Hybrid inertial viscosity-type forward-backward splitting algorithms for variational inclusion problems

Doaa Filali; Mohammad Dilshad; Mohammad Akram; Mohd. Falahat Khan; Syed Shakaib Irfan; Doaa Filali; Mohammad Dilshad; Mohammad Akram; Mohd. Falahat Khan; Syed Shakaib Irfan

doi:10.3934/math.20251269

AIMS Mathematics

2025, Volume 10, Issue 12: 28829-28860. doi: 10.3934/math.20251269

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Hybrid inertial viscosity-type forward-backward splitting algorithms for variational inclusion problems

1.
Department of Mathematical Science, College of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk-71491, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 170, KSA
4.
Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, U.P., India

Received: 18 September 2025 Revised: 10 November 2025 Accepted: 25 November 2025 Published: 09 December 2025
MSC : 49J53, 49K99

Herein, we aimed to develop two hybrid inertial viscosity-type forward-backward splitting algorithms for estimating the common solution of the variational inclusion problem and fixed point problem in Hilbert spaces. At the beginning of each iteration, the first algorithm estimated viscosity, fixed point, and inertial extrapolation. On the other hand, the second method estimated viscosity and inertial extrapolation alone. We demonstrated that the sequence induced by the proposed hybrid algorithms has strong convergence. We discussed a few special cases of the proposed algorithms and also presented theoretical applications of our findings. We furnished suitable numerical examples to validate the effectiveness of the recommended approaches.
- Hilbert space,
- inertial,
- viscosity approximation,
- variational inclusion,
- fixed point,
- convergence
Citation: Doaa Filali, Mohammad Dilshad, Mohammad Akram, Mohd. Falahat Khan, Syed Shakaib Irfan. Hybrid inertial viscosity-type forward-backward splitting algorithms for variational inclusion problems[J]. AIMS Mathematics, 2025, 10(12): 28829-28860. doi: 10.3934/math.20251269

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Abstract

Herein, we aimed to develop two hybrid inertial viscosity-type forward-backward splitting algorithms for estimating the common solution of the variational inclusion problem and fixed point problem in Hilbert spaces. At the beginning of each iteration, the first algorithm estimated viscosity, fixed point, and inertial extrapolation. On the other hand, the second method estimated viscosity and inertial extrapolation alone. We demonstrated that the sequence induced by the proposed hybrid algorithms has strong convergence. We discussed a few special cases of the proposed algorithms and also presented theoretical applications of our findings. We furnished suitable numerical examples to validate the effectiveness of the recommended approaches.

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