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A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space

Lateef Olakunle Jolaoso, Adeolu Taiwo, Timilehin Opeyemi Alakoya, Oluwatosin Temitope Mewomo ORCID logo and Qiao-Li Dong

Abstract

In this paper, we introduce a Totally Relaxed Self-adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of a Variational Inequality Problem (VIP) and the fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation of projection onto the feasible set of the VIP; instead, it uses a projection onto a finite intersection of sub-level sets of convex functions. The advantage of this is that any general convex feasible set can be involved in the VIP. We also introduce a modified TRSSEM which involves the projection onto the set of a convex combination of some convex functions. Under some mild conditions, we prove a strong convergence theorem for our algorithm and also present an application of our theorem to the approximation of a solution of nonlinear integral equations of Hammerstein’s type. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature. Our algorithm is simple and easy to implement for computation.

MSC 2010: 65K15; 47J25; 65J15; 90C33

Funding source: National Research Foundation

Award Identifier / Grant number: 119903

Funding statement: The first author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF CoE-MaSS) Doctoral Bursary. The second author acknowledges with thanks the International Mathematical Union (IMU) Breakout Graduate Fellowship Award for his doctoral study. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the IMU, CoE-MaSS and NRF.

Acknowledgements

The authors sincerely thank the reviewers for their careful reading, constructive comments and fruitful suggestions that improved the manuscript.

  1. Conflict of Interest: The authors declare that they have no competing interests.

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Received: 2020-10-26
Revised: 2021-07-13
Accepted: 2021-09-18
Published Online: 2021-10-16

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