A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory

Corvellec, J.. "A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory." Serdica Mathematical Journal 22.1 (1996): 57-68. <http://eudml.org/doc/11630>.

@article{Corvellec1996,
abstract = {The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.},
author = {Corvellec, J.},
journal = {Serdica Mathematical Journal},
keywords = {Slope; Variational Principle; Coercivity; Weak Slope; Nonsmooth Critical Point Theory; variational principle; nonsmooth critical point theory; coercivity; slope},
language = {eng},
number = {1},
pages = {57-68},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory},
url = {http://eudml.org/doc/11630},
volume = {22},
year = {1996},
}

TY - JOUR
AU - Corvellec, J.
TI - A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory
JO - Serdica Mathematical Journal
PY - 1996
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 22
IS - 1
SP - 57
EP - 68
AB - The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.
LA - eng
KW - Slope; Variational Principle; Coercivity; Weak Slope; Nonsmooth Critical Point Theory; variational principle; nonsmooth critical point theory; coercivity; slope
UR - http://eudml.org/doc/11630
ER -