Niveloids
type: Item
instance of: scholarly article
title: Niveloids
zbMATH Open document ID: 0839.49010
DOI: 10.12775/TMNA.1995.001
author: Szymon Dolecki
author: Gabriele H. Greco
published in: Topological Methods in Nonlinear Analysis
publication date: 1996-06-19T00:00:00Z
review text: The authors introduce a class of functionals named niveloids, defined on the set of all extended real-valued functions on a set \(X\), as follows: \[ {\mathcal N}(X) = \{T: \overline{\mathbb{R}}^X\mapsto \overline{\mathbb{R}};\;T\text{ is isotone and vertically invariant}\}, \] where the both defining notions mean, respectively, the properties: \[ f\leq g\Rightarrow Tf\leq Tg,\tag{1} \] \[ \forall r\in \mathbb{R}\;\;T(f+ r)= T(f)+ r.\tag{2} \] Among the niveloids one can find, for instance, limitoids of G. H. Grecco, \(\Gamma\)-limits of E. De Giorgi, \(Q\)-limits of L. Cesari, etc. Some operations (like upper and lower projections) on functionals as elements of the complete lattice \({\mathcal F}(X)\) of all the functionals ordered pointwise are defined and some representation theorems for niveloids are given. Other representation theorems for niveloids are obtained by using the inf- and sup-convolutions on functions from \(\overline{\mathbb{R}}^X\). Besides, some uniform and improper topologies on functions from \(\overline{\mathbb{R}}^X\) are introduced so that niveloids would be upper (resp. lower) semicontinuous. Also, several other notions are introduced and various links between them are established as, for instance, a lattice structure for the set of niveloids is considered and an isomorphism with the complete lattice of closed sets with respect to the uniform topology is proved.
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 807911
zbMATH Keywords: semicontinuity
zbMATH Keywords: niveloids
zbMATH Keywords: limitoids
zbMATH Keywords: \(\Gamma\)-limits
zbMATH Keywords: \(Q\)-limits
zbMATH Keywords: representation theorems
zbMATH Keywords: uniform and improper topologies
zbMATH Keywords: lattice structure
reviewed by: Zdzisław Denkowski
MaRDI profile type: Publication
full work available at URL: tmna.1995.001
OpenAlex ID: W4210936209
description: scientific article; zbMATH DE number 807911