An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in hilbert spaces

Habib ur Rehman; Poom Kumam; Yusuf I. Suleiman; Wiyada Kumam

doi:10.3934/naco.2022007

Article Contents

2023, Volume 13, Issue 2: 273-298. Doi: 10.3934/naco.2022007

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An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in hilbert spaces

1.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Department of Mathematics, Kano University of Science and Technology, Wudil 713101, Nigeria
3.
Applied Mathematics for Science and Engineering Research, Unit (AMSERU) Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani 12110, Thailand

^* Corresponding author
^* Corresponding author

Received: July 2021

Revised: March 2022

Early access: April 2022

Published: January 2023

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Abstract

In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.

Note: The name of the fourth author has been corrected from Widaya Kumam to Wiyada Kumam.

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Mathematics Subject Classification: Primary: 47H05, 47H09, 49M37, 65K10.

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Figure 1. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $

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Figure 2. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 5) $ and TOL = $ 10^{-5} $

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Figure 3. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $

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Figure 4. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (5; 10) $ and TOL = $ 10^{-5} $

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Figure 5. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $

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Figure 6. Numerical illustration of Algorithm 3.1 with S(N; M) = $ (10; 10) $ and TOL = $ 10^{-5} $

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Figure 7. Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $

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Figure 8. Numerical illustration of Algorithm 3.1 with different values of $ \kappa_{n} = \frac{1}{k(n+2)} $

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Figure 9. Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $

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Figure 10. Numerical illustration of Algorithm 3.1 with different values of $ \rho_{n} = k $

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Figure 11. Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $

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Figure 12. Numerical illustration of Algorithm 3.1 with $ x_{1} = (1, 3) $ and TOL = $ 10^{-4} $

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Figure 13. Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $

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Figure 14. Numerical illustration of Algorithm 3.1 with $ x_{1} = (3, 4) $ and TOL = $ 10^{-4} $

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Table 1. Numerical data for Experiment 1

N.P	S(N; M)	TOL.	CPU(s)	ITER.
$ 1 $	(5; 5)	$ 10^{-5} $	4.66201370000000	68
$ 2 $	(5; 10)	$ 10^{-5} $	8.59728030000000	76
$ 3 $	(10; 10)	$ 10^{-5} $	9.57778720000000	92
$ 4 $	(10; 20)	$ 10^{-5} $	13.4638439200000	108
$ 5 $	(20; 30)	$ 10^{-5} $	20.5758686930000	123

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Table 2. Numerical data for Experiment 2

k	S(N; M)	TOL.	CPU(s)	ITER.
$ 1 $	(5; 10)	$ 10^{-5} $	6.29621610000000	80
$ 3 $	(5; 10)	$ 10^{-5} $	7.10547680000000	92
$ 10 $	(5; 10)	$ 10^{-5} $	10.2573273000000	113
$ 20 $	(5; 10)	$ 10^{-5} $	18.1375856000000	155
$ 50 $	(5; 10)	$ 10^{-5} $	30.5698815000000	226

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Table 3. Numerical data for Experiment 3

k	S(N; M)	TOL.	CPU(s)	ITER.
$ 0.15 $	(5; 10)	$ 10^{-5} $	18.6105856000000	208
$ 0.65 $	(5; 10)	$ 10^{-5} $	16.1703019000000	167
$ 1.15 $	(5; 10)	$ 10^{-5} $	11.3304408000000	128
$ 1.64 $	(5; 10)	$ 10^{-5} $	9.50234320000000	106
$ 1.89 $	(5; 10)	$ 10^{-5} $	6.33439070000000	67

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Table 4. Numerical data for Example 4.2

SAlgorithm		Algorithm 3.1
$ x_{1} $	CPU(s)	ITER.	CPU(s)	ITER.
$ (1, 3) $	10.7807425000000	577	3.90895790000000	282
$ (3, 4) $	40.4251408000000	1926	8.90772100000000	507

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