Set-valued pseudo-metric families and Ekeland's variational principles in fuzzy metric spaces

In this paper, we introduce a set-valued pseudo-metric family on a fuzzy metric space and the notion of compatibility between the set-valued pseudo-metric family and the original fuzzy metric. By means of this notion, we prove a general set-valued EVP, where the perturbation involves a set-valued pseudo-metric family compatible with the original fuzzy metric. From the general EVP, we deduce several particular EVPs, which extend the EVPs in Qiu (2013) [36] and in Gutiérrez et al. (2008) [20] to fuzzy metric spaces. By using set-valued pseudo-metric families and using the unified approach for approximate solutions introduced by Gutiérrez, Jiménez and Novo, we deduce a general version of set-valued EVP based on (C,ϵ)(C,ϵ)-efficient solutions in fuzzy metric spaces, where C is a coradiant set contained in the order cone. By choosing two specific versions of the coradiant set C in the general version of EVP, we obtain several particular set-valued EVPs for ϵ-efficient solutions in the sense of Németh and of Dentcheva and Helbig, respectively. These EVPs improve and generalize the related known results.