  Many iterative optimization algorithms involve compositions of special cases
of Lipschitz continuous operators, namely firmly nonexpansive, averaged and
nonexpansive operators. The structure and properties of the compositions are of
particular importance in the proofs of convergence of such algorithms. In this
paper, we systematically study the compositions of further special cases of
Lipschitz continuous operators. Applications of our results include
compositions of scaled conically nonexpansive mappings, as well as the
Douglas--Rachford and forward-backward operators, when applied to solve certain
structured monotone inclusion and optimization problems. Several examples
illustrate and tighten our conclusions.
