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Let X be a Banach space, C a closed bounded convex subset of X. If U is a mapping (generally nonlinear) of C into X, U is said to be nonexpansive if for each pair of elements x and y of C, we have IlUx -UyIl . 11x -YH.In the present note, we give the proof of the following two theorems: THEOREM 1. Let X be a uniformly convex Banach space, U a nonexpansive mapping of the bounded closed convex subset C of X into C. Then U has a fixed point in C.THEOREM 2. Let X be a uniformly convex Banach space, { Ux I a commuting family of nonexpansive mappings of a given bounded closed convex subset C of X into C. Then the family of mappings I U, I has a common fixed point in C.In two recent notes" 2 in these PROCEEDINGS, the writer proved Theorem 1 for the case when X is a Hilbert space using the concepts of the theory of monotone operators in Hilbert space as developed by the writer' and G. J. Minty.4 The writer extended this result to the class of spaces having weakly continuous duality mappings5 and T. Kato gave an independent proof for the spaces 1P, 1 < p < co (for which weakly continuous duality mappings exist). The result we give here is much more general, since it is valid for the L7'-spaces, 1 < p < + c, for which the duality mappings are not weakly continuous (p $ 2). Theorem 2 is a nonlinear extension of the Markov-Kakutani theorems for linear mappings and an extension of the result of De Marr7 for C compact.If C is compact or U is compact, Theorem 1 is a consequence of the Schauder fixed-point theorem, while if U is weakly continuous, it is a special case of the Tychonoff fixed-point theorem since in a reflexive space, every bounded closed convex

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