The strong Ekeland variational principle in quasi-pseudometric spaces

Abstract

Roughly speaking, Ekeland's Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324--353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. \textbf{131} (1988), no.~1, 1--21) and Tomonari Suzuki (J. Math. Anal. Appl. \textbf{320} (2006), no.~2, 787--794 and Nonlinear Anal. \textbf{72} (2010), no.~5, 2204--2209)), proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key-role played by reflexivity. In the last years an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, that is, to quasi-metric spaces (see the references). Applications to optimization, behavioral sciences, and others, were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev and Suzuki versions, to the quasi-pseudometric case. At the end we ask for the possibility to extend it to asymmetric normed spaces (i.e., the extension of Suzuki's results).