Nonlinear ergodic theorem for semitopological semigroups of non-Lipschitzian mappings in Banach spaces

摘要

Recently Lau and Takahashi proved the ergodic theorem for right reversible semigroup of nonexpansive mappings in the uniformly convex Banach space with the uniformly Frechet differentiable norm by using the methods of invariant means. In this note, weprove the nonlinear ergodic theorem for semitopological semigroups of non-Lipschitzian mappings in the uniformly convex Banach space with the Frechet differentiable norm without using the concept of mean. This result extends the result in ref. to the case where @ is semitopological semigroup of non-Lipschitzian mappings. Moreover, we prove that some key conditions that are assumed in ref. are not necessary.