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
SN - 1609-4840
VL - 22
SP - 73
EP - 95
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
IS - 1
ER -