On the solvability of the boundary-value problem for second-order equations in Hilbert space with an operator coefficient in the boundary condition

We consider the boundary-value problem on a finite interval for a class of second-order operator-differential equations with a linear operator in one of its boundary conditions. We obtain sufficient conditions for the regular solvability of the boundary-value problem under consideration; these conditions are expressed only in terms of its operator coefficients.     

Completeness of elementary solutions to a class of second order operator-differential equations

We present conditions of solvability of a boundary value problem for a class of second order operator-differential equations on a finite segment, study the behavior of the resolvent of the corresponding operator pencil, prove the double completeness of a system of the derived chains of eigenvectors and associated vectors corresponding to a boundary value problem on a segment, and establish the completeness of elementary solutions to the homogeneous equation in the solution space.     

Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation

We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1146–1152, August, 2006.     

Two-fold completeness of root vectors of a system of quadratic pencils

We consider a spectral problem for a system of second order (in the spectral parameter) abstract pencils in a Hilbert space and prove the completeness and the Abel basis property of a system of eigenvectors and associated vectors. In some special cases, we obtain the expansion of vectors with respect to eigenvectors. Further, it is considered a relevant application of these abstract results to boundary-value problems for second and fourth order ordinary differential equations with a quadratic spectral parameter both in the equation and in boundary-value conditions.     

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.     

On solutions of one class of second-order operator differential equations in the class of holomorphic vector functions

We establish sufficient conditions for the completeness of a part of root vectors of one class of the second-order operator bundles corresponding to the characteristic numbers from a certain sector and prove the theorem on completeness of a system of elementary holomorphic solutions of the corresponding second-order homogeneous operator differential equations. We also indicate the conditions of correct and unique solvability of a boundary-value problem for the analyzed equation with linear operator in the boundary condition and estimate the norm of the operator of the intermediate derivative in the perturbed part of the equation.     

On the well-posedness of a boundary value problem for a class of fourth-order operator-differential equations

We find sufficient coefficient conditions for the well-posed solvability of a boundary value problem for a class of fourth-order operator-differential equations with multiple characteristics. Furthermore, we indicate the sharp values of norms of operators of intermediate derivatives in a Sobolev-type space. In addition, for the corresponding polynomial operator pencil, we prove the completeness of the part of its eigenvectors and associated vectors corresponding to the eigenvalues in the left half-plane.     

On boundary value problem solvability theory for a class of high-order operator-differential equations

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.     

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.     

Equivalence of certain principal vectors of polynomial operator sheaves

The equivalence of derived chains constructed from the principal vectors of polynomial sheafs of operators acting in Hilbert space is studied. These derived chains correspond to different boundary-value problems on the semi-axis for operator-differential equations whose symbol is these operator sheafs. On the basis of equivalence tests assertions are deduced concerning the minimality of derived chains corresponding to a boundary-value problem on the semi-axis in the case in which the initial conditions of the vector solution at zero are known, and the solution itself obeys radiation-type conditions at infinity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 948–970, July, 1992.     

Normal solvability of boundary value problems for a class of fourth-order operator-differential equations in a weighted space

In a weighted space, we find sufficient conditions for the normal solvability of a boundary value problem for a class of fourth-order operator-differential equations whose leading part has multiple characteristics. Note that these conditions are expressed in terms of properties of operator coefficients of the equation considered.     

What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

Spectral properties of boundary-value problem with a shift for wave equation

We consider a differential operator determined by wave equation with potential in characteristic triangle, and boundary-value conditions with shift on the characteristics, and with oblique derivative on non-characteristic part of a boundary. We obtain condition for validity of the Volterra property, and show completeness of the root functions in the rest cases. We study basis property for the system of root functions under assumption that the potential depends on a single variable.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

A class of evolution equations: Existence of solutions with functional boundary conditions

We consider the evolution equation whose right-hand side is the sum of a linear unbounded operator generating a compact strongly continuous semigroup and a continuous operator acting in function spaces. We prove the existence of a solution that stays within a given closed convex set and moreover, satisfies a functional boundary condition, particular cases of which are the Cauchy initial condition, periodicity condition, mixed condition including continuous transformations of spatial variables, etc. The main result is illustrated by using an example of the boundary-value problem for a partial operator-differential equation. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 48–60, January, 1999.     

On a solution of a boundary-value problem for a hyperbolic equation on a disk with random initial conditions

We establish conditions under which a solution of a boundary-value problem for a hyperbolic equation on a disk with random initial conditions can be represented as a series uniformly convergent with probability one.     

On one boundary-value problem for an equation of higher even order

We study boundary-value problem for an equation of even order in a rectangular domain. By the spectralmethod we obtain necessary and sufficient conditions of uniqueness of a solution. The solution is constructed in the formof infinite series in eigenfunctions. We obtain sufficient conditions under which this series is a regular solution.     

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.     

On Differential Operators with Singularity and Discontinuity Conditions inside an Interval

We investigate boundary-value problems for differential equations with singularity and discontinuity conditions inside an interval. We describe properties of the spectrum, prove a theorem on the completeness of eigenfunctions and associated functions, and study the inverse spectral problem.     

On a weighted boundary-value problem in an infinite half-strip for a biaxisymmetric Helmholtz equation

We study a boundary-value problem for a generalized biaxisymmetric Helmholtz equation. Boundary conditions in this problem depend on equation parameters. By the method of separation of variables, using the Fourier-Bessel series expansion and the Hankel transform, we prove the unique solvability of the problem and establish explicit formulas for its solution.