From Caristi’s Theorem to Ekeland’s Variational Principle in0σ-Complete Metric-Like Spaces

We discuss the extension of some fundamental results in nonlinear analysis to the setting of0σ-complete metric-like spaces. Then, we show that these extensions can be obtained via the corresponding results in standard metric spaces.

Transfinite methods in metric fixed-point theory

This is a brief survey of the use of transfinite induction in metric fixed-point theory. Among the results discussed in some detail is the author's 1989 result on directionally nonexpansive mappings (which is somewhat sharpened), a result of Kulesza and Lim giving conditions when countable compactness implies compactness, a recent inwardness result for contractions due to Lim, and a recent extension of Caristi's theorem due to Saliga and the author. In each instance, transfinite methods seem necessary.

A minimum condition and some realted fixed-point theorems

The classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weakended hypothesis may be applied in Caristi's theorem, and how combinatorial arguments may be used in proving fixed-point theorems.