An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space

J. B. Baillon C. R. Acad. Sci. Paris Ser. A.280 (1975), 1511–1514] proved an ergodic theorem for a single nonexpansive mapping in a Hilbert space, which is a nonlinear version of von Neumann's mean ergodic theorem. In this paper, we study the ergodic behavior of a semigroup of nonexpansive mappings. We try to find a sequence of means on the semigroup, generalizing the Cesàro means on $\text{N}$, such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed-points. Our method of proof is an appropriate modification of A. Pazy's proof Israel J. Math.26 (1977), 197–204] of Baillon's theorem.