Abstract
What happens to the conclusion of the Ekeland variational principle (briefly, EVP) if a considered function is lower semicontinuous not on the whole metric space X but only on its domain? We provide a straightforward proof showing that it still holds but only for
varying in some interval
, where
is a quantity expressing quantitatively the violation in the lower semicontinuity of f outside its domain. The obtained result extends EVP to a larger class of functions. As applications, we obtain some results about properties of Gâteaux differentiable functions on Banach spaces.
Acknowledgements
The author would like to thank the two anonymous referees for useful suggestions.
Notes
1 This has been recently corrected, see https://www.carma.newcastle.edu.au/jon/ToVA/errata.pdf, p.7.