The Asymptotic Behaviour of Convex Combinations of Firmly Nonexpansive Mappings

We show that in the framework of CAT(0) spaces, any convex combination of two mappings which are firmly nonexpansive -- or which satisfy the more general property (P2) -- is asymptotically regular, conditional on its fixed point set being nonempty, and, in addition, also Δ-convergent to such a fixed point. These results are established by the construction and study of a convex combination metric on the Cartesian square of a CAT(0) space. We also derive a uniform rate of asymptotic regularity in the sense of proof mining. All these results are then interpreted in the special case of the mappings being projections onto closed, convex sets.