  In this paper we generalize the strongly convergent Krasnoselskii- Mann-type
iteration for families of nonexpansive mappings defined recently by Bo\c{t} and
Meier in Hilbert spaces to the abstract setting of W - hyperbolic spaces and we
compute effective rates of asymptotic regularity for our generalization. This
also generalizes recent results by Leu\c{s}tean and the author on the
Tikhonov-Mann iteration from single mappings to families of mappings.
