# American Institute of Mathematical Sciences

September  2022, 18(5): 3735-3748. doi: 10.3934/jimo.2021133

## Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

 1 Institute of Theoretical and Applied Research, Hanoi, 100000 2 Faculty of Information Technology, Duy Tan University, Da Nang, 550000, Vietnam 3 Vietnam Academy of Science and Technology, Institute of Information Technology, 18, Hoang Quoc Viet, Hanoi, Vietnam

Received  December 2020 Revised  April 2021 Published  September 2022 Early access  September 2021

Fund Project: The author is supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.02-2017.305

In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.

Citation: Nguyen Buong. Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3735-3748. doi: 10.3934/jimo.2021133
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##### References:
Computational results by the first method with (28)-(30)
 $k$ $x_1^{k+1}$ $x_2^{k+1}$ $x_3^{k+1}$ $x_4^{k+1}$ $x_5^{k+1}$ 10 0.0478319229 0.0597899037 0.0717478844 0.0837058651 0.1610527635 20 0.0065913104 0.0082391379 0.0098869655 0.0115347931 0.0455817835 30 0.0011746538 0.0014683173 0.0017619807 0.0020556442 0.0239721493 40 0.0002336513 0.0002920642 0.0003504770 0.0004088898 0.0167508304 50 0.0000494154 0.0000617693 0.0000741233 0.0000864770 0.0131756678
 $k$ $x_1^{k+1}$ $x_2^{k+1}$ $x_3^{k+1}$ $x_4^{k+1}$ $x_5^{k+1}$ 10 0.0478319229 0.0597899037 0.0717478844 0.0837058651 0.1610527635 20 0.0065913104 0.0082391379 0.0098869655 0.0115347931 0.0455817835 30 0.0011746538 0.0014683173 0.0017619807 0.0020556442 0.0239721493 40 0.0002336513 0.0002920642 0.0003504770 0.0004088898 0.0167508304 50 0.0000494154 0.0000617693 0.0000741233 0.0000864770 0.0131756678
Computational results by the second method with (30)-(31)
 $k$ $x_1^{k+1}$ $x_2^{k+1}$ $x_3^{k+1}$ $x_4^{k+1}$ $x_5^{k+1}$ 10 0.0162388209 0.0127985261 0.0153582313 0.0179179365 0.0874825391 20 0.0003020202 0.0003775252 0.0004530303 0.0005285353 0.0325388347 30 0.0000115214 0.0000144018 0.0000172821 0.0000201625 0.0215356201 40 0.0000004906 0.0000006132 0.0000007354 0.0000008585 0.0162614503 50 0.0000000222 0.0000000277 0.0000000333 0.0000000389 0.0130719537
 $k$ $x_1^{k+1}$ $x_2^{k+1}$ $x_3^{k+1}$ $x_4^{k+1}$ $x_5^{k+1}$ 10 0.0162388209 0.0127985261 0.0153582313 0.0179179365 0.0874825391 20 0.0003020202 0.0003775252 0.0004530303 0.0005285353 0.0325388347 30 0.0000115214 0.0000144018 0.0000172821 0.0000201625 0.0215356201 40 0.0000004906 0.0000006132 0.0000007354 0.0000008585 0.0162614503 50 0.0000000222 0.0000000277 0.0000000333 0.0000000389 0.0130719537
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