Slope Variational Principle Coercivity Weak Slope Nonsmooth Critical Point Theory
Issue Date:
1996
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 22, No 1, (1996), 57p-68p
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.