On the structure of fixed-point sets of asymptotically regular semigroups

We extend a few recent results of G'{o}rnicki (2011) asserting that the set of fixed points of an asymptotically regular mapping is a retract of its domain. In particular, we prove that in some cases the resulting retraction is H"{o}lder continuous. We also characterise Bynum’s coefficients and the Opial modulus in terms of nets.