A remark on the lower semicontinuity assumption in the Ekeland variational principle

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.

Keywords:

• Ekeland variational principle
• lower semicontinuity
• Gâteaux differentiability

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.

Additional information

Funding

This work was partly carried out at the Vietnam Institute for Advanced Study of Mathematics and supported in part by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01-2014.27].