A common fixed-point theorem in reflexive locally uniformly convex Banach spaces
Michael L. Edelstein, Mo Tak Kiang · Proceedings of the American Mathematical Society · 1985
Let X X be a reflexive locally uniformly convex Banach space and G G an ultimately nonexpansive commutative semigroup of continuous self-maps of X X . If there exists a point x x in X X recurrent under G G such that G ( x ) G(x) is bounded, then G G has a common fixed point in co ¯ ( G ( x ) ) \overline {{\text {co}}} (G(x)) . If X X is a Hilbert space then there is exactly one such point in co ¯ ( G ( x ) ) \overline {{\text {co}}} (G(x)) .