A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces

The reflected–forward–backward splitting (RFBS) method is well-established for solving monotone inclusion problems involving Lipschitz continuous operators in Hilbert spaces, where it converges weakly under mild assumptions. Extending this method to Banach spaces presents significant challenges, primarily due to the nonlinearity of the duality mapping. In this paper, we propose and analyze an RFBS algorithm in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. To the best of our knowledge, this work presents the first strong (R-linear) convergence result for the RFBS method in such Banach spaces, achieved under a newly adapted notion of strong monotonicity. Our results thus establish a foundational theoretical guarantee for RFBS in Banach spaces under strengthened monotonicity conditions, while highlighting the open problem of proving weak convergence for the general monotone case.