Around the nonlinear Ryll-Nardzewski theorem

Suppose that Q is a weak* compact convex subset of a dual Banach space with the Radon-Nikodym property. We show that if (S, Q) is a nonexpansive and norm-distal dynamical system, then there is a fixed point of S in Q and the set of fixed points is a nonexpansive retract of Q. As a consequence we obtain a nonlinear extension of the Bader-Gelander-Monod theorem concerning isometries in L-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.