Univalent polynomials and fractional order differences of their coefficients

Using fractional order differences, we find sufficient conditions on the coefficients of certain polynomials defined on the unit disk such that the zeros of these polynomials does not lie in D.     

Existence results for fractional order functional differential equations with infinite delay

The Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.     

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution \({\mu _ \in } \in C(\bar \Omega )\) of the Dirichlet problem

$$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$

where Ω is a bounded, open set and \({f_ \in } \in C(\bar \Omega )\) for all ? ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator \(\mathcal{I}_{\in}\) is explicitly given by

$${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$

which is an approximation of the well-known fractional Laplacian of order σ, as ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.

     

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.     

Monotonicity of order preserving operator functions

We discuss monotonicity of order preserving operator functions and related order preserving operator inequalities.LetA?B?0withA>0,t∈[0,1]andp?1. Let

     

Monotonicity of the mean order of subtrees

This paper is concerned with the average number of nodes in certain families of subtrees of a tree. It is shown that this average increases when the underlying tree is enlarged and decreases if the family itself is enlarged within the same underlying tree.     

The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces

We continue the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schr?dinger operators and their exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order ${\vartheta}$ for ${\vartheta < 3/4}$ . The bound 3/4 is best possible and can neither be reached nor surpassed. Such results are important for the study of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons.     

Monotonicity and supermodularity results for the Erlang loss system

For the Erlang loss system with s servers and offered load a, we show that: (i) the load carried by the last server is strictly increasing in a; (ii) the carried load of the whole system is strictly supermodular on {(s,a)|s=0,1,… and a>0}.     

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.     

Second order duality for minmax fractional programming

In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The concept of η-bonvexity/generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems. The research of Z. Husain is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral Fellowship Program No. 40/9/2005-R&D II/1739.     

Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

In this paper, the existence of solutions of an anti-periodic fractional boundary value problem for nonlinear fractional differential equations is investigated. The contraction mapping principle and Leray-Schauder’s fixed point theorem are applied to establish the results.