Differential Equations for Closed Sets in a Banach Space

Abstract

Nonempty subsets of a vector space suggest themselves for describing shapes (without restrictions of regularity) or states (together with their deterministic uncertainty). Hence, there have been developed several suggestions how to extend differential and integral equations respectively to time-dependent subsets although subsets do not form a linear space in an obvious way. In this article, we summarize three central approaches which can handle possibly non-convex closed subsets: integral funnel solution (a.k.a. R-solutions), morphological primitives, and reachable sets (generalizing Aumann integrals). They are extended to a separable Banach space and characterized by means of semilinear evolution inclusions.We formulate conditions sufficient for the equivalence of these generalized concepts and, then this joint basis is used for specifying differential equations for closed (not necessarily convex or compact) subsets. Several further approaches in the literature prove to be special cases. In this purely metric setting, the counterpart of the Picard–Lindelof theorem (a.k.a. Cauchy–Lipschitz theorem) ensures the existence and uniqueness of set-valued solutions to initial value problems.