A minimum condition and some related fixed-point theorems

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