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Baire's category in existence problems for non-convex lower semicontinuous evolution differential inclusions

Part of: General theory in ordinary differential equations Existence theories Differential equations in abstract spaces

Published online by Cambridge University Press:  15 June 2011

F.S. De Blasi  and
G. Pianigiani
F.S. De Blasi
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Roma, Italy ([email protected])
G. Pianigiani
Affiliation:
Dipartimento di Matematica per le Decisioni, Università di Firenze, Via Lombroso 6/17, 50134 Firenze, Italy ([email protected])
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Abstract

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The existence of mild solutions to the non-convex Cauchy problem

is investigated. Here A is the infinitesimal generator of a C0-semigroup in a reflexive and separable Banach space , F is a Pompeiu–Hausdorff lower semicontinuous multifunction whose values are closed convex and bounded sets with non-empty interior contained in , and ∂F(t, x(t)) denotes the boundary of F(t, x(t)). Our approach is based on the Baire category method, with appropriate modifications which are actually necessary because, under our assumptions, the underlying metric space that naturally enters in the Baire method, i.e. the solution set of the convexified Cauchy problem (CF), can fail to be a complete metric space.

Keywords

Baire methodBanach spacesinfinitesimal generatorC0-semigroupnon-convex evolution inclusionsPompeiu–Hausdorff lower semicontinuity

MSC classification

Secondary: 34G25: Evolution inclusions 34A60: Differential inclusions 49J27: Problems in abstract spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 3 , October 2011 , pp. 645 - 667
Copyright
Copyright © Edinburgh Mathematical Society 2011

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