On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

基于统一灰色生成算子的三参数离散灰色预测模型及其应用

现有三参数离散灰色预测模型的累加阶数取值范围局限于正实数,导致模型建模能力和作用空间受限。为此,论文首先引入实数域统一灰色生成算子;然后,基于统一灰色生成算子构造了新型三参数离散灰色预测模型,实现了其阶数从正实数到全体实数的拓展与优化,从而使得新型模型具备挖掘时序数据积分特性与差异信息的双重功能;最后,将该新模型应用于某装甲装备维修经费的建模,结果显示其精度优于其它同类灰色模型。本研究成果对完善灰色算子基础理论及提高灰色预测模型建模能力具有重要价值。     

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.     

分块算子矩阵的加权EP

本文在加权广义Schur补的基础上, 引入并研究了Hilbert空间上分块算子矩阵的加权Moore-Penrose逆和加权EP. 进一步, 给出了加权EP算子在算子方程中的一个应用.     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.     

Products and transforms of white-noise functionals (in general setting)

Some sharp results about Weiner and Wick products of whitenoise functionals are obtained. Using the inequality of Wick products we show to what extent scaling transformations, translations, and Sobolev differentiations can be performed on white-noise functionals.This work was supported by the National Natural Science Foundation of China. This paper is an enlargement and revised version of the paper entitled Products and Transforms of White-Noise Functionals (preprint, 1990).     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

ON THE ESSENTIAL SPECTRUM OF PSEUDODIFFERENTIAL OPERATORS （Ⅱ）

This paper coutinues the studies of the essential spectrum of nonsemi-bounded pseudodifferential operators. The author improves the results in [5] in some sense. For the relativistic Schrodinger operator,√(-Δ+m^2)+v(x),complete results are obtained.     

关于B-凸空间及其上黎斯算子West分解

给出 Ｂａｎａｃｈ空间列｛Ｘｉ｝ｉ＝１∞的 ｌｐ乘积Ｂ－凸的特征刻划, 证明Ｂ－凸空间上的每个黎斯算子可Ｗｅｓｔ分解,即分解成一个紧算子和一个拟幂 零算子的和．