A stationary point theorem characterizing metric completeness
We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.
Some generalizations of Ekeland's variational principle with applications to fixed point theory
In this paper, some extensions of the Ekeland variational principle in metric spaces are given for a generalized pseudodistance. As an application we obtain Caristi's fixed point theorem. Then, by using this result, we establish some fixed point theorems for set-valued contractive mappings.
Asymptotic fixed points for nonlinear contractions
Recently, W. A. Kirk proved an asymptotic fixed point theorem for nonlinear contractions by using ultrafilter methods. In this paper, we prove his theorem under weaker assumptions. Furthermore, our proof does not use ultrafilter methods.