On some Banach space properties sufficient for weak normal structure and their permanence properties

Abstract

Abstract. We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.

On the fixed points of nonexpansive mappings in direct sums of Banach spaces

“…The following lemma was proved in [28, Lemma 4]. Similar arguments can be found in [11,29]. Then lim n→∞ y n = 0.…”

“…A Banach space X is said to have the generalized Gossez-Lami Dozo property (GGLD, in short) if lim sup m→∞ lim sup n→∞ x n − x m > 1 whenever (x n ) converges weakly to 0 and lim n→∞ x n = 1. It is known that N(X) > 1 ⇒ WCS(X) > 1 ⇒ GGLD ⇒ weak normal structure and that the GGLD property is equivalent to the so-called property asymptotic (P) (see, e.g., [29]).…”

“…In 1984, Landes [23] showed that normal structure is preserved under a large class of direct sums including all ℓ N p -sums, 1 < p ≤ ∞, but not under ℓ N 1 -direct sums (see [24]). Nowadays, there are many results concerning permanence properties of conditions which imply normal structure, see [7,26,29] and references therein. Several recent papers consider the general case, but always under additional geometrical assumptions, see [4-6, 9, 10, 18, 19, 28, 30].…”

Hyper-extensions in metric fixed point theory

“…A Banach space X is said to have the generalized Gossez-Lami Dozo property (GGLD, in short) if lim sup m→∞ lim sup n→∞ x n − x m > 1 whenever (x n ) converges weakly to 0 and lim n→∞ x n = 1. It is known that the GGLD property is weaker than weak uniform normal structure (see, e.g., [28]).…”

Fixed point theorems and demiclosedness principle for mappings of asymptotically nonexpansive type in Banach spaces

“…The definition of WCS(X) above does not make sense if the space X has the Schur property but in that case we may say by convention that WCS(X) = 2. In this article, we use the following equivalent formulation (see [18] WCS (X) = inf lim n,m;n =m…”