Vectorial variational principle with variable set-valued perturbation

We give a general vectorial Ekeland.s variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland.s variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the objective function values. The general version includes and improves a number of known versions of vectorial Ekeland.s variational principle. From the general Ekeland.s principle, we deduce the corresponding versions of Caristi-Kirk.s fixed point theorem and Takahashi.s nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.