Vietnam Journal of Mathematics

, Volume 45, Issue 1–2, pp 5–49 | Cite as

Differential Equations for Closed Sets in a Banach Space

Survey and Extension
  • Thomas LorenzEmail author


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–Lindelöf theorem (a.k.a. Cauchy–Lipschitz theorem) ensures the existence and uniqueness of set-valued solutions to initial value problems.


Evolution inclusion in a Banach space Reachable set Integral funnel equation Morphological equation Set differential equation Set evolution equation 

Mathematics Subject Classification (2010)

34G25 34A60 49J27 49J53 93B03 37B55 28B20 45N05 



Prof. Willi Jäger was my academic teacher at Heidelberg University from my very first semester until the habilitation. Infected by the “virus” of analysis, I enjoyed following his courses, full of insights into mathematical relations between diverse fields. As a part of his scientific support, he drew my attention to set-valued maps quite early and gave me the opportunity to gain research experience very autonomously. Consequences of his initial inspirations almost 20 years ago are still reflected in this article. Hence, I would like to express my deep gratitude to Prof. Jäger.

Some of the results presented here were developed in connection with a research stay at University Paris 1 Panthéon-Sorbonne. I thank Prof. Joël Blot and Prof. Georges Haddad for their invitation and the hospitality. Last, but not least I would also like to express my gratitude to Prof. Jean-Pierre Aubin especially for the interesting discussions opening me new perspectives how the notions of viability theory and mutational analysis can be applied beyond maths.


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© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Applied MathematicsRheinMain University of Applied SciencesWiesbadenGermany

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