Fixed points and stable balls in ultrametric spaces (Q1412981)

Let \(X\) be a spherically complete ultrametric space; the metric is allowed to take values in a partially ordered set with a smallest element. Let \(f\) be a nonexpansive (contracting in the terminology of the author) map of \(X\) into itself. Then \(f\) has a fixed point or a stable ball, that is a closed ball \(B\) such that \(d(x, f(x))\) is equal to the radius of \(B\) for every \(x\) in \(B\). The result is illustrated with three examples.