Eur. Phys. J. Special Topics 226, 3473–3487 (2017)
https://doi.org/10.1140/epjst/e2018-00077-9

Regular Article

On fractional systems with Riemann-Liouville derivatives and distributed delays – Choice of initial conditions, existence and uniqueness of the solutions

Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria

^a e-mail: kiskinov@uni-plovdiv.bg

Received: 27 August 2017
Received in final form: 30 December 2017
Published online: 25 July 2018

Abstract

A comparative analysis among the possible types of initial conditions including (or not) derivatives in the Riemann-Liouville sense for incommensurate fractional differential systems with distributed delays is proposed. The provided analysis is essentially based on the possibility to attribute physical meaning to the initial conditions expressed in terms of Riemann-Liouville fractional derivatives. This allows the values of the initial functions for the mentioned initial conditions to be obtained by appropriate measurements or observations. In addition, an initial problem with non-continuous initial conditions partially expressed in terms of Riemann-Liouville fractional derivatives is considered and existence and uniqueness of a (1 − α)-continuous solution of this initial problem is proved.