. Lincei Rend. 68 (1980), 278-280. Zbl0481.54011
11. [11] A. F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1962), 76-84. Zbl0139.05102
12. [12] A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova 83 (1990), 33-44. Zbl0708.28005
13. [13] A. A. Tolstonogov, Extreme continuous selectors of multivalued maps and their applications, preprint, 1992.

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