Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions We study (set-valued) mappings of bounded Φ-variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the Φ-variation of a metric space valued mapping. We show that the linear span GV Φ(I;X) of the set of all mappings of bounded Φ-variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X → Y is a given mapping and the composition operator ℋ is defined by (ℋf)(t)=h(t,f(t)), where t∈I and f:I → X, we show that ℋ:GV Φ(I;X)→ GV Ψ(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, t∈I, x∈X. This result is further extended to multivalued composition operators ℋ with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded Φ-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Φ-variation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011879221347
Publisher site
See Article on Publisher Site

Abstract

We study (set-valued) mappings of bounded Φ-variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the Φ-variation of a metric space valued mapping. We show that the linear span GV Φ(I;X) of the set of all mappings of bounded Φ-variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X → Y is a given mapping and the composition operator ℋ is defined by (ℋf)(t)=h(t,f(t)), where t∈I and f:I → X, we show that ℋ:GV Φ(I;X)→ GV Ψ(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, t∈I, x∈X. This result is further extended to multivalued composition operators ℋ with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded Φ-variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded Φ-variation.

Journal

PositivitySpringer Journals

Published: Oct 3, 2004

References

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