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Multivalued Exponentiation Analysis. Part III: Forward Exponentials

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Abstract

We continue our work on exponentiability of multivalued maps on Banach spaces. In Part I we studied the exponentiability of a map F : X ⇉ X by using a Maclaurin expansion approach. In Part II we studied the recursive exponentiation approach. Recursive exponentials are built by using the trajectories of a discrete-time evolution system governed by F. We now focus the attention on forward exponentiability. The forward exponential of F at the point x is defined as the Kuratowski limit

$$e^{F}(x):=\lim\limits_{n\to\infty}\; \left(I+ n^{-1} F \right)^{n}(x).$$

This type of exponential arises in connection with the Euler discretization scheme for solving the first-order differential inclusion \(\dot \psi (t)\in F(\psi (t))\).

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Authors and Affiliations

  1. Université de Montpellier II, Laboratoire I3M, Place E. Bataillon, 34095, Montpellier, France

    Alexandre Cabot

  2. Université d’Avignon, Département de Mathématiques, 33 rue Louis Pasteur, 84000, Avignon, France

    Alberto Seeger

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  1. Alexandre Cabot
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  2. Alberto Seeger
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Correspondence to Alexandre Cabot.

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Cabot, A., Seeger, A. Multivalued Exponentiation Analysis. Part III: Forward Exponentials. Set-Valued Var. Anal 22, 617–638 (2014). https://doi.org/10.1007/s11228-014-0273-8

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  • DOI: https://doi.org/10.1007/s11228-014-0273-8

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