Ergodic Theory Of Non-linear Mapping In Banach Spaces

The research of nonlinear ergodic theory began in the mid-seventies. Consequently, it got great development because it was widely used in many questions such as the numerical solution of differential equation, the existence theory of positive solution, control theory, optimization. Baillon proved the first nonlinear ergodic theorem for nonexpansive mappings in the framework of Hubert space. Bruck, Hirano and Reich extended Baillon抯 theorem to a uniformly convex Banach space with a Frechet differentiable norm. Hirano-Kido-Takahashi, Oka, Park and Jenong proved the ergodic theorem for commutative semigroups of nonexpansive mappings and asymptotically nonexpansive mappings in the uniformly convex I3anach space with the Frechet differentiable nonn.When G is not a commutative semitopological semigroup, Lau and Takahashi proved the ergodic theorem for right reversible semigroups of nonexpansive mappings by using the methods of invariant means. Moreover, when X is a Hilbert space, Miaxoguchi and Takahashi provided the ergodic retraction theorem for general semigroups of asymptotically nonexpansive mapping by using the concept of invariant submean. Li [101 proved the nonlinear ergodic theorem for semitopological semigroups of Lipschitzian mappings in the uniformly convex Banach space with the Frechet differentiable norm without using the concept of invariant mean and submean. In Chapter 1, we extend the results in [23] [24] to the case of almost-orbit. We proved the nonlinear ergodic theorem for semitopological semigroups of asymptotically nonexpansive mappings in the uniformly convex Banach space with the Frechet differentiable norm. We also point out that some key conditions in [23] [24] are unnecessary. Bose [4], Feathers & Dotson [5] gave the weak convergence theorem of asymptotically nonexpansive mappings in a uniformly convex Banach space with weak continuous duality mapping by using Opial抯 Lemma [6]. Using Bruck抯 Lemma [2], Passty [7] extended to the results of [4,5] to a uniformly convex Banach space with a Frechet differentiable norm. But when X has not a Frechet deffrentiable norm, similar results have not been known for many years. The objective of Chapter 2 is to give the weak convergence theorem of S = {T1 :t E G) of asymptotically nonexpansive mappings in a uniformly convex Banach space without assuming that X has a Frechet differentiable norm. We should notice that the condition X has the (KK) property is strictly weaker ~i4~: Banach~I拁A~ 3 than the condition X has a Frechet differentiable norm. We also provide the ergodic convergence theorems for semitopological nonexpansive type mappings in the reflexive Banach in Chapter 3. By this theorem we can get the main results in [13]14][15][16], and we should notice that their methods do not extend beyond Lipschitzian mappings. Our results are new in the case of nonexpansive semigroups.