A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

Deville, Robert. "A Mean Value Theorem for non Differentiable Mappings in Banach Spaces." Serdica Mathematical Journal 21.1 (1995): 59-66. <http://eudml.org/doc/11657>.

@article{Deville1995,
abstract = {We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.},
author = {Deville, Robert},
journal = {Serdica Mathematical Journal},
keywords = {Mean Value Theorem; Smooth Variational Principle; Non Smooth Analysis; Viscosity Solutions; smooth variational principle; lower semicontinuous function; Banach space; Lipschitz continuous function; subgradients; regularity; viscosity supersolutions; Hamilton-Jacobi equations},
language = {eng},
number = {1},
pages = {59-66},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {A Mean Value Theorem for non Differentiable Mappings in Banach Spaces},
url = {http://eudml.org/doc/11657},
volume = {21},
year = {1995},
}

TY - JOUR
AU - Deville, Robert
TI - A Mean Value Theorem for non Differentiable Mappings in Banach Spaces
JO - Serdica Mathematical Journal
PY - 1995
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 21
IS - 1
SP - 59
EP - 66
AB - We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition. This last result slightly improves some earlier work by G. Barles and H. Ishii.
LA - eng
KW - Mean Value Theorem; Smooth Variational Principle; Non Smooth Analysis; Viscosity Solutions; smooth variational principle; lower semicontinuous function; Banach space; Lipschitz continuous function; subgradients; regularity; viscosity supersolutions; Hamilton-Jacobi equations
UR - http://eudml.org/doc/11657
ER -