Some properties of viability problems depending on a parameter
type: Item
instance of: scholarly article
title: Some properties of viability problems depending on a parameter
zbMATH Open document ID: 0823.34020
DOI: 10.1007/BF01194011
author: Vladimir V. Goncharov
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1995-11-06T00:00:00Z
review text: Let \(X\) be a Banach space, \(\Xi\) a separable metric space and \(K : \Xi - \circ X\) a compact-valued and lower semicontinuous multimap admitting a continuous selection \(x^ 0 (\cdot)\). Let \(\Gamma : [0,a] \times grK - \circ X\) be a closed, bounded-valued multimap such that \(\Gamma (\cdot, x, \xi)\) is measurable, \(\Gamma (t, \cdot, \cdot)\) is l.s.c for a.e. \(t \in [0,a]\), \(\Gamma (t, \cdot, \xi)\) satisfies some Lipschitz condition and the tangential condition \(\Gamma (t,x, \xi) \subset T_{K(\xi)} (x)\) where \(T\) denotes the Bouligand contingent cone is fulfilled. The author proves the existence and the continuous dependence on the parameter \(\xi\) of solutions of the problem \(x'(t, \xi) \in \Gamma (t,x, \xi)\), \(x(t, \xi) \in K (\xi)\), \(x(0, \xi) = x^ 0 (\xi)\). The relaxation property for this problem is also considered.
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 736847
zbMATH Keywords: viability problem
zbMATH Keywords: differential inclusion
zbMATH Keywords: Banach space
zbMATH Keywords: multimap
zbMATH Keywords: continuous selection
zbMATH Keywords: existence
zbMATH Keywords: continuous dependence on the parameter
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
cites work: On the set of solutions to Lipschitzian differential inclusions
cites work: Representation of the attainable set for Lipschitzian differential inclusions
cites work: Q3752168
cites work: A continuous version of the Filippov-Gronwall inequality for differential inclusions
cites work: Q3989360
cites work: Well posedness for differential inclusions on closed sets
cites work: Continuous version of Filippov-Wažewski relaxation theorem
cites work: Continuous selections for a class of non-convex multivalued maps
cites work: Extensions and selections of maps with decomposable values
cites work: A continuous version of Liapunov's convexity theorem
cites work: Continuous selections of Aumann integrals
cites work: Q3985290
cites work: Viable trajectories of nonconvex differential inclusions
full work available at URL: https://doi.org/10.1007/bf01194011
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description: scientific article; zbMATH DE number 736847