Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 970579, 20 pages
doi:10.1155/2010/970579

Equivalent extensions to Caristi-Kirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem

Abstract

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.