V. I. Ul’yanova

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.

General Nonlinear Self-Adjoint Eigenvalue Problem for Systems of Ordinary Differential Equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.

Nonlinear Eigenvalue Problem for Second-Order Hamiltonian Systems

The nonlinear self-adjoint eigenvalue problem for a Hamiltonian system of two ordinary differential equations is examined under the assumption that the matrix of the system is a monotone function of the spectral parameter. Certain properties of eigenvalues that were previously established by the authors for Hamitonian systems of arbitrary order are now worked out in detail and made more precise for the above system. In particular, a single second-order ordinary differential equation is analyzed.

On the general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations with singularities

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.

On the Index of the Boundary Value Problem for a Homogeneous Hamiltonian System of Differential Equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.

Determination of the Number of an Eigenvalue of a Singular Nonlinear Self-Adjoint Spectral Problem for a Linear Hamiltonian System of Differential Equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.