Nonlinear Mapping Fixed Point Theorem And Ergodic Theory

2 Abstract 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 Hilbert space. A series extensions of Baillon抯 result have been given, e.g., Takahashi and Zhang [4] , Tan and Xu [5] proved the ergodic theorem for asymptotically nonexpansive and asymptotically nonexpansive type semigroup respectively. Futher, Li [6], Li and Ma [7] indicated that some key conditions of previous results, such as C is convex and closed, are not necessary. Rouhani [8] first introduced the notion of almost nonexpansive curves, which contains all the almost-orbits of nonexpansive semigroups. He got some asymptotic behaviours of almost nonexpansive curves. In [9], Li and Ma introduced the concept of asymptotically almost nonexpansive curves(AANC) on [0, + a3], which is much more general than almost nonexpansive curves, and contains all the almost-orbits of asymptotically nonexpansive type semigroups. By discussing the asymptotically behavior and ergodic theorem of AANC, they got the corresponding results of asymptotically nonexpansive type semigroups when C is not necessary convex and closed. In the present paper, we first prove the edgodic retraction theorem for any functions on 3. This result contains many previous work. After giving the definition of asymptotically nonexpansive type curves, which contains all the almost-orbits of asymptotically nonexpansive type semigroups, we will prove the ergodic theorem of this curves. In 1965 , Kirk [20] proved that if C is a weakly compat convex subset of a Banach space with normal structure, then every nonexpansive self-mapping T of C has a fixed poind. Seven years late, in 1972, Goebel and Kirk [5] proved that if the space X is assumed to be uniformly convex, then every asymptotically nonexpansive self-mapping T of C has a fixed point. This was extended to mappings of asymptotically nonexpansvie type by Kirk in [12]. More recently these results have been extended to wider classes of spaces, see for example [8], [4], [10], [7] and [9]. Li and Sims [25] solve the open question that whether normal structure implies the existence of fixed points for mappings of asymptotically nonexpansive type. This present paper point out the conditions when fixed points exist and so is a futher step toward answering the above question.. We also represents an extension of the results of [10] and [9] to uniformly k-Lipsehitzian semigroup. 3...