Accelerated double step-length method for solving monotone nonlinear equations with convex-constraint and application

Muhammad Abdullahi; Abubakar Sani Halilu; Mohammed A. Saleh; Abdulgader Z. Almaymuni; Seyed Yaser Mousavi Siamakani; Tiamiyu Abd'gafar Tunde; Sulaiman Mohammed Ibrahim; Muhammad Abdullahi; Abubakar Sani Halilu; Mohammed A. Saleh; Abdulgader Z. Almaymuni; Seyed Yaser Mousavi Siamakani; Tiamiyu Abd'gafar Tunde; Sulaiman Mohammed Ibrahim

doi:10.3934/math.2026448

AIMS Mathematics

2026, Volume 11, Issue 4: 10908-10935. doi: 10.3934/math.2026448

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Research article

Accelerated double step-length method for solving monotone nonlinear equations with convex-constraint and application

1.
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China
2.
Department of Mathematics, Sule Lamido University Kafin Hausa, Nigeria
3.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Kuala Terengganu 21300, Malaysia
4.
Department of Cybersecurity, College of Computer, Qassim University, Saudi Arabia
5.
Department of Civil Engineering, College of Engineering, Rangsit University, Mueang, Pathum Thani 12000, Thailand
6.
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China
7.
College of Applied and Health Sciences, A'Sharqiyah University, Ibra 400, Sultanate of Oman

Received: 14 December 2025 Revised: 16 March 2026 Accepted: 31 March 2026 Published: 20 April 2026
MSC : 65K05, 90C52, 90C53, 90C56

The hybrid procedure is an efficient technique for improving the global and numerical performance of iterative algorithms for large-scale monotone nonlinear problems. This is achieved by integrating two or more methods into a unified framework. In this study, we present a hybrid of the double step-length method and the Picard-Mann iterative technique for solving monotone nonlinear equations with convex constraints. By combining the Picard-Mann approach with a newly proposed scheme, we obtain an iterative method that reduces computational cost and achieves faster convergence. The acceleration parameter is determined by evaluating the difference between the Broyden update and its approximation using the Frobenius norm. The global convergence of the proposed method is proved, and a Q-linear convergence rate is also established. Numerical experiments demonstrate that the proposed approach is computationally efficient for solving large-scale nonlinear equations compared with existing methods. Finally, the method is applied to signal processing and image restoration problems, highlighting its practical relevance.
- Frobenius norm,
- Picard-Mann,
- optimization method,
- signal reconstruction,
- double step length
Citation: Muhammad Abdullahi, Abubakar Sani Halilu, Mohammed A. Saleh, Abdulgader Z. Almaymuni, Seyed Yaser Mousavi Siamakani, Tiamiyu Abd'gafar Tunde, Sulaiman Mohammed Ibrahim. Accelerated double step-length method for solving monotone nonlinear equations with convex-constraint and application[J]. AIMS Mathematics, 2026, 11(4): 10908-10935. doi: 10.3934/math.2026448

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Abstract

The hybrid procedure is an efficient technique for improving the global and numerical performance of iterative algorithms for large-scale monotone nonlinear problems. This is achieved by integrating two or more methods into a unified framework. In this study, we present a hybrid of the double step-length method and the Picard-Mann iterative technique for solving monotone nonlinear equations with convex constraints. By combining the Picard-Mann approach with a newly proposed scheme, we obtain an iterative method that reduces computational cost and achieves faster convergence. The acceleration parameter is determined by evaluating the difference between the Broyden update and its approximation using the Frobenius norm. The global convergence of the proposed method is proved, and a Q-linear convergence rate is also established. Numerical experiments demonstrate that the proposed approach is computationally efficient for solving large-scale nonlinear equations compared with existing methods. Finally, the method is applied to signal processing and image restoration problems, highlighting its practical relevance.

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