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

Huang, Changchi; Peng, Jigen; Qin, Liqian; Tang, Yuchao

doi:10.3390/math14020245

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

by

Changchi Huang

,

Jigen Peng

,

Liqian Qin

and

Yuchao Tang

^*

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(2), 245; https://doi.org/10.3390/math14020245

Submission received: 26 November 2025 / Revised: 24 December 2025 / Accepted: 5 January 2026 / Published: 8 January 2026

(This article belongs to the Special Issue Nonlinear Functional Analysis: Theory, Methods, and Applications)

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Abstract

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.

Keywords: maximal monotone operator; strong monotone operator; Lipschitz continuous operator; reflected–forward–backward splitting method; Banach space

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MDPI and ACS Style

Huang, C.; Peng, J.; Qin, L.; Tang, Y. A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces. Mathematics 2026, 14, 245. https://doi.org/10.3390/math14020245

AMA Style

Huang C, Peng J, Qin L, Tang Y. A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces. Mathematics. 2026; 14(2):245. https://doi.org/10.3390/math14020245

Chicago/Turabian Style

Huang, Changchi, Jigen Peng, Liqian Qin, and Yuchao Tang. 2026. "A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces" Mathematics 14, no. 2: 245. https://doi.org/10.3390/math14020245

APA Style

Huang, C., Peng, J., Qin, L., & Tang, Y. (2026). A Reflected–Forward–Backward Splitting Method for Monotone Inclusions Involving Lipschitz Operators in Banach Spaces. Mathematics, 14(2), 245. https://doi.org/10.3390/math14020245

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

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