A reinterpretation of set differential equations as differential equations in a Banach space. Issue 2 (20th November 2017)

• Record Type: Journal Article
• Authors: Rasmussen, Martin; Rieger, Janosch; Webster, Kevin
• Is Part Of: Proceedings. Volume 148:Issue 2(2018)
• Page Start: 429
• Page End: 446
• Publication Date: 2017-11-20
• Subjects: set differential equations, -- ordinary differential equations in Banach spaces, -- existence and uniqueness; Primary 58D25, -- 34G20, -- 26E25; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210517000178
• Languages: English
• ISSNs: 0308-2105

Abstract

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 ℝ d 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.