Strong convergence of the composition of firmly nonexpansive mappings

In this paper, we provide a necessary and sufficient condition under which the compositions of finitely many firmly nonexpansive mappings in a p-uniformly convex space converge strongly. In particular, we obtain convergence results for the method of alternating projections in CAT( \(\kappa \) $\kappa$ ) spaces, with \(\kappa \ge 0\) $\kappa \ge 0$ . We also study the rate of convergence of the sequence generated by the method in cases where the firmly nonexpansive mappings satisfy an error bound, and the set of fixed points of these mappings is boundedly linearly regular.