  This paper introduces the concept of quasi $\alpha$-firmly nonexpansive
mappings in Wasserstein spaces over $\mathbb R^d$ and analyzes properties of
these mappings. We prove that for quasi $\alpha$-firmly nonexpansive mappings
satisfying a certain quadratic growth condition, the fixed point iterations
converge in the narrow topology. As a byproduct, we will get the known
convergence of the proximal point algorithm in Wasserstein spaces. We apply our
results to show for the first time that cyclic proximal point algorithms for
minimizing the sum of certain functionals on Wasserstein spaces converge under
appropriate assumptions.
