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Subdifferential Rolle's and mean value inequality theorems

Published online by Cambridge University Press:  17 April 2009

D. Azagra  and
R. Deville
D. Azagra
Affiliation:
Dpto. de Análisis MatemáticoFacultad de MatemáticasUniversidad Complutense de Madrid28040 MadridSpain e-mail: daniel@sunam1.mat.ucm.es
R. Deville
Affiliation:
Département de Mathématiques Pures et AppliquéesUniversitá: Bordeaux I351, cours de la Libération33405 Talence CedexFrance e-mail: deville@math.u-bordeaux.fr
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Abstract

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In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdiferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle's theorem satating that if a subdifferentiable function oscilllates between −ɛ and ɛ on the boundary of the unit ball then there exists a subgradient of the function at an interior point of the ball which has norm less than or equal to 2ɛ.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 319 - 329
Copyright
Copyright © Australian Mathematical Society 1997

References

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