On the weak convergence theorem for nonexpansive semigroups in Banach spaces

Abstract

Assume that $K$ is a closed convex subset of a uniformly convex Banach space $E$, and assume that $\left\{T\right(s){\}}_{s>0}$ is a nonexpansive semigroup on $K$. By using the following implicit iteration sequence $\left\{x_n\right\}$ defined by $$x_n=(1-{\alpha }_n)x_{n-1}+{\alpha }_n\cdot \frac{1}{t_n}{\int }_0^{t_n}T\left(s\right)x_{n}ds,\forall n\ge 1,$$ the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup $\left\{T\right(s){\}}_{s>0}$ in uniformly convex Banach spaces without the Opial property. Our results are different from some recently announced results.