Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Abstract EN

Let X be a uniformly convex Banach space and ?={T(s):0≤s<∞} be a nonexpansive semigroup such that F(?)=⋂s>0F(T(s))≠∅.

Consider the iterative method that generates the sequence {xn} by the algorithm xn+1=αnf(xn)+βnxn+(1-αn-βn)(1/sn)∫0snT(s)xnds,n≥0, where {αn}, {βn}, and {sn} are three sequences satisfying certain conditions, f:C→C is a contraction mapping.

Strong convergence of the algorithm {xn} is proved assuming X either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.