Vol. 30, No. 3, 1969

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Mean value iteration of nonexpansive mappings in a Banach space

Curtis L. Outlaw

Vol. 30 (1969), No. 3, 747–750

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 TxTyxy 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.

Mathematical Subject Classification
Primary: 47.85
Received: 12 March 1968
Published: 1 September 1969
Curtis L. Outlaw