# American Institute of Mathematical Sciences

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

Received  July 2021 Revised  March 2022 Early access April 2022

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.

Citation: Habib ur Rehman, Poom Kumam, Yusuf I. Suleiman, Widaya Kumam. An adaptive block iterative process for a class of multiple sets split variational inequality problems and common fixed point problems in Hilbert spaces. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022007
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Numerical illustration of Algorithm 3.1 with S(N; M) = $(5; 5)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with S(N; M) = $(5; 5)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with S(N; M) = $(5; 10)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with S(N; M) = $(5; 10)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with S(N; M) = $(10; 10)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with S(N; M) = $(10; 10)$ and TOL = $10^{-5}$
Numerical illustration of Algorithm 3.1 with different values of $\kappa_{n} = \frac{1}{k(n+2)}$
Numerical illustration of Algorithm 3.1 with different values of $\kappa_{n} = \frac{1}{k(n+2)}$
Numerical illustration of Algorithm 3.1 with different values of $\rho_{n} = k$
Numerical illustration of Algorithm 3.1 with different values of $\rho_{n} = k$
Numerical illustration of Algorithm 3.1 with $x_{1} = (1, 3)$ and TOL = $10^{-4}$
Numerical illustration of Algorithm 3.1 with $x_{1} = (1, 3)$ and TOL = $10^{-4}$
Numerical illustration of Algorithm 3.1 with $x_{1} = (3, 4)$ and TOL = $10^{-4}$
Numerical illustration of Algorithm 3.1 with $x_{1} = (3, 4)$ and TOL = $10^{-4}$
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
 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
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
 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
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
 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
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
 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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