Further Study on R-Sets Operator in Acyclic Fashion

Authors

  • Salihah Alwadani Yanbu Industrial College

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i4.6837

Keywords:

Nonexpansive mapping; Acyclic Douglas–Rachford method; cyclic Douglas–Rachford operator; fixed point theory; convex analysis; projection algorithms; alternating projections; weak convergence; asymptotic regularity.

Abstract

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.

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Published

2025-11-05

Issue

Vol. 18 No. 4: (October 2025)

Section

Topology

License

Copyright (c) 2025 Salihah Alwadani

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How to Cite

Further Study on R-Sets Operator in Acyclic Fashion. (2025). European Journal of Pure and Applied Mathematics, 18(4), 6837. https://doi.org/10.29020/nybg.ejpam.v18i4.6837