# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021160
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## An efficient iterative method for solving split variational inclusion problem with applications

 1 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand 2 Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto 840244, Nigeria 3 Departments of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

* Corresponding author: poom.kum@kmutt.ac.th

Received  October 2020 Revised  May 2021 Early access September 2021

A new strong convergence iterative method for solving a split variational inclusion problem involving a bounded linear operator and two maximally monotone mappings is proposed in this article. The study considers an iterative scheme comprised of inertial extrapolation step together with the Mann-type step. A strong convergence theorem of the iterates generated by the proposed iterative scheme is given under suitable conditions. In addition, methods for solving variational inequality problems and split convex feasibility problems are derived from the proposed method. Applications of solving Nash-equilibrium problems and image restoration problems are solved using the derived methods to demonstrate the implementation of the proposed methods. Numerical comparisons with some existing iterative methods are also presented.

Citation: Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021160
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##### References:
Comparative results for random instances.
Results of the compared algorithms with different cases of initial points.
Original test images of Monarch, Flowers and Colorchecker.
Original cropped test images of Monarch, Flowers and Colorchecker.
Degraded and restored Monarch images by the compared algorithms.
Degraded and restored cropped Monarch images by the compared algorithms.
Degraded and restored Flowers images by the compared algorithms.
Degraded and restored cropped Flowers images by the compared algorithms.
Degraded and restored Colorchecker images by the compared algorithms.
Degraded and restored cropped Colorchecker images by the compared algorithms.
The PNSR and SSIM values of the compared algorithms
 Scheme 51 Algorithm 4.4 Images SNR SSIM SNR SSIM Monarch 43.3255 0.9684 39.5788 0.9624 Flowers 40.6001 0.9116 36.7745 0.8660 Colorchecker 41.7454 0.8996 39.0134 0.9061
 Scheme 51 Algorithm 4.4 Images SNR SSIM SNR SSIM Monarch 43.3255 0.9684 39.5788 0.9624 Flowers 40.6001 0.9116 36.7745 0.8660 Colorchecker 41.7454 0.8996 39.0134 0.9061
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