Isopiestic Technique for Measuring Leaf Water Potentials with a Thermocouple Psychrometer

Banach spaces have been given by Edelstein but under hypotheses on the existence of strong limit points for the sequence I Unx I which are difficult to verify. Cf. Edelstein, M., "On fixed and periodic points under contractive mappings," J. London Math. Soc., 37, 74-79 (1962), "On nonexpansive mappings of Banach spaces," Proc. Cambridge Philos. Soc., 60, 439-447 (1964). 9 Added in proof: After the present note was transmitted for publication, the writer received copies of the manuscripts of two unpublished papers, by W. A. Kirk, and by W. A. Kirk and R. Belluce, respectively, obtaining similar results. In Kirk's paper, it is remarked that the general line of argument is due to Brodskii and Milman (Dokl. (1948)], who proved Theorem 1 for isometries. It should also be remarked that in another recent note by the writer [Browder, F. E., "Mapping theorems for noncompact nonlinear operators in Banach spaces," these PROCEEDINGS, 54, 337-342 (1965)], results are obtained which in the case of Hilbert spaces imply that the conclusion of Theorem 1 is also valid for semicontractive mappings, i.e., mappings U of C into C where Ux = S(x,x) and S is a mapping of C X C into C which is nonexpansive in its first variable and completely continuous in the second variable. As we show elsewhere, this extension remains valid for Banach spaces having weakly continuous duality mappings. It is not clear at the time of writing whether this extension is also valid for the general class of uniformly convex spaces.