Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces

Journal of inequalities and applications

Lu-Chuan Ceng¹ 

关键词: three-step Mann iterations;    general system of variational inequalities;    infinitely many nonexpansive mappings;    sunny nonexpansive retraction;    fixed point;    strictly convex Banach space;    uniformly smooth Banach space;    reflexive Banach space with weakly continuous duality map;   

DOI  :  10.1186/1029-242X-2013-539

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We introduce and consider three-step Mann iterations for finding a common solution of a general system of variational inequalities (GSVI) and a fixed point problem (FPP) of an infinite family of nonexpansive mappings in X. Here three-step Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for three-step Mann iterations involving the GSVI and the FPP in a Hilbert space. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature. MSC:49J30, 47H09, 47J20.

【 授权许可】

CC BY