Ascoli-Arzela定理的一个推广及应用

给出了无穷区间上一类抽象连续函数族相对紧性判断的一个充分必要条件，并应用它获得了抽象空间上一阶微分方程的终值问题和Ｆｒｅｄｈｏｌｍ型积分方程解的存在性。     

无穷区间上一类抽象函数族相对紧的判定及其应用

给出了无穷区间上一类抽象连续可微函数族相对紧性判定的一个充要条件,并应用它获得了二阶微分方程终值问题解的存在性.     

无穷区间上二阶微分方程的边值问题

利用Schauder不动点定理讨论了一类非线性二阶微分方程在无穷区间上的边值问题无界解的存在性，部分改进了郭大钧教授最近得到的结果。     

Banach空间二阶脉冲积微分方程终值问题

利用上下解方法讨论了抽象空间中定义在无穷区间上且有无穷个脉冲点的一类二阶脉冲积微分方程终值问题,获得了其最小最大解的存在性.     

瑕积分中易出现的错误

<正> 积分学告诉我们,被积函数在有限区间[a,b]上出现无穷型间断点,这样的积分为无界函数的广义积分,又名瑕积分。该无穷型间断点,称为瑕积分的瑕点(或叫奇点)。     

R~N上拟线性椭圆方程的无穷多解

运用喷泉定理研究了无界区域上拟线性椭圆方程无穷多解的存在性,推广了一些结果.     

时变观测算子族的无限时容许性

对Hilbert空间中的无界时变观测算子族引入无限时容许性概念,并利用发展半群理论,给出无界时变观测算子族的无限时容许性刻画.最后,对一类具混合边界条件的非自治系统,构造了一类无限时容许观测算子族.     

在无界区域上拟线性散度型椭圆型方程的Dirichlet问题

研究在无界区域上的二阶拟线性散度型椭圆型方程Dirichlet问题在无穷远处径向收敛的古典解存在性和唯一性。     

收敛且等值的无界函数与无穷区间广积分的积分变量间关系

本讨论收敛且等值的无界函数与无穷区间广义积分的积分变量间关系，提出并证明了在一定的条件下，这两个变量间存在唯一单调增加和唯一单调减少的函数关系。     

Banach空间非线性脉冲Volterra积分方程组的整体解

研究Banach空间中定义在无穷区间R+上具有无穷多个脉冲点的非线性脉冲Volterra积分方程组解的存在性。给出了若干极值解的存在定理,改进了定义在有限区间上具有有限个脉冲点情形时该类方程的相应结果,并利用该结果讨论了一个无穷维积分方程组。     

Utility maximization problem in the case of unbounded endowment

We consider a problem of expected utility maximization with an utility function finite on ?₊ and with an unbounded random endowment in an abstract model of financial market. We formulate a dual problem to the primal one and prove duality relations between them. In addition, we study necessary conditions to the existence of solutions to the primal problem. Finally, we reduce the dual problem to a form more convenient for practice.     

Approximate solutions of abstract differential equations

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Asymptotically exponential solutions in nonlinear integral and differential equations

In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age-dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results.     

Iterative solution of linear equations with unbounded operators

A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space, including equations with unbounded, closed, densely defined linear operators. The method is proved to be stable towards small perturbation of the data. Some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators. The dynamical systems method (DSM) is justified for unbounded, closed, densely defined linear operators. The stopping time is chosen by a discrepancy principle. Equations with selfadjoint operators are considered separately. Numerical examples, illustrating the efficiency of the proposed method, are given.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

非自反Banach空间无界控制算子的容许性

本文利用共轭Ｃ_０半群的扰动理论研究了无界容许控制算子，在太阳自反和非太阳自反Ｂａｎａｃｈ空间分别导出了一些容许性判据，并把这些抽象结果应用到了有限和无限延滞方程．     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.     

Correct Solvability of Integrodifferential Equations in Spaces of Vector Functions Holomorphic in an Angular Domain

Doklady Mathematics - Integrodifferential equations with unbounded operator coefficients in a Hilbert space are studied. The main part of an equation of this kind is an abstract parabolic equation...