Converses
Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959:
Let be a map of an abstract set such that each iterate ƒn has a unique fixed point. Let q be a real number, 0 < q < 1. Then there exists a complete metric on X such that ƒ is contractive, and q is the contraction constant.
Indeed, very weak assumptions suffice to obtain such a kind of converse. E.g. if is a map on a T1 topological space (i.e., a T1 space) with a unique fixed point a, such that for each we have that ƒn(x) converges to a, then there already exists a metric on X with respect to which f satisfies the conditions of the Banach contraction principle with contraction constant . In this case the metric is in fact an ultrametric.
Read more about this topic: Banach Fixed-point Theorem
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