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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

  • painleve-kuratowski limit

  • series expansion

  • exponentiation

  • banach space equipped

  • òp painleve-kuratowski

  • recursive exponentiation method

  • euclidean space


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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.