Regular variation and fractional operators

Abstract The paper can be understood as a contribution to the study of relationships between the notion of regular variation on one hand and fractional calculus on the other hand, along with their consequences. We present a full fractional extension of some of the fundamental statements from the theory of regularly varying functions, thereby providing a characterization of regular variation in terms of fractional operators. Various types of fractional integral and differential operators are included. Asymptotics at infinity as well as at the initial point is studied. We also establish new versions of the fractional L’Hospital rule. The obtained results will find applications in the asymptotic theory of fractional differential equations. As a by-product, we obtain information on the solutions to the equations which, in the limiting case, lead to viscoelastic models.