Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

Do Sang Kim; Nguyen Ngoc Hai; Bui Van Dinh

doi:10.3934/naco.2021051

Article Contents

2022, Volume 12, Issue 1: 63-78. Doi: 10.3934/naco.2021051

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Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

1.
Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea
2.
Department of Scientific Fundamentals, Vietnam Trade Union University, Hanoi, Vietnam
3.
Department of Mathematics, Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam

^*Corresponding author
^*Corresponding author

Received: March 2020

Revised: October 2021

Early access: November 12, 2021

Published online: November 12, 2021

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2019R1A2C1008672).

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Abstract

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

Keywords:

Mathematics Subject Classification: 47H06, 47H09, 47H10, 47J05, 47J25.

Citation:

Full Text(HTML)

Table 1. Results computed with Algorithm 2

N.P	Size (n)	Average Times	Average Iterations
10	5	2.3766	152
10	10	4.2141	223
10	20	6.7813	457
10	30	10.5266	515
10	50	17.4891	567
10	100	29.2406	674

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Table 2. Results computed with Algorithm 3

N.P	Size (n)	Average Times	Average Iterations
10	5	2.4656	99
10	10	4.1422	132
10	20	6.6375	164
10	30	8.0672	170
10	50	11.8828	192
10	100	21.4953	210

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Table 3. Results computed with Algorithm 1 in [6]

N.P	Size (n)	Average Times	Average Iterations
10	5	23.2484	826
10	10	34.7438	1445
10	20	87.1016	2346
10	30	157.5781	2715
10	50	255.4578	3839

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Table 4. Results computed with Algorithm 2 in [6]

N.P	Size (n)	Average Times	Average Iterations
10	5	38.5938	904
10	10	106.3172	2242
10	20	163.1266	3050
10	30	250.9313	3001
10	50	359.1094	3592

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