Set-valued Ekeland variational principles in fuzzy metric spaces

In this paper, we establish a general set-valued Ekelandʹs variational principle in fuzzy metric spaces, where the objective function is a set-valued map defined on a fuzzy metric space and taking values in a pre-ordered locally convex space, and the perturbation involves a quasi-metric family generating the fuzzy topology of the domain space. Moreover, the direction of the perturbation is a convex subset of the positive cone instead of a single positive vector. In our general version, the assumption that the objective function is lower semi-continuous and one that the range of the function is lower bounded are both weakened. From the general Ekelandʹs variational principle, we obtain several particular set-valued Ekelandʹs variational principles in fuzzy metric spaces, which generalize and improve some related known results. From these, we deduce the corresponding Caristiʹs fixed point theorems for set-valued maps and the corresponding Takahashiʹs non-convex minimization theorems in set-valued optimization. Finally, we extend the obtained results to F-type topological spaces.