TY - JOUR

T1 - A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space

AU - Jolaoso, Lateef Olakunle

AU - Taiwo, Adeolu

AU - Alakoya, Timilehin Opeyemi

AU - Mewomo, Oluwatosin Temitope

AU - Dong, Qiao Li

N1 - Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston 2022.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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.

AB - 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.

KW - Extragradient Method

KW - Hammerstein Equation

KW - Nonexpansive Mappings

KW - Subgradient Method

KW - Total Relaxed

KW - Variational Inequality

UR - http://www.scopus.com/inward/record.url?scp=85123316281&partnerID=8YFLogxK

U2 - 10.1515/cmam-2020-0174

DO - 10.1515/cmam-2020-0174

M3 - Article

AN - SCOPUS:85123316281

VL - 22

SP - 73

EP - 95

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 1

ER -