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.