Two-step projection methods for a system of variational inequality problems in Banach spaces

• Published in: Journal of global optimization Vol. 55; no. 4; pp. 801 - 811
• Main Authors: Yao, Yonghong, Liou, Yeong-Cheng, Kang, Shin Min
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.04.2013; Springer; Springer Nature B.V
• Subjects: Banach space; Banach spaces; Computer Science; Constants; Hilbert space; Inequalities; Mapping; Mathematical models; Mathematics; Mathematics and Statistics; Methods; Operations Research/Decision Theory; Optimization; Projection; Real Functions; Studies; Projection method; 47H10; Variational inequality; Accretive mapping; Banach spaces; 47H05; 47J25
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-011-9804-0

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let Π C be a sunny nonexpansive retraction from E onto C . Let the mappings be γ 1 -strongly accretive,  μ 1 -Lipschitz continuous and γ 2 -strongly accretive,  μ 2 -Lipschitz continuous, respectively. For arbitrarily chosen initial point , compute the sequences { x k } and { y k } such that where { α k } is a sequence in [0,1] and ρ , η are two positive constants. Under some mild conditions, we prove that the sequences { x k } and { y k } converge to x * and y *, respectively, where ( x *, y *) is a solution of the following system of variational inequality problems in Banach spaces: Our results extend the main results in Verma (Appl Math Lett 18:1286–1292, 2005 ) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.