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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 148; no. 2; pp. 429 - 446
• Main Authors: Rasmussen, Martin, Rieger, Janosch, Webster, Kevin
• Format: Journal Article
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.04.2018; Cambridge University Press
• Subjects: Banach spaces; Calculus; Differential calculus; Differential equations; Differential geometry; Existence theorems; Mathematical analysis; Ordinary differential equations; Set theory; Uniqueness theorems; Primary 58D25; set differential equations; 34G20; ordinary differential equations in Banach spaces; existence and uniqueness; 26E25
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210517000178

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.