Volume 61, Nº 5 (2025)

Various qualitative properties of a differential system related to the behavior of its solutions starting near zero are considered: stability and asymptotic stability, complete oscillation, wandering, and rotation, as well as complete negations of each of these properties. Logical connections of their various radial and general radial varieties are studied both with each other and with the corresponding complete properties, as well as with the measures of these properties.

The classical solution of the first mixed problem for the wave equation in the cylindrical area in the space with odd number of dimensions is considered. The solution is constructed using the spherical averages operators which allows the reduction of the equation to the first mixed problem for the one-dimensional wave equation. Using the characteristics method the explicit solution is obtained and necessary and sufficient matching conditions for the existance of the unique classical solution are proved.

For a controlled nonlinear affine system, a piecewise linear approximation in the form of a switched affine system is considered. The concept of consistency of a piecewise linear approximation is introduced when the system is closed by a variable structure controller. The consistency of the approximation ensures the equality of the graphs of discrete states of the nonlinear system itself and its piecewise linear approximation. A sufficient condition for the consistency of the approximation is obtained and an approach to numerical verification of its fulfillment is proposed.

In the paper, functional identities for the difference between a given function and exact solution of the generalized Stokes problem are derived. The restrictions on the form of such a function are minimal. Actually, they are reduced to the requirement that it must belong to the same functional class as the solution of the problem. The left part of the identity represents a weighted sum of norms and characterizes deviations from the exact velocity and stresses fields. The right part includes a number of summands. Some of them can be directly computed from the problem data and known approximate solutions. Other terms contain unknown functions but can be efficiently estimated. Hence the identity is a basis for getting fully computable two-sided bounds of the distance to the solution of the problem. The identities and the estimates derived from them can be used to estimate errors of approximations generated by various numerical methods. They are true both for solenoidal approximations, as well as for those that satisfy the incompressibility condition only with a certain degree of accuracy. In addition, identities and estimates make it possible to compare exact solutions to problems with different data. Therefore, they open a way to evaluate errors of mathematical models, e.g., those that arise after changing (simplification) of the differential equation or due to replacing the incompressibility condition by weaker conditions.