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2023, Volume 19, Issue 9: 6781-6805. Doi: 10.3934/jimo.2022239

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An inertial subgradient extragradient method with Armijo type step size for pseudomonotone variational inequalities with non-Lipschitz operators in Banach spaces

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

^*Corresponding author: Oluwatosin Temitope Mewomo
^*Corresponding author: Oluwatosin Temitope Mewomo

Received: June 2022

Revised: October 2022

Early access: December 2022

Published: September 2023

The first author is supported by the University of KwaZulu-Natal Doctoral Scholarship. The second author is funded by the University of KwaZulu-Natal Postdoctoral Fellowship. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903)

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Abstract

In this paper, we study the pseudomonotone variational inequality problems with non-Lipschitz operators. We propose an inertial subgradient extragradient method with Halpern technique and Armijo type step size for approximating the solution of the problem in the framework of 2-uniformly convex real Banach spaces. We prove that the sequence generated by our proposed method converges strongly to the solution of the problem under some mild conditions and without the weak sequential continuity condition often assumed by authors in solving pseudomonotone variational inequality problems. Finally, we provide some numerical experiments for the proposed method in comparison with other existing methods in the literature. Our result extends and improves several of the existing results in the current literature in this direction.

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Mathematics Subject Classification: 47H09, 47H10, 49J20, 49J40.

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Figure 1. Top left: m = 15; Top right: m = 30; Bottom left: m = 45; Bottom right: m = 60

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Figure 2. Top left: Case Ⅰ; Top right: Case II; Bottom left: Case Ⅲ; Bottom right: Case Ⅳ

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Table 1. Numerical results for Example 5.1

Cases		Alg. 1.5	Alg. 1.7	App.7.1	App. 7.2	Alg. 3.2
m = 15	No. of Iter.	19	31	24	33	14
	CPU time (sec)	8.2375	5.3746	3.7576	5.3048	5.8052
m = 30	No. of Iter.	23	33	26	44	18
	CPU time (sec)	11.6430	6.3741	4.7479	7.8673	6.9292
m = 45	No. of Iter.	25	40	29	46	20
	CPU time (sec)	9.6496	8.3209	5.6280	9.5082	6.3000
m = 60	No. of Iter.	26	43	31	51	24
	CPU time (sec)	9.5374	10.1644	6.5321	11.4672	6.5011

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Table 2. Numerical results for Example 5.2

		Alg. 1.5	Alg. 1.7	App.7.1	App. 7.2	Alg. 3.2
Case Ⅰ	No. of Iter.	30	20	39	26	9
	CPU time (sec)	0.0169	0.0104	0.0102	0.0102	0.0107
Case Ⅱ	No. of Iter.	19	18	26	33	8
	CPU time (sec)	0.0164	0.0099	0.0091	0.0096	0.0111
Case Ⅲ	No. of Iter.	16	19	33	26	9
	CPU time (sec)	0.0181	0.0106	0.0124	0.0132	0.0144
Case Ⅳ	No. of Iter.	35	20	39	32	9
	CPU time (sec)	0.0171	0.0093	0.0094	0.0098	0.0104

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