Some unified axiomatic theories of generalized derivatives (Q1978833)

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.