A note on property of the Mittag-Leffler function

Recently the authors have found in some publications that the following property (0.1) of Mittag-Leffler function is taken for granted and used to derive other properties.

(0.1)     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights is defined as

     

Topologically transitive semigroup actions of real linear fractional transformations

Let , where I is a proper closed subinterval of R. Our main goal in this paper is to describe all f,g∈FI such that the action of the semigroup generated by f and g is topologically transitive on I. We also prove that this action is never weakly topologically mixing. Finally, we describe all pairs of affine functions on R whose generated semigroup action is weakly topologically mixing.     

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.     

Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method.     

Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup

Under the standard assumptions on the variable exponent p(x) (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space Bα[Lp(⋅)(Rn)] in terms of the rate of convergence of the Poisson semigroup Pt. We show that the existence of the Riesz fractional derivative Dαf in the space Lp(⋅)(Rn) is equivalent to the existence of the limit . In the pre-limiting case we show that the Bessel potential space is characterized by the condition ‖α(I−Pε)f_‖p(⋅)≦Cεα.     

Parabolic composition operators on the ball

We give a classification, up to automorphisms, of parabolic linear fractional maps of the ball. We then show that this classification is very convenient to study the geometric properties of these maps, as well as the associated composition operators.     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation

In this paper, a Cauchy problem for the time fractional advection-dispersion equation (TFADE) is investigated. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order . We show that the Cauchy problem of TFADE is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. The convergence estimate is obtained under a priori bound assumptions for the exact solution. Numerical examples are given to show the effectiveness of the proposed numerical method.