This paper applies a certain
method of iteration, of the mean value type introduced by W. R. Mann, to obtain
two theorems on the approximation of a fixed point of a mapping of a Banach space
into itself which is nonexpansive (i.e., a mapping which satisfies ∥Tx−Ty∥≦∥x−y∥
for each x and y).
The first theorem obtains convergence of the iterates to a fixed point of a
nonexpansive mapping which maps a compact convex subset of a rotund Banach
space into itself.
The second theorem obtains convergence to a fixed point provided that
the Banach space is uniformly convex and the iterating transformation is
nonexpansive, maps a closed bounded convex subset of the space into itself,
and satisfies a certain restriction on the distance between any point and its