On fractional deviation operators

The so called fractional deviation operators are introduced. This class of integral transforms appears naturally from the study of iteration of fractional integrals of Riemann-Liouville type. Since B. Ross' formulation on fractional iteration process, among other problems selected by T. Osler [5] toward 1974, several authors have been working on this subject. In particular, are worth mentioning contributions of B. Rubin [7] that allowed an intrinsic connection between fractional integrals with different limits of integration (Love's question, see [5] also) and the corresponding Ross' problem for Chen fractional integrals, handled by A. Nahushev [4] and M. Salahitdinov, with broad applications to non local boundary value problems. In this article we consider deviation operators as integral transforms, their connection with operators of Rubin type and mapping properties between classical and weighted Lebesgue spaces

Fractional integrals and derivatives: mapping properties

This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Holder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions

Variable order Mittag-Leffler fractional operators on isolated time scales and application to the calculus of variations

We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler-Lagrange type.Comment: This is a preprint of a paper whose final and definite form is with Springe

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850