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Volume 14
Issue 11
10.3390/axioms14110783
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Article

Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators

by
Ali Berrailes
1,2,* and
Abdallah Beddani
1,3
1
Department of Mathematics, University of Relizane, Bourmadia 48000, Algeria
2
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran 31000, Algeria
3
Laboratory of Analysis, Geometry and Its Applications (LAGA), Ahmed Zabana University, Relizane 48000, Algeria
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(11), 783; https://doi.org/10.3390/axioms14110783 (registering DOI)
Submission received: 2 September 2025 / Revised: 22 October 2025 / Accepted: 23 October 2025 / Published: 25 October 2025
(This article belongs to the Special Issue Modern Problems of Analysis, Optimization, Approximation and Their Applications)
Versions Notes

Abstract

In this paper, we address a variational problem involving the sum of two maximal monotone operators combined with a finite family of nonexpansive operators. To solve this problem, we propose iterative algorithms based on single-valued mappings. First, we examine cases involving two or three maximal monotone operators, introducing novel algorithms to obtain their solutions. Secondly, we extend our analysis by applying the Ishikawa iterative scheme within the framework of fixed-point theory. This allows us to establish strong convergence results. Finally, we provide an illustrative example to demonstrate the effectiveness and applicability of the proposed methods.
Keywords: maximal monotone operator; α-strongly monotone; Yosida approximation; fixed point maximal monotone operator; α-strongly monotone; Yosida approximation; fixed point

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MDPI and ACS Style

Berrailes, A.; Beddani, A. Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators. Axioms 2025, 14, 783. https://doi.org/10.3390/axioms14110783

AMA Style

Berrailes A, Beddani A. Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators. Axioms. 2025; 14(11):783. https://doi.org/10.3390/axioms14110783

Chicago/Turabian Style

Berrailes, Ali, and Abdallah Beddani. 2025. "Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators" Axioms 14, no. 11: 783. https://doi.org/10.3390/axioms14110783

APA Style

Berrailes, A., & Beddani, A. (2025). Algorithms for Solving the Resolvent of the Sum of Two Maximal Monotone Operators with a Finite Family of Nonexpansive Operators. Axioms, 14(11), 783. https://doi.org/10.3390/axioms14110783

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