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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On continuous and measurable selections and the existence of solutions of generalized differential equations


Author: Henry Hermes
Journal: Proc. Amer. Math. Soc. 29 (1971), 535-542
MSC: Primary 34.04
DOI: https://doi.org/10.1090/S0002-9939-1971-0277794-3
MathSciNet review: 0277794
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Abstract: Let $ \mathcal{C}({B^n})$ denote the space of nonempty compact subsets of some bounded set $ {B^n}$ in Euclidean n dimensional space $ {E^n}$, topologized with the Hausdorff metric topology. The existence of a solution to the initial value problem for the generalized differential equation $ dx(t)/dt \in R(x(t))$ is shown under the assumption that $ R:{E^n} \to \mathcal{C}({B^n})$ has bounded variation in some neighborhood of the initial value, and under a less restrictive condition on the variation of R. Included are continuous and Lipschitz continuous selection results for mappings $ Q:{E^1} \to \mathcal{C}({B^n})$ which are, respectively, of bounded variation and Lipschitz continuous.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0277794-3
Keywords: Generalized differential equation, continuous selections
Article copyright: © Copyright 1971 American Mathematical Society