Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial’s property

Abstract

Let $X$ be a Banach space with Opial’s property, $C$ a weakly compact subset of $X$, $x \in C$ and $S$ a nonexpansive semigroup on $C$. Then ${\left \{ {S(t)x} \right \}_{t \geqslant 0}}$ converges weakly to a common fixed point of $S$ iff $S(t + h)x - S(t)x \rightharpoonup 0$ as $t \to \infty$ for all $h > 0$.