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

Abstract

Let $X$ be a uniformly convex Banach space and be a nonexpansive semigroup such that $F\left(\mathcal{S}\right)={\bigcap }_{s>0}F\left(T\right(s\left)\right)\ne \varnothing$. Consider the iterative method that generates the sequence $\left\{x_n\right\}$ by the algorithm $x_{n+1}={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+(1-{\alpha }_n-{\beta }_n)(1/s_n){\int }_0^{s_n}T\left(s\right)x_{n}ds,n\ge 0$, where $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$, and $\left\{s_n\right\}$ are three sequences satisfying certain conditions, $f:C\to C$ is a contraction mapping. Strong convergence of the algorithm $\left\{x_n\right\}$ is proved assuming $X$ either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.