Some unified axiomatic theories of generalized derivatives
type: Item
instance of: scholarly article
title: Some unified axiomatic theories of generalized derivatives
zbMATH Open document ID: 0944.26500
author: K. M. Garg
published in: Real Analysis Exchange
publication date: 2000-05-21T00:00:00Z
review text: This is an announcement of theories of generalized derivatives to be presented in the forthcoming author's monograph ``Theory of differentiation. A unified theory of differentiation via new derivate theorems and new derivatives'' (1998; Zbl 0918.26003). New derivatives are, in general, set valued functions. E.g., a~function defined on an open interval \(I\) is said to be lower derivable at \(x\in I\), if \(D^{-}f(x)\leq D_{+}f(x)\). Then the corresponding new derivative is \(Lf'(x)=[D^{-}f(x),D_{+}f(x)]\). Further, the notions of upper derivable, semiderivable, weakly derivable are introduced. For a~generalized (unilateral) derivates, the author defines (by a~set of axioms) its admissibility and s-admissibility and investigates these notions for derivates induced by generalized limits of generalized quotients. (Known generalized derivates are of this type.) Regularity of generalized derivates and status of new generalized derivates are discussed. Mean value theorems and monotonicity theorems are also investigated. Similar questions for generalized symmetric derivates are treated in the second part of the paper. No proofs are given.
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26E25
zbMATH DE Number: 1449391
zbMATH Keywords: generalized derivatives
zbMATH Keywords: generalized symmetric derivatives
reviewed by: Ivan Netuka
MaRDI profile type: Publication
Recommended article: Q3617531
Recommended article: Q3418308
Recommended article: Some properties of generalized derivatives
Recommended article: Q4217270
Recommended article: Generalized Derivative and Generalized Continuity
Recommended article: Remarks on Various Generalized Derivatives
Recommended article: Generalized differentiation of parameter-dependent sets and mappings†
Recommended article: Q5203265
Recommended article: Q3709004
Recommended article: Q4788804
description: scientific article; zbMATH DE number 1449391