A common fixed-point theorem in reflexive locally uniformly convex Banach spaces

Abstract:

Let $X$ be a reflexive locally uniformly convex Banach space and $G$ an ultimately nonexpansive commutative semigroup of continuous self-maps of $X$. If there exists a point $x$ in $X$ recurrent under $G$ such that $G(x)$ is bounded, then $G$ has a common fixed point in $\overline {{\text {co}}} (G(x))$. If $X$ is a Hilbert space then there is exactly one such point in $\overline {{\text {co}}} (G(x))$.