Uniform asymptotic normal structure, the uniform semi‐Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

Abstract

Abstract. In this paper we introduce the uniform asymptotic normal structure and the uniform semi-Opial properties of Banach spaces. This part is devoted to a study of the spaces with these properties. We also compare them with those spaces which have uniform normal structure and with spaces with W CS(X) > 1.

Fixed points of asymptotically regular nonexpansive mappings onnonconvex sets

“…The Opial property and its modifications and generalizations (see [8,9,32,44,45]) have many applications in problems of weak convergence of a sequence, either of iterates {T n (x)} of a nonexpansive (asymptotically nonexpansive) mapping or averaging iterates (see, e.g., [1,5,7,10,16,17,23,38,39,40,43]). This property is also a crucial assumption in theorems about the behavior of some products of nonexpansive mappings [16,17] and in ergodic results (see the literature given in [29,30]).…”

Geometrical Background of Metric Fixed Point Theory

“…Kadec-Klee and Opial properties were extended to the abstract hyperbolic metric setting in [60]. Refined versions of Opial properties can be found for instance in [101], [17] and [18]. Property L(τ, ρ) has its origin in [70] (see also [60] and [32]).…”

Opial’s modulus and fixed points of semigroups of mappings

“…A semi-Opial coefficient with respect to the weak topology, w-SOC (X) for short, is defined as follows (see [3]):…”