# American Institute of Mathematical Sciences

December  2022, 12(4): 815-836. doi: 10.3934/naco.2021037

## Inertial method for split null point problems with pseudomonotone variational inequality problems

 1 School of Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa 2 Mathematics Institute African University of Science and Technology, Abuja Nigeria 3 Department of Mathematics, Federal University Dutsinma, Kastina State Nigeria 4 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94 Medunsa 0204, Pretoria, South Africa

*Corresponding author: Christian Chibueze Okeke

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

This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.

Citation: Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2022, 12 (4) : 815-836. doi: 10.3934/naco.2021037
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##### References:
Example 5.1, Top Left: Case I; Top Right: Case II; Bottom: Case III
Example 5.2, Top Left: Case I; Top Right: Case II; Bottom: Case III
Computational result for Example 5.1
 Case I Case II Case III Algo. 3.3 ($\alpha =5$) No of Iter. 7 8 8 CPU time (sec) 0.034 0.0050 0.0034 Algo. 3.3 ($\alpha =3$) No of Iter. 13 14 14 CPU time (sec) 0.0109 0.0160 0.0076 NUMA No of Iter. 51 54 58 CPU time (sec) 0.0249 0.0296 0.0288
 Case I Case II Case III Algo. 3.3 ($\alpha =5$) No of Iter. 7 8 8 CPU time (sec) 0.034 0.0050 0.0034 Algo. 3.3 ($\alpha =3$) No of Iter. 13 14 14 CPU time (sec) 0.0109 0.0160 0.0076 NUMA No of Iter. 51 54 58 CPU time (sec) 0.0249 0.0296 0.0288
Computational result for Example 5.2
 Case I Case II Case III Algo. 3.3 ($\alpha =5$) No of Iter. 8 16 90 CPU time (sec) 0.0096 0.0223 0.0425 Algo. 3.3 ($\alpha =3$) No of Iter. 8 19 323 CPU time (sec) 0.0035 0.00089 0.4394 NUMA No of Iter. 8 17 69 CPU time (sec) 0.0051 0.0084 0.0257
 Case I Case II Case III Algo. 3.3 ($\alpha =5$) No of Iter. 8 16 90 CPU time (sec) 0.0096 0.0223 0.0425 Algo. 3.3 ($\alpha =3$) No of Iter. 8 19 323 CPU time (sec) 0.0035 0.00089 0.4394 NUMA No of Iter. 8 17 69 CPU time (sec) 0.0051 0.0084 0.0257

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