On some generalizations of Ekeland’s principle and inward contractions in gauge spaces

Abstract

We give generalizations in complete gauge spaces of the following results: Bishop-Phelps’ theorem, Ekeland’s variational principle, Caristi’s fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent. We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete gauge space and satisfying a generalized inwardness condition.