Periodic functions with difference functions
Pointwise convergence to the initial data for nonlocal dyadic diffusions
We solve the initial value problem for the diffusion induced by dyadic fractional derivative in . First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator....
Polynomial approximation and twenty years later.
Positive solutions for boundary value problem of nonlinear fractional differential equation.
Positive solutions for boundary value problem of nonlinear fractional differential equation.
Positive solutions for boundary value problems of -dimension nonlinear fractional differential system.
Positive solutions for higher-order nonlinear fractional differential equation with integral boundary condition.
Positive solutions for multipoint boundary value problem of fractional differential equations.
Positive solutions for three-point nonlinear fractional boundary value problems.
Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order
Classical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as...
Positivity and stabilization of fractional 2D linear systems described by the Roesser model
A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation...
Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains
The purpose of the paper is to extend results of the potential theory of the classical Schrödinger operator to the α-stable case. To obtain this we analyze a weak version of the Schrödinger operator based on the fractional Laplacian and we prove the Conditional Gauge Theorem.
Product rule and chain rule estimates for the Hajłasz gradient on doubling metric measure spaces
We use the Calderón Maximal Function to prove the Kato-Ponce Product Rule Estimate and the Christ-Weinstein Chain Rule Estimate for the Hajłasz gradient on doubling measure metric spaces.
Professor Rudolf Gorenflo and his Contribution to Fractional Calculus
MSC 2010: 26A33 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryThis paper presents a brief overview of the life story and professional career of Prof. R. Gorenflo - a well-known mathematician, an expert in the field of Differential and Integral Equations, Numerical Mathematics, Fractional Calculus and Applied Analysis, an interesting conversational partner, an experienced colleague, and a real friend. Especially his role in the modern Fractional Calculus and its applications...
Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces
In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic -mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic -mild solution of a two-dimensional impulsive fractional...