Volume 20, Issue 4

This paper deals with the existence of infinitely many solutions for a class of Dirichlet elliptic problems driven by a bi–nonlocal operator u ↦ M (∥ u ∥ 2 )(−Δ) s u , where M models a Kirchhoff–type coefficient while (−Δ) s denotes the fractional Laplace operator. More precisely, by adapting to our bi–nonlocal framework the variational and topological tools introduced in [16], we establish the existence of infinitely many solutions. The main feature and difficulty of our problems is due to the possible degenerate nature of the Kirchhoff term M .

In this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2.

In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H -function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β .

Our general aim is to give sufficient conditions for robustness behavior and convergence to the equilibrium point of linear time-varying fractional system’s solutions. We approach this problem using as a framework a series of recent results due to Cong et al. We establish theorems that generalize in several ways many previously published results, including those of Cong et al. We use the proposed theorems in control and adaptive systems, proving convergence and robustness of such schemes, that up to date remain as unsolved problems, showing the wide scope of applications of our results.

In the present work we study linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivatives, with respect to a nonnegative density function. For the initial problem of this kind of systems, existence, uniqueness and a priory estimate of the solution are proved. As an application of the obtained results, we establish sufficient conditions for global asymptotic stability of the zero solution of the investigated types of systems.

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space BV of functions of bounded variation, whose derivatives are not functions but measures and the space SBV , say the space of bounded variation functions whose derivative has no Cantor part. We prove that SBV is included in W s ,1 for every s ∈ (0, 1) while the result remains open for BV . We study examples and address open questions.

This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on the norm continuity and compactness of some resolvent operators (also called solution operators). And then via the obtained properties on resolvent operators and fixed point technique, we give some approximate controllability results for Sobolev type fractional differential systems in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. Particularly, the existence or compactness of an operator E −1 is not necessarily needed in our results.

We consider two types of fractional integral moduli of smoothness, which are widely used in the theory of functions and approximation theory. In particular, we obtain new equivalences between these moduli of smoothness and the classical moduli of smoothness. It turns out that for the fractional integral moduli of smoothness some pathological effects arise.

In the numerical approximation of fractional order derivatives, the crucial point is to balance the computing complexity and the computing accuracy. We proposed a piecewise memory principle for fractional derivatives, in which the past history is divided into several segments instead of discarded. The piecewise approximation is performed on each segment. Error estimation of piecewise memory principle is analyzed also. Numerical examples show that the contradiction of computing accuracy and complexity is effectively relaxed and the piecewise memory principle is superior to the existing short, variable and equal-weight memory principles. The impacts of the memory length, step size and segment size are also discussed.

A new computational approach for approximating of variable-order fractional derivatives is proposed. The technique is based on piecewise cubic spline interpolation. The method is extended to a class of nonlinear variable-order fractional integro-differential equation with weakly singular kernels. Illustrative examples are discussed, demonstrating the performance of the numerical scheme.

In this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

This is an erratum to the paper “Fractional integral on martingale Hardy spaces with variable exponents”, Fract. Calc. Appl. Anal . 18 , No 5 (2015), 1128–1145; DOI: 10.1515/fca-2015-0065.