算子和右端都近似给定的第一类算子方程的广义Arcangeli方法

本文利用正则化方法解算子和右端都是近似给定的第一类算子方程,利用广义Arcangeli准则决定正则参数,给出正则解的收敛性和渐近收敛阶估计,以及算子为Fredholm积分算子时的正则解的一致收敛性。     

Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator

We study the Cauchy problem for an equation with singular Bessel operator. Unlike traditional methods to solve this problem, we apply Erde´ lyi–Kober fractional operator and find an explicit formula for the desired solution. We prove that the resulting formula is a unique classical solution to the problem.     

一类两相模型的导数估计

将一类两相模型作了一个推广．将模型中的Laplace算子换成P—Laplace算子后,研究了方程的解,得出解的一个导数估计．     

Spectral aspects of regularization of the Cauchy problem for a degenerate equation

We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.     

On the Index Formula for an Isometric Diffeomorphism

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah–Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold.     

Existence of solutions to nonlinear Neumann boundary value problems with <Emphasis Type="Italic">p</Emphasis>-Laplacian operator and iterative construction

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the p-Laplacian operator. We also construct an iterative scheme converging strongly to this solution.     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.     

Cauchy problem for evolution equations with infinite-order Bessel operators

We study the properties of an infinite-order Bessel operator with variable symbol and the structure of the fundamental solution of the Cauchy problem for the evolution equation with such an operator.     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

A new concept of a generalized solution of operator equations in a Banach space

We define a new notion of a generalized solution of an operator equation with a closed linear operator in a Banach space as an element of the completion of this space with respect to some locally convex topology. We prove a theorem on the existence and uniqueness of the generalized solution. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 93–99.     

Semiclassical asymptotic approximation of the two-dimensional Hartree operator spectrum near the upper boundaries of spectral clusters

We consider an eigenvalue problem for the two-dimensional Hartree operator with a small parameter at the nonlinearity. We obtain the asymptotic eigenvalues and the asymptotic eigenfunctions near the upper boundaries of the spectral clusters formed near the energy levels of the unperturbed operator and construct an asymptotic expansion around the circle where the solution is localized.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

On the concept of generalized solution of operator equations in banach spaces

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9. pp. 1286–1290, September, 1996.     

Non‐compactness of the solution operator to $ \bar \partial $ on the Fock‐space in several dimensions

We consider the canonical solution operator to restricted to (0, 1)‐forms with coefficients in the generalized Fock‐spaces (1) We will show that the canonical solution operator restricted to (0, 1)‐forms with ‐coefficients can be interpreted as a Hankel‐operator. Furthermore we will show that the canonical solution operator is not compact for m ≥ 2. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-point problem for a class of evolution equations: I

We study the properties of a pseudo-Bessel operator of infinite order constructed on the basis of a symbol nonsmooth at the point 0 and analyze the structure of the fundamental solution of a two-point problem for the equation with such an operator.     

Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.     

On the solvability of a complete two-dimensional hypersingular integral equation

We study the solvability of a complete two-dimensional linear hypersingular integral equation that contains a hypersingular integral operator in which the integral is understood in the sense of Hadamard finite value as well as an integral operator in which the integral is understood in the sense of principal value, an integral operator with a weakly singular kernel, and an integral-free term. We consider smooth solutions in the class of functions that have Hölder continuous derivatives outside a neighborhood of the boundary. We prove the Fredholm alternative and estimate the norm of the solution in a special metric.     

New approach to vacuum string field theory

We discuss the possibility of applying a purely ghost kinetic operator in open string field theory from the perspective of a modern analytic method based on the KBc subalgebra. A purely ghost kinetic operator is obtained as a result of gauge fixing string field theory around the identity-based tachyon vacuum solution. We show that the obtained kinetic operator is not equivalent to the midpoint insertion of a conformal ghost, to which an extensive literature is devoted. We also show that the equation of motion does not admit nontrivial solutions.     

An adiabatic theorem for section determinants of spectral projections

We derive an adiabatic‐type theorem that expresses the section determinants of spectral projections of a selfadjoint operator through the solution to an operator‐valued Wiener‐Hopf equation. The solution theory of this equation is developed and for a special case a concrete criterion that ensures uniqueness of the solution is presented. Furthermore, for a special class of operators a dichotomy criterion, which is used in the proof of the adiabatic theorem, is proved. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)