MEAN ERGODIC THEOREM FOR SEMIGROUPS OF NONEXPANSIVE LINEAR OR NONLINEAR OPERATORS

Let E be a reflexive Banach space, S a linear operator in E such that parallel to S-n parallel to less than or equal to M<infinity, For All n greater than or equal to 1 (respectively: S is a nonlinear operator from a bounded closed convex and separable subset C of E, into C, such that parallel to S-n y-S-n z parallel to less than or equal to M parallel to y-z parallel to, For All y, z epsilon C, For All n greater than or equal to 1). Then for every x epsilon E (reap. every x epsilon C), we show that: (i) The trajectory of x defines an asymptotical measure on E (supposed to be separable) equipped with its weak topology. (ii) For every xi epsilon E' dual of E, every q greater than or equal to 1, the limit [GRAPHICS] exists, uniformly with respect to p greater than or equal to 1. In particular, the weak limit sigma (x) of sigma(N), (x) = (1/N) [GRAPHICS] S-n x exists in E (resp. in C). (iii) If E is unformly convex, and S a nonlinear contraction from C to C (bounded closed convex and separable subset of E) then sigma (x) is a fixed point for S. Results analogous to (i), (ii) and (iii) are true for strongly continuous semigroups (S-t). Property (iii) generalises results from [10], [11], [12] obtained for E uniformly convex with Frechet-differentiable norm (but without separability of C). Detailed proofs of this Note will be given in a forthcoming paper.