A new weak convergence non-monotonic self-adaptive iterative scheme for solving equilibrium problems

Habib ur Rehman; Wiyada Kumam; Poom Kumam; Meshal Shutaywi; Habib ur Rehman; Wiyada Kumam; Poom Kumam; Meshal Shutaywi

doi:10.3934/math.2021332

AIMS Mathematics

2021, Volume 6, Issue 6: 5612-5638. doi: 10.3934/math.2021332

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Research article

A new weak convergence non-monotonic self-adaptive iterative scheme for solving equilibrium problems

1.
Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathumthani 12110, Thailand
3.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics College of Science & Arts, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia

Received: 13 November 2020 Accepted: 21 February 2021 Published: 23 March 2021
MSC : 47J25, 47H09, 47H06, 47J05

A number of methods have been proposed to solve the equilibrium problems, one of which is an extragradient method that is particularly interesting and effective. In this paper, we introduce a modified subgradient extragradient method to solve the equilibrium problems in a real Hilbert space. The proposed method uses a non-monotonic step size rule based on local bi-function information instead of its Lipschitz-type constant or other line search method and is capable of solving pseudo-monotone equilibrium problems. Our method only needs to solve a strongly convex programming problem per iteration. Applications of the designed algorithm are presented in order to solve fixed-point problems and variational inequalities. Finally, several computational experiments are studied to confirm the effectiveness of the proposed method. The results of our study include many similar literature studies and detailed numerical studies also show their potential usefulness.
- equilibrium problem,
- non-monotonic step size rule,
- Lipschitz-type conditions,
- subgradient extragradient method,
- fixed point problems,
- variational inequalities
Citation: Habib ur Rehman, Wiyada Kumam, Poom Kumam, Meshal Shutaywi. A new weak convergence non-monotonic self-adaptive iterative scheme for solving equilibrium problems[J]. AIMS Mathematics, 2021, 6(6): 5612-5638. doi: 10.3934/math.2021332

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Abstract

A number of methods have been proposed to solve the equilibrium problems, one of which is an extragradient method that is particularly interesting and effective. In this paper, we introduce a modified subgradient extragradient method to solve the equilibrium problems in a real Hilbert space. The proposed method uses a non-monotonic step size rule based on local bi-function information instead of its Lipschitz-type constant or other line search method and is capable of solving pseudo-monotone equilibrium problems. Our method only needs to solve a strongly convex programming problem per iteration. Applications of the designed algorithm are presented in order to solve fixed-point problems and variational inequalities. Finally, several computational experiments are studied to confirm the effectiveness of the proposed method. The results of our study include many similar literature studies and detailed numerical studies also show their potential usefulness.

References

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