A mean ergodic theorem for mappings which are asymptotically nonexpansive in the intermediate sense

Convergence Theorems for Mappings Which Are Asymptotically Nonexpansive in the Intermediate Sense

“…Theorem 3.5 extends Theorem 1 of Oka [22] and Kaczor et al [17] to the more general class of non-self maps. Under the additional hypothesis that the dual E* of E has the Kadec-Klee property, Theorem 3.…”

“…5 of Oka [22] (see also, Kaczor et. al [17]). For completeness and because of more general nature of our map, we sketch the details.…”

Existence and convergence of fixed points for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces

“…In 1993, Bruck et al [2] introduced the notion of mappings which are asymptotically nonexpansive in the intermediate sense (continuous mappings of asymptotically nonexpansive type) and obtained the weak convergence theorems of averaging iteration for mappings of asymptotically nonexpansive in the intermediate sense in uniformly convex Banach space with the Opial property. Since then many authors have studied on the existence and convergence theorems of fixed points for these two classes of mappings in Banach spaces, for example, Xu [3], Kaczor [4,5], Rhoades [6], etc.…”

Asymptotical smoothness and its applications

“…In some papers concerning nonlinear ergodic theory, to get weak convergence of suitable sequences or nets in a Banach space X, authors use the Kadec-Klee property and the Kadec property (respectively) of the dual space X* (see, for example, [7] and [10,11,12]). In the next section of our paper we introduce the notion of asymptotical smoothness and prove that it is partially dual to the Kadec property.…”