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Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces
Author(s) -
Buong Nguyen
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600240
Subject(s) - mathematics , variational inequality , banach space , hilbert space , differentiable function , regular polygon , method of steepest descent , norm (philosophy) , convergence (economics) , regularization (linguistics) , pure mathematics , extension (predicate logic) , class (philosophy) , mathematical analysis , computer science , geometry , artificial intelligence , political science , law , economics , programming language , economic growth
In this paper, we present a new approach to the problem of finding a common zero for a system of m ‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods.