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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.

Information

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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