Nonlinear ergodic theorems and weak convergence theorems
Norimichi Hirano · Journal of the Mathematical Society of Japan · 1982
In this paper we study the asymptotic behavior of nonexpansive mappings and of one parameter semigroups of nonexpansive mappings in Banach spaces.In [1], Baillon proved the first nonlinear ergodic theorem for nonexpansive mappings in Hilbert spaces.Reich [16] extended Baillon's result to uniformly convex Banach spaces which have Fr\'echet differentiable norms and Bruck [8] simplified the original argument of Reich.The weak convergence of trajecto- ries of one parameter semigroups of nonexpansive mappings was studied by Baillon [2], Bruck [7], Pasy [16], Miyadera [12] and Reich [17].In section 2, we give ergodic theorems for nonexpansive mappings in uniformly convex Banach spaces which satisfy Opial's condition.In section 3, we consider a necessary and sufficient condition for the weak convergence of trajectories of nonexpansive mappings and one parameter semigroups of nonexpansive mappings in Banach spaces.