  The correspondence between the class of nonexpansive mappings and the class
of maximally monotone operators via the reflected resolvents of the latter has
played an instrumental role in the convergence analysis of the splitting
methods. Indeed, the performance of some of these methods, e.g.,
Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the
so-called splitting operator associated with the individual operators. This
splitting operator is a function of the composition of the reflected resolvents
of the underlying operators. In this paper, we provide a comprehensive study of
the class of uniformly monotone operators and their corresponding reflected
resolvents. We show that the latter is closely related to the class of the
strongly nonexpansive operators introduced by Bruck and Reich. Connections to
duality via inverse operators are systematically studied. We provide
applications to Douglas-Rachford and Peaceman-Rachford methods. Examples that
illustrate and tighten our results are presented.
