On the Ekeland–Ghoussoub–Preiss and Stuart criteria for locating Cerami sequences

Abstract

A classical result of Ekeland, based on an idea of Ghoussoub and Preiss, asserts that under technical conditions, Cerami sequences for a real valued functional ${\Phi}$ on a Banach space X can be found in the vicinity of suitable closed subsets W of X. In this statement, “vicinity” is defined by means of an associated geodesic distance on X. Recently, under virtually the same hypotheses, Stuart discovered a similar property with a much clearer content since it can very simply be expressed in terms of the norm of X and the corresponding standard distance. In this paper, we prove that the Ekeland–Ghoussoub–Preiss and Stuart criteria are in fact equivalent. We also show that this equivalence need not be true when Cerami sequences are replaced by more general, yet admissible sequences, but that the equivalence is preserved, in part or in totality, under simple additional conditions. These results are also applicable to more general linking geometries than considered by Ekeland–Ghoussoub–Preiss or Stuart and to nonsmooth functionals.