Equivalents to Ekeland's variational principle in uniform spaces

A minimal element theorem on sequentially complete uniform spaces is presented that generalizes earlier results of [Pacific J. Math. 55(2) (1974) 335-341; Proc. Amer. Math. Soc. 108(3) (1990) 707-714]. Two more equivalent formulations of the minimal element theorem are given. It is applied to derive new versions of Ekeland-type theorems which improve results of [J. Math. Anal. Appl. 202 (1996) 398-412]. Finally, the minimal element theorem is used to obtain an Ekeland-type theorem for functions with values in a linear space without topological structure.