  We show that the set of fixed points of the average of two resolvents can be
found from the set of fixed points for compositions of two resolvents
associated with scaled monotone operators. Recently, the proximal average has
attracted considerable attention in convex analysis. Our results imply that the
minimizers of proximal-average functions can be found from the set of fixed
points for compositions of two proximal mappings associated with scaled convex
functions. When both convex functions in the proximal average are indicator
functions of convex sets, least squares solutions can be completely recovered
from the limiting cycles given by compositions of two projection mappings. This
provides a partial answer to a question posed by C. Byrne. A novelty of our
approach is to use the notion of resolvent average and proximal average.
