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A reinterpretation of set differential equations as differential equations in a Banach space

Published online by Cambridge University Press:  20 November 2017

Martin Rasmussen
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
Janosch Rieger
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
Kevin Webster
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK

Abstract

Set differential equations are usually formulated in terms of the Hukuhara differential. As a consequence, the theory of set differential equations is perceived as an independent subject, in which all results are proved within the framework of the Hukuhara calculus. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of ℝd with their support functions. Using this representation, standard existence and uniqueness theorems for ordinary differential equations can be applied to set differential equations. We provide a geometric interpretation of the main result, and demonstrate that our approach overcomes the heavy restrictions that the use of the Hukuhara differential implies for the nature of a solution.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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