Abstract: In this paper, based on equivalent formulations of various types of Ekeland's variational principle, we consider the equivalence on vectorial Ekeland's variational principle for monotonically semicontinuous mappings with perturbations given by a convex bounded subset of directions multiplied by the distance function in locally convex spaces. Firstly, by using a vectorial Ekeland's variational principle in locally convex spaces, we present a simple proof of vectorial Caristi-Kirk's fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and vectorial Oettli-Th\'{e}ra's theorem. Furthermore, we study the equivalence among the vectorial Ekeland's variational principl, the vectorial Caristi-Kirk's fixed-point theorem, the vectorial Takahashi's nonconvex minimization theorem and the vectorial Oettli-Th\'{e}ra's theorem.

Key words: vectorial Ekeland's variational principle, vectorial Caristi-Kirk's fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem, vectorial Oettli-Th\'{e}ra's theorem, equivalence