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

We show that the set of fixed points of an asymptotically regular mapping acting on a convex and weakly compact subset of a Banach space is, in some cases, a Hölder continuous retract of its domain. Our results qualitatively complement the corresponding fixed point existence theorems and extend a few recent results of Górnicki [15–17]. We also characterize Bynum’s coefficients and the Opial modulus in terms of nets.