Abstract. Let C be a closed convex subset of a Hilbert space and {Tn} a sequence
of nonexpansive self-mappings of C. Then we consider the following iterative sequence
{zn}: x1 = x �� C, xn+1 = Tnxn, and zn = 1/n ��
k=1 xk for n �� 
. In this paper,
we obtain a weak convergence theorem for such a sequence {zn}. Using our result, we
get a nonlinear ergodic theorem which is a generalization of Baillon [2]. Further we
apply our result to the problem of finding a common fixed point of a countable family
of nonexpansive mappings.