Asymptotical smoothness and its applications

Abstract

In this paper we introduce the notion of asymptotical smoothness of a Banach space and show that it is strongly related to the Kadec-Klee property. This notion is then applied to obtain new theorems about weak convergence of almost orbits of three various types of semigroups of mappings.

The Γ-opial property

“…Therefore in such cases the basic question is whether the Opial property for nets is equivalent to the Opial property for sequences. For the weak topology this equivalence was proved by W. Kaczor and S. Prus in [7]. In our paper we show that if (X, || • ||) is a Banach space, F a norming set for X and C is a nonempty, bounded and sequentially compact, in the F-topology, subset of X, then in C the T-Opial condition for nets is equivalent to the F-Opial condition.…”

“…In our paper we show that if (X, || • ||) is a Banach space, F a norming set for X and C is a nonempty, bounded and sequentially compact, in the F-topology, subset of X, then in C the T-Opial condition for nets is equivalent to the F-Opial condition. However, in our proof we use a different method than that of [7].…”

On the structure of fixed-point sets of asymptotically regular semigroups

“…Kaczor and Prus [19] initiated a systematic study of assumptions under which one can replace sequences by nets in a given condition. We follow the arguments from that paper and use the well known method of constructing basic sequences attributed to S. Mazur (see [25]).…”

“…t i x i for any integers p > q ≥ 1 and any sequence of scalars (t i ). In the proof of the next lemma, based on Mazur's technique, we follow in part the reasoning given in [19,Cor. 2.6]…”

“…Recall that a sequence (x n ) is basic if and only if there exists a number c > 0 such that q i=1 t i x i ≤ c p i=1 t i x i for any integers p > q ≥ 1 and any sequence of scalars (t i ). In the proof of the next lemma, based on Mazur's technique, we follow in part the reasoning given in [19,Cor. 2.6].…”