A remark on the lower semicontinuity assumption in the Ekeland variational principle. (2nd October 2016)

• Record Type: Journal Article
• Authors: Ha, Truong Xuan Duc
• Is Part Of: Optimization. Volume 65:Number 10(2016)
• Page Start: 1781
• Page End: 1789
• Publication Date: 2016-10-02
• Subjects: Ekeland variational principle -- lower semicontinuity -- Gâteaux differentiability; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1195830
• Languages: English
• ISSNs: 0233-1934

Abstract

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.