Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping Tμ for a Banach space with uniformly Gâteaux differentiable norm when μ is a multiplicative left invariant mean on l∞(S). As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial's condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.