Time-band limiting matrices and lamé's equation

The continuity and differentiability for integral operator in Banach spaces are proved with respect to L^p–topology. The weak normal integrand is defined, so that a generalized measurable selection theorem is derived, and the conjugate functional for integral functional is formulated. The duality theorem of optimization problem for integral functional is established by using the conjugate functional.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Signal Processing in Relaxation Experiments

Signal processing problems arising in the study of the linearly viscoelastic behavior of polymers and composites are considered. It is shown that the great amount of data conversions is associated with integral transforms using kernels which depend on the ratio or product of arguments for monotonic long-time-interval and wide-frequency-band functions (signals). A unified method of carrying out these integral transforms is developed by combining a logarithmic transformation of the signal time scale with digital filtering. For integral transforms leading to ill-conditioned inverse problems, a method of regularization is proposed based on choosing a sampling rate which ensures an acceptable error variance of the output signal. The specific features of the functional filters used for performing the functional (integral) transforms are discussed. Examples of performing the Heaviside-Carson sine transform and an inherently ill-conditioned problem of inverting the integral transform for determining the relaxation spectrum are represented by digital functional filters.     

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

A new approach to the theory of functional integral equations of fractional order

The aim of the paper is to present a new approach to the theory of functional integral equations of fractional order. That approach depends on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type. It turns out that the study of functional integral equations of Volterra-Stieltjes type is more convenient and effective than the study of functional integral equations of fractional order. An example illustrating our approach is also discussed.     

Classical solutions of hyperbolic functional differential systems

The paper deals with a generalized Cauchy problem for quasi-linear hyperbolic functional differential systems. The unknown function is the functional variable in the system of equations and the partial derivatives appear in the classical sense. A theorem on the local existence of a solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. A method of bicharacteristics and integral inequalites are applied.     

Diffraction by a wedge at skew incidence: integral representations of Cauchy-Carleman for the electromagnetic fields

An electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional difference equations by means of Sommerfeld integrals and Malyuzhinets theorem. By introducing an integral operator it is shown that the solutions of this system of functional equations can be defined in terms of integral representations whose kernels are solutions of a singular integral equation of Cauchy-Carleman type for which an explicit solution is given.     

函数列的黎曼积分极限定理的应用

利用函数列的极限理论方法,研究函数列积分极限中积分和极限可交换次序的问题.对一致收敛的可积函数列给出积分的极限定理,对一致有界局部一致收敛函数列给出积分控制收敛定理,通过大量实例表明该理论的意义所在.     

On parabolic distributed optimal control problems with restrictions on the gradient

Convex optimal control problems for parabolic distributed parameter systems with an integral constraint for the gradient of the state are considered. The objective function is a general convex integral functional in which the control constraints are incorporated. Necessary and sufficient conditions for optimality are derived.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

Conditions for Choquet integral representation of the comonotonically additive and monotone functional

If the universal set X is not compact but locally compact, a comonotonically additive and monotone functional (for short c.m.) on the class of continuous functions with compact support is not represented by one Choquet integral, but represented by the difference of two Choquet integrals. The conditions for which a c.m. functional can be represented by one Choquet integral are discussed.     

Existence of solutions to Hamilton-Jacobi functional differential equations

This paper deals with the Cauchy problem for nonlinear first order partial functional differential equations. The unknown function is the functional variable in the equation and the partial derivatives appear in a classical sense. A theorem on the local existence of a generalized solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. A method of bicharacteristics and integral inequalities are applied.     

Stieltjes integral inequalities of Gronwall type and applications

Summary We obtain estimates for solutions of integral inequalities of Gronwall type involving Stieltjes integrals and their inverse inequalities. From these we obtain some new results for integral inequalities for Riemann integrals and functional integral inequalities. Extensions are also given to Bihari type integral inequalities.Research supported by NSERC Canada.     

F-Willmore曲面的间隙现象

对于空间形式中的2维曲面,定义了F-Willmore泛函,此泛函包括经典的Willmore泛函作为特殊情形.F-Willmore泛函的临界点称为F-Willmore曲面.推导了第1变分公式并由此构造了F-Willmore曲面的典型例子.利用自伴算子作用于特殊的实验函数,得到了Simons类积分不等式,讨论了F-Willmore曲面的间隙现象,定出了间隙端点对应的特殊曲面.     

Smooth functional derivatives in Feynman path integrals by time slicing approximation

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].     

Introduction to the improved functional epsilon algorithm

This paper introduces the improved functional epsilon algorithm. We have defined this new method in principle of the modified Aitken Δ² algorithm. Moreover, we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral Padé approximant. The use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated. The relationship of the improved functional epsilon algorithm with the integral Padé approximant is also demonstrated. Moreover, we illustrate the similarity between the integral Padé approximant and the modified Aitken Δ² algorithm; thus we have shown that the integral Padé approximant is a natural generalisation of modified Aitken Δ² algorithm.     

Infinite Systems of Hyperbolic Functional Differential Equations

We consider initial-value problems for infinite systems of first-order partial functional differential equations. The unknown function is the functional argument in equations and the partial derivations appear in the classical sense. A theorem on the existence of a solution and its continuous dependence upon initial data is proved. The Cauchy problem is transformed into a system of functional integral equations. The existence of a solution of this system is proved by using integral inequalities and the iterative method. Infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators.     

Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations

In this article, we investigate a class of stochastic neutral partial functional differential equations. By establishing new integral inequalities, the attracting and quasi-invariant sets of stochastic neutral partial functional differential equations are obtained. The results in [15, 16] are generalized and improved.     

The attractivity of functional hereditary integral equations

In this paper, the problem of attractivity of solutions for functional hereditary integral equations is initiated. A hereditary integral equation is converted into a second kind Volterra integral equation of convolution type by use of the property of convolution. Sufficient conditions for the existence of attractive solutions are established according to the fixed point theorem. The conclusions presented in this paper extend some published results. The effectiveness of the proposed results is illustrated by three numerical examples.     

On the boundary integral equations for the crack opening displacement of flat cracks

The boundary integral equations for the crack opening displacement in acoustic and elastic scattering problems are discussed in the case of flat cracks by means of the Fourier analysis technique. The pseudo-differential nature of the hypersingular integral operators is shown and their symbols explicited. It is then proved that the variational problems assocaited with these BIE are well-posed in a Sobolev functional framework which is closely linked with the elastic energy. A decomposition of the vector integral equation in the elastic case into scalar integral equations is obtained as a by-product of the variational formulation.