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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mean value inequalities in Hilbert space


Authors: F. H. Clarke and Yu. S. Ledyaev
Journal: Trans. Amer. Math. Soc. 344 (1994), 307-324
MSC: Primary 49J52; Secondary 26A24, 47H99, 47N10, 49L25
MathSciNet review: 1227093
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Abstract: We establish a new mean value theorem applicable to lower semi-continuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance, and generalized solutions to partial differential equations.

Résumé. On établit un nouveau théorème de la valeur moyenne qui s'applique aux fonctions semicontinues inférieurement sur un espace de Hilbert. On déduit plusieurs conséquences du résultat portant, par exemple, sur les fonctions monotones et sur les solutions généralisées des équations aux dérivées partielles.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1227093-8
PII: S 0002-9947(1994)1227093-8
Keywords: Mean value theorem, nonsmooth analysis, flow invariance, monotone functions, generalized solutions of partial differential equations
Article copyright: © Copyright 1994 American Mathematical Society