Том 54, № 4 (2018)

We consider the spectral problem y'''(x) = λy(x) with general two-point boundary conditions that do not contain the spectral parameter λ. We prove that the boundary conditions in this problem are degenerate if and only if their 3 × 6 coefficient matrix can be reduced by a linear row transformation to a matrix consisting of two diagonal 3 × 3 matrices one of which is the identity matrix and the diagonal entries of the other are all cubic roots of some number. Further, the characteristic determinant of the problem is identically zero if and only if that number is −1. We also show that the problem in question cannot have finite spectrum.

We study a fourth-order differential operator with matrix coefficients whose domain is determined by the Dirichlet boundary conditions. An asymptotics of the weighted average of the eigenvalues of this operator is obtained in the general case. As a consequence of this result, we present the asymptotics of the eigenvalues in several special cases. The obtained results significantly improve the already known asymptotic formulas for the eigenvalues of a one-dimensional fourth-order differential operator.

We consider the traces on submanifolds of G-operators generated by pseudodifferential operators and operators of shift along the orbits of a discrete group G. Such operators arise in various problems in differential equations and mathematical physics, for example, in Sobolev problems. We show that the trace of a G-operator on a submanifold is the sum of a pseudodifferential operator on the submanifold and a G-operator concentrated on a sub-submanifold.

We show that the oblique derivative problem for the Helmholtz equation in a disk is uniquely solvable under certain restrictions on the parameter.

Based on previously stated approaches, we propose a method for solving point-topoint steering problems in the case where a 2n-parametric family of solutions of a nonlinear system is known and for Liouville systems. Two examples of the helicopter motions in the vertical plane and the Kapitsa pendulum are considered to demonstrate the efficiency of the proposed method.

We consider the construction of the interval Taylor model used to prove the existence of periodic trajectories in systems of ordinary differential equations. Our model differs from the ones available in the literature in the method for describing the algorithms for the computation of arithmetic operations over Taylor models. In the framework of the current model, this permits reducing the computational expenditures for obtaining interval estimates on computers. We prove an assertion that permits establishing the existence of a periodic solution of a system of ordinary differential equations by verifying the convergence of the Picard iterations in the sense of embedding of the proposed Taylor models. An example illustrating how the resulting assertion can be used to prove the existence of a closed trajectory in the van der Pol system is given.

We describe the Krein extension of the minimal operator associated with the expression A:= (−1)ⁿd²ⁿ/dx²ⁿ on the interval [a, b] in terms of boundary conditions. We also describe all nonnegative extensions of the operator A and extensions with finitely many negative squares.

We prove that the Dirichlet problem for a mixed-type equation of the second kind with an integer negative coefficient has a solution in a rectangular domain. The corresponding estimates are constructed and used to determine sufficient conditions for the problem to have a solution. The cases of nonuniqueness of the solution are distinguished, and solvability conditions in these cases are obtained.