The adults of Trichosurolaelaps dixous Domrow, 1972 are redescribed from a population of Trichosurus cunninghami Lindenmayer, Dubach et Viggers, 2002 in south-eastern Australia. The nymphal stages are described for the first time. Morphologically, T. dixous is similar to Trichosurolaelaps crassipes Womersley, 1956. Morphological differences between the pre-female deutonymphs and adult females of the two mite species are the presence of a single large ventral spur on tibia I of T. dixous. Males of T. dixous could not be distinguished from T. crassipes morphologically and the idiosomal length of male T. dixous was variable (475-683 μm). Protonymphs of the two mite species differed only in size, with that of T. dixous being larger. Although T. crassipes was prevalent in a sympatric population of Trichosurus vulpecula and has been reported from other populations of T. cunninghami in southern Australia, it was never recovered from the population of T. cunninghami studied.
We consider the functional equation f(xf(x)) = ϕ(f(x)) where ϕ: J → J is a given homeomorphism of an open interval J ⊂ (0, ∞) and f : (0, ∞) → J is an unknown continuous function. A characterization of the class S(J,ϕ) of continuous solutions f is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when ϕ is increasing. In the present paper we solve the converse problem, for which continuous maps f : (0, ∞) → J, where J is an interval, there is an increasing homeomorphism ϕ of J such that f ∈ S(J,ϕ). We also show why the similar problem for decreasing ϕ is difficult.