Generalized quasi-Banach sequence spaces and measures of noncompactness

Abstract

Given 0 < s ≤ 1 and à an s-convex function, s − à -sequence spaces are introduced. Several quasi-Banach sequence spaces are thus characterized as a particular case of s − Ã-spaces. For these spaces, new measures of noncompactness are also defi ned, related to the Hausdorff measure of noncompactness. As an application, compact sets in s − Ã-interpolation spaces of a quasi-Banach couple are studied.

Nonlinear analysis by applying best approximation method in p-vector spaces

“…From now on, we denote by V 0 the set of all shrinkable zero neighborhoods in E. We have the following result which is Theorem 1 of Machrafi and Oubbi [73], saying that in the general setting of locally p-convex spaces, the measure of noncompactness α for U given by Definition 4.4 is stable from U to its p-convex hull C p (A) of the subset A in E, which is key for us to establish the fixed points for condensing mappings in locally pconvex spaces for 0 < p ≤ 1. This also means that the key property for the Kuratowski and Hausdorff measures of noncompactness in normed (or p-seminorm) spaces also holds for the measure of noncompactness by Definition 4.4 in the setting of locally p-convex spaces with (0 < p ≤ 1) (see more similar and related discussion in detail by Alghamdi et al [5] and Silva et al [112]).…”

“…In order to establish fixed point theorems for the classes of 1-set contractive and condensing mappings in p-vector spaces by using the concept of the measure of noncompactness (or the noncompactness measures), which were introduced and widely accepted in mathematical community by Kuratowski [65], Darbo [30] (see related references therein), we first need to have a brief introduction for the concept of noncompactness measures for the so-called Kuratowski or Hausdorff measures of noncompactness in normed spaces (see Alghamdi et al [5], Machrafi and Oubbi [73], Nussbaum [78], Sadovskii [106], Silva et al [112], Xiao and Lu [123] for the general concepts under the framework of p-seminorm or just for locally convex p-convex settings for 0 < p ≤ 1, which will be discussed below, too).…”

“…In order to establish the fixed points of set-valued condensing mappings in p-vector spaces for 0 < p ≤ 1, we need to recall some notions introduced by Machrafi and Oubbi [73] for the measure of noncompactness in locally p-convex vector spaces, which also satisfies some necessary (common) properties of the classical measures of noncompactness such as β K and β H mentioned above introduced by Kuratowski [65], Sadovskii [106](see also the related discussion by Alghamdi et al [5], Nussbaum [78], Silva et al [112], Xiao and Lu [123] and the references therein). In particular, the measures of noncompactness in locally p-convex spaces (for 0 < p ≤ 1) should have the stable property, which means the measure of noncompactness A is the same by transition to the (closure) for the p-convex hull of subset A.…”

“…Jarchow [54], Kalton [55,56], Kalton et al [57], Machrafi and Oubbi [73], Park [90], Qiu and Rolewicz [99], Rolewicz [103], Silva et al [112], Simons [113], Tabor et al [116], Tan [117], Wang [120], Xiao and Lu [123], Xiao and Zhu [124,125], Yuan [133], and many others). However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis have not been well developed.…”

The Banach Space of Quasinorms on a Finite-Dimensional Space

“…For the reader interested in interpolation, we suggest [2]. A nice paper on interpolation in quasinormed spaces is [9].…”