Further Study on R-Sets Operator in Acyclic Fashion (European Journal of Pure and Applied Mathematics, 2025)
We provide a different and detalied proof of the weak convergence of the acyclic Douglas–Rachford iteration to a point whose nearest-point projections onto each of $N$ convex sets coincide. The analysis shows that the cyclic Douglas–Rachford operator is asymptotically regular, its fixed point set matches the intersection of the individual fixed point sets when nonempty, and the iteration converges weakly to such a point. Special cases reveal when the method coincides with alternating projections and when it diverges from von Neumann’s scheme.
Keywords: Nonexpansive mapping; Acyclic Douglas–Rachford method; cyclic Douglas–Rachford operator; fixed point theory; convex analysis; projection algorithms; alternating projections; weak convergence; asymptotic regularity.