Mean value iteration of nonexpansive mappings in a Banach space
Curtis Outlaw · Pacific Journal of Mathematics · 1969
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 11 Tx -Ty \ | ^ 11 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 image.We note that a rotation T about zero of the closed unit disc in the complex plane satisfies the conditions of Theorems 1 and 2, but the usual sequence {T n x} of iterates of x does not converge unless x is zero.DEFINITIONS.If Y is a Banach space, T is a mapping from Y into itself, and xe Y, then M(x, T) is the sequence {v n } defined by v x = x and v n+1 = {Ij2){v n + Tv n ).Following Wilansky [3, pp.107-111], we say that a Banach space Y is rotund provided that if w e Y, y e Y, w Φ y, and \\w\\ = \\y\\ <ί 1, then (1/2) || w + y\\