Vol 61, No 2 (2025)

An ordinary second-order differential equation with positive parameter and discontinuous right-hand side which changes its sign at the point of the jump is considered. Various boundary value problems for it are formulated, including mixed and periodic boundary conditions. Theorems on existence of periodic solutions of the studied boundary value problems are established. The obtained results are illustrated by examples.

For the Sturm–Liouville operator on the half-axis with a complex decreasing potential that allows analytical continuation to some neighborhood of zero, an analogue of Ambarzumyan’s theorem is obtained.

For the system of ordinary differential equations of the second order with the main positively homogeneous nonlinearity, an a priori estimate of periodic solutions of a fixed period is investigated. New conditions of a priori estimate are found, in which the influence of the properties of the main nonlinear part, including its set of zeros, is mediated by functional estimates from above and below. The feasibility of the new conditions is investigated for three types of nonlinearities.

The asymptotic behavior for large values of the independent variable of the fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of arbitrary odd order is investigated, depending on the coefficients of the highest derivative and the free term.

The singular ultrahyperbolic equation (Δ𝐵𝛽)𝑦𝑢= (Δ𝐵𝛾)𝑥𝑢 is considered under the assumption that the I.A. Kipriyanov condition is satisfied, where fractional dimensions of the Δ𝐵𝛾 -operators included in the equation are equal to the same positive number 𝜎. Three types of solutions to the radial Cauchy problem are studied, one of them is based on the T-pseudoshift operator, generalized T-shift and S.A. Tersenov’s method for determining solutions to equations that degenerate on the boundary. Poisson formulas for solving the Cauchy problem for the Euler–Poisson–Darboux equation are given for various values of the parameters in this equation.

The paper considers the numerical solution of the Cauchy problem for systems of ordinary differential equations. Special attention is paid to problems with limiting singular points on integral curves. It is known that traditional explicit methods for solving the Cauchy problem are ineffective for this class of problems. Implicit methods are much more difficult to use and do not always lead to the result of the desired accuracy. Therefore, along with traditional methods of numerical integration of the Cauchy problem authors use the method of solution continuation with respect to the best argument (also known as the best parameterization and the arc length method). The best argument is calculated tangentially along the integral curve of the problem under consideration. For the Cauchy problems transformed to the best argument, the authors in this paper present the results of a study of the local error for the numerical solution obtained by the explicit Euler method. An estimate of the numerical solution local error of the numerical solution for the Cauchy problem transformed to the best argument is obtained for the explicit Euler method. Using it, an upper estimate of the local error was obtained and the effectiveness of using the best argument was proved. This is reflected in a decrease of the solution local error for the transformed problem in the neighborhood of the limiting singular points. The theoretical results are compatible with the numerical solution of the ill-conditioned initial value problem of deformable solid mechanics with one limiting singular point.

The question of the existence of a positive solution to a two-point boundary value problem with homogeneous almost symmetric boundary conditions for one nonlinear fourth-order ordinary differential equation is investigated.