Volume 19, Issue 3

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 Hölder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.

We introduce conditions on the construction of grand Lebesgue spaces on ℝ n which imply the validity of the Sobolev theorem for the Riesz fractional integrals I α and the boundedness of the maximal operator, in such spaces. We also give an inversion of the operator I α by means of hypersingular integrals, within the frameworks of the introduced spaces. We also proof the denseness of $C_0^\infty {(\mathbb R)^n}$C0∞(ℝ)n in a subspace of the considered grand space.

New fractional operators of non-integer and integer orders for N -dimensional lattice ℤ N are suggested. The proposed lattice fractional integro-differentiation of positive order can be considered as a lattice analog of partial derivatives of non-integer orders. For negative values of order, it can be considered as a lattice fractional integration. For integer orders, the continuum limit of the united lattice derivatives gives the usual partial derivatives of integer orders. In contrast to the usual lattice derivatives, which are proposed in our recent paper (“Lattice fractional calculus”, Appl. Math. Comp . 257 (2015), 12–33), the suggested lattice operators give the usual partial derivatives for all integer orders. The united lattice integro-differentiation is represented by finite differences with infinite series that describe long-range lattice interactions. The suggested lattice operators of positive orders can be considered as an exact discretization of derivatives of corresponding orders. The basic mathematical feature of these united lattice operators is an implementation of the same algebraic properties that have the differentiation operator.

We discuss solvability of a nonlinear Riemann-Liouville integral equation in Lebesgue spaces. We treat the Volterra equations of the first and the second types by applying boundedness criteria for the Riemann-Liouville integral operator. The existence of a solution to integral equations will follow from the Leray-Schauder Nonlinear Alternative.

In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

The boundedness of multi(sub)linear Hardy–Littlewood maximal, Calderón–Zygmund and fractional integral operators defined on metric measure spaces is established in weighted grand Lebesgue spaces. In particular, we derive the one-weight inequality for maximal and singular integrals under the Muckenhoupt type conditions, weighted Sobolev type theorem and trace type inequality for fractional integrals.

In this paper we consider the differential-difference reflection operator associated with a finite cyclic group, $$Y_\nu f(x)=\frac{df(x)}{dx}+\sum_{i=1}^{m-1}\frac{m\nu_i+m-i}{x}\sum_{j=0}^{m-1}\varepsilon^{-ij}f(\varepsilon^jx).$$Yν f(x)=df(x)dx+∑ i=1m− 1mν i+m− ix∑ j=0m− 1ε − ijf(ε jx). First we show that the Dimovski ([5], [6]) hyper–Bessel differential operator of arbitrary integer order m is close in frame of the algebra similar to U (sl(2;C)). Secondly, we introduce a difference-differential operator associated to finite cyclic group in the rank one case, and then by using a Poisson-type integral transform proposed by Dimovski and Kiryakova ([7], [11]), we construct a new explicit intertwining (transmutation) operator between the operator Yν and the derivative operator d/dx . It is to emphasize that both hyper–Bessel operators and the so-called Poisson–Dimovski transformation (transmutation) are typical examples of the operators of generalized fractional calculus [11, 12].

In this paper, we investigate the existence and uniqueness of ( μ,ν )-pseudo almost automorphic mild solutions to semilinear fractional differential equation with Riemann-Liouville derivative in Banach space, where the nonlinear perturbation is of ( μ,ν )-pseudo almost automorphic type or Stepanov-like ( μ,ν )-pseudo almost automorphic type. As application, we explore the same topic for a fractional relaxation-oscillation equation.

In this paper we consider initial value problems for fractional differential equations involving Caputo differential operators. By establishing a new property of the Mittag-Leffler function and using the Schauder fixed point theorem, we obtain new sufficient conditions for the existence and uniqueness of global solutions of initial value problems.

Perfect nonlinear fractional descriptor observers for fractional descriptor continuous-time nonlinear systems are proposed. Necessary and sufficient conditions for the existence of the observers are established. The design procedure of the nonlinear fractional observers is given. It is based on the elementary row (column) operations and reducing the singular matrix of the system to upper (lower) triangular form. The effectiveness of the procedure is demonstrated on a numerical example.