We discuss the Earle-Hamilton fixed-point theorem and show how it can be applied when restrictions are known on the numerical range of a holomorphic function. In particular, we extend the Earle-Hamilton theorem to holomorphic functions with numerical range having real part strictly less than 1. We also extend the Lumer-Phillips theorem estimating resolvents to dissipative holomorphic functions.

We discuss a family of holomorphic self-maps of $\mathbb{C}^3$ tangent to the identity at the origin presenting dynamical phenomena not appearing for lower-dimensional maps.

We apply some continuous selection theorems to establish coincidence theorems for a family of multimaps under various conditions. Then we apply these coincidence theorems to study the equilibrium problem with $m$ families of players and $2m$ families of constraints on strategy sets. We establish the existence theorems of equilibria of this problem and existence theorem of equilibria of abstract economics with two families of players.

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.