Equivalent Extensions to Caristi-Kirk’s Fixed Point Theorem, Ekeland’s Variational Principle, and Takahashi’s Minimization Theorem

With a recent result of Suzuki 2001 we extend Caristi-Kirk’s fixed point theorem, Ekeland’s variational principle, and Takahashi’s minimization theorem in a complete metric space by replacing the distance with a τ-distance. In addition, these extensions are shown to be equivalent. When the τ-distance is l.s.c. in its second variable, they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds, which are in turn useful for extending some existing results such as the petal theorem.