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- painleve-kuratowski limit
- series expansion
- exponentiation
- banach space equipped
- òp painleve-kuratowski
- recursive exponentiation method
- euclidean space
Publié par
Langue
English
XLIM
UMR CNRS 6172
Département
Mathématiques-Informatique
Multivalued Exponentiation Analysis.
Part I: Maclaurin Exponentials
Alexandre Cabot & Alberto Seeger
Rapport de recherche n° 2006-06
Déposé le 4 avril 2006
Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex
Tél. (33) 5 55 45 73 23 - Fax. (33) 5 55 45 73 22
http://www.xlim.fr
http://www.unilim.fr/laco
Université de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex
Tél. (33) 5 55 45 73 23 - Fax. (33) 5 55 45 73 22
http://www.xlim.fr
http://www.unilim.fr/laco
To appear in SET-VALUED ANALYSIS
1
MULTIVALUED EXPONENTIATION ANALYSIS.
PART I: MACLAURIN EXPONENTIALS
Alexandre Cabot
and
Alberto Seeger
Abstract.The exponentiation theory of linear continuous operators on Banach spaces can be
extended in manifold ways to a multivalued context.In this paper we explore the Maclaurin
exponentiation technique which is based on the use of a suitable power series.More precisely, we
discuss about the existence and characterization of the Painlev´-Kuratowski limit
n
X
1
p
[ExpF](x) =limF(x)
p!
n→∞
p=0
under different assumptions on the multivalued mapF:X X. InPart II of this work we study
−→
the so-called recursive exponentiation method which uses as ingredient the set of trajectories
associated to a discrete time evolution system governed byF.
Mathematics Subject Classifications.26E25, 33B10, 34A60.
Key Words.Exponentiation, multivalued map, differential inclusion, power series,
Painlev´Kuratowski convergence.
Introduction
1.1 Formulationof the Problem
Throughout this work,Xis assumed to be a real Banach space equipped with a norm denoted by| ∙ |. The
closed unit ball inXis represented by the symbolBXa couple of occasions we will ask. InXto be a Hilbert
space or even a finite dimensional Euclidean space, but this will be explicitly mentioned in the appropriate
place.
What does it mean exponentiating a multivalued operatorF:X X? Morethan the nature of the
−→
underlying spaceX, what is important to stress here is the multivalued character ofFare using the. We
double arrow notation for emphasizing thatF(x) is a subset ofXand not just a single point.
The above question arises, for instance, when it comes to study a Cauchy problem of the form
z˙(t)∈F(z(t)) fora.e.t∈[0,1]
(1)
z(0) =x,
with trajectories being sought in a suitable space of functions, say
AC([0,1], X) ={z: [0,1]→X|zis absolutely continuous}.
By analogy with the concept of velocity field employed in the context of ordinary differential equations, the
operatorFon the right-hand side of (1) is sometimes referred to as avelocity map.
