Differential inclusions of arbitrary fractional order with anti-periodic conditions in Banach spaces

In this paper, we establish various existence results of solutions for fractional differential equations and inclusions of arbitrary order $q\in (m-1,m)$, where $m$ is an arbitrary natural number greater than or equal to two, in infinite dimensional Banach spaces, and involving the Caputo derivative in the generalized sense (via the Liouville-Riemann sense). We study the existence of solutions under both convexity and nonconvexity conditions on the multivalued side. Some examples of fractional differential inclusions on lattices are given to illustrate the obtained abstract results

A Bayesian Approach to Estimate the Size and Structure of the Broad-line Region in Active Galactic Nuclei Using Reverberation Mapping Data

This is the first paper in a series devoted to systematic study of the size and structure of the broad-line region (BLR) in active galactic nuclei (AGNs) using reverberation mapping (RM) data. We employ a recently developed Bayesian approach that statistically describes the variabibility as a damped random walk process and delineates the BLR structure using a flexible disk geometry that can account for a variety of shapes, including disks, rings, shells, and spheres. We allow for the possibility that the line emission may respond non-linearly to the continuum, and we detrend the light curves when there is clear evidence for secular variation. We use a Markov Chain Monte Carlo implementation based on Bayesian statistics to recover the parameters and uncertainties for the BLR model. The corresponding transfer function is obtained self-consistently. We tentatively constrain the virial factor used to estimate black hole masses; more accurate determinations will have to await velocity-resolved RM data. Application of our method to RM data with Hbeta monitoring for about 40 objects shows that the assumed BLR geometry can reproduce quite well the observed emission-line fluxes from the continuum light curves. We find that the Hbeta BLR sizes obtained from our method are on average ~20% larger than those derived from the traditional cross-correlation method. Nevertheless, we still find a tight BLR size-luminosity relation with a slope of alpha=0.55\pm0.03 and an intrinsic scatter of ~0.18 dex. In particular, we demonstrate that our approach yields appropriate BLR sizes for some objects (such as Mrk 142 and PG 2130+099) where traditional methods previously encountered difficulties.Comment: 17 pages, 10 figures, 2 tables; minor reversion to match the published versio

A note on the fractional Cauchy problems with nonlocal initial conditions

AbstractOf concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered