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

Serdica Mathematical Journal (1996)

- Volume: 22, Issue: 1, page 57-68
- ISSN: 1310-6600

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topCorvellec, 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 -

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