A Parallel Hybrid Bregman Subgradient Extragradient Method for a System of Pseudomonotone Equilibrium and Fixed Point Problems

Annel Thembinkosi Bokodisa,Lateef Olakunle Jolaoso,Maggie Aphane

doi:10.3390/sym13020216

Annel Thembinkosi Bokodisa, Lateef Olakunle Jolaoso + Show 1 more

Open Access

https://doi.org/10.3390/sym13020216

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Journal: Symmetry	Publication Date: Jan 28, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Sefako Makgatho Health Sciences University

Abstract

We introduce a new parallel hybrid subgradient extragradient method for solving the system of the pseudomonotone equilibrium problem and common fixed point problem in real reflexive Banach spaces. The algorithm is designed such that its convergence does not require prior estimation of the Lipschitz-like constants of the finite bifunctions underlying the equilibrium problems. Moreover, a strong convergence result is proven without imposing strong conditions on the control sequences. We further provide some numerical experiments to illustrate the performance of the proposed algorithm and compare with some existing methods.

Highlights

In this paper, we consider the Equilibrium Problem (EP) in the framework of a real reflexive Banach space
Persuaded by the outcomes above, in this present paper, we provide another subgradient extragradient technique for finding a common element of the set of solutions of equilibrium problems for a finite family of pseudomonotone bifunctions and the set of common fixed points of a finite family of Bregman relatively nonexpansive mappings in the framework of reflexive Banach spaces
For i = 1, 2, . . . , N, let gi : E × E → R be a finite family of bifunctions satisfying Assumptions (A1)–(A5)

Summary

Introduction

We consider the Equilibrium Problem (EP) in the framework of a real reflexive Banach space. Eskandani et al [19], using the Hybrid Parallel extragradient method (HPA), introduced a Bregman–Lipschitz-type condition for a pseudomonotone bifunction For estimating this common point p∗ for a finite family of multi-valued Bregman relatively nonexpansive mappings in reflexive Banach spaces, the following algorithm, called HPA, was presented: x0. Persuaded by the outcomes above, in this present paper, we provide another subgradient extragradient technique for finding a common element of the set of solutions of equilibrium problems for a finite family of pseudomonotone bifunctions and the set of common fixed points of a finite family of Bregman relatively nonexpansive mappings in the framework of reflexive Banach spaces. We give some numerical examples to demonstrate the proficiency, competitiveness and efficiency of our algorithm with respect to other algorithms in the literature

Preliminaries

Results

Numerical Examples

Conclusions