Electron. J. Diff. Equ., Vol. 2013 (2013), No. 74, pp. 1-19.

Generalized Riemann derivative

Sorin Radulescu, Petrus Alexandrescu, Diana-Olimpia Alexandrescu

Initiated by Marshall Ash in 1966, the study of generalized Riemann derivative draw significant attention of the mathematical community and numerous studies where carried out since then. One of the major areas that benefits from these developments is the numerical analysis, as the use of generalized Riemann derivatives leads to solving a wider class of problems that are not solvable with the classical tools. This article studies the generalized Riemann derivative and its properties and establishes relationships between Riemann generalized derivative and the classical one. The existence of classical derivative implies the existence of the Riemann generalized derivative, and we study conditions necessary for the generalized Riemann derivative to imply the existence of the classical derivative. Furthermore, we provide conditions on the generalized Riemann derivative that are sufficient for the existence of the classical derivative.

Submitted January 10, 2013. Published March 18, 2013.
Math Subject Classifications: 26A24, 28A15.
Key Words: Riemann generalized derivative; symmetric derivative; Schwarz derivative; (sigma,tau) differentiable function.

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Sorin Radulescu
Institute of Mathematical Statistics and Applied Mathematics
Calea 13 Septembrie, no. 13, Bucharest 5, RO-050711, Romania
email: xsradulescu@gmail.com
Petrus Alexandrescu
Institute of Sociology, Casa Academiei Romane
Calea 13 Septembrie, no. 13, Bucharest 5, RO-050711, Romania
email: alexandrescu_petrus@yahoo.com
Diana-Olimpia Alexandrescu
Department of Mathematics, University of Craiova
200585 Craiova, Romania
email: alexandrescudiana@yahoo.com

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