Convergence of proximal splitting algorithms in $\operatorname{CAT}(\kappa)$ spaces and beyond

Abstract In the setting of $\operatorname{CAT}(\kappa)$ CAT(?) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under assumption linear metric subregularity. Linear subregularity is in any case necessary for convergent sequences, so result tight. To show this, we develop a theory that violate usual assumptions nonexpansiveness and firm p -uniformly convex spaces.