Vectorial Variational Principle with Variable Set-Valued Perturbation

摘要

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 ob jective 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.