Vol 60, No 3 (2024)

The Sturm–Liouville operator with a singular potential on an interval with conjugation conditions at the interior point of the interval is considered. The operator potential may have a non-integrable singularity. For a strong solution of the Cauchy problem for an equation with a parameter, asymptotic formulas and estimates are obtained on each of the smoothness segments of the solution.

In this paper is investigated the solvability of a periodic problem for a system of nonlinear ordinary differential equations second order with the main positively homogeneous part. New conditions have been found that provide an a priori estimate solutions of the periodic problem under consideration. The conditions for the a priori estimate are formulated in terms of the properties of the main positively homogeneous part of the system of equations. Under the conditions of an a priori estimate, using and developing methods for calculating the mapping degree, a theorem on the solvability of the periodic problem is proven. The proven theorem generalizes previously obtained the authors’ results on the study of a periodic problem for systems of nonlinear ordinary differential equations of the second order.

In the Banach space for functional-differential equation, generalizing the Euler–Poisson–Darboux equation, the Cauchy problem and boundary value problems of Dirichlet and Neumann are consider. A sufficient condition for solvability is proved Cauchy problem and an explicit form of the resolving operator is indicated, which is written using the ones introduced by the author Bessel and Struve operator functions. For boundary value problems in the hyperbolic case, sufficient conditions for their unique solvability imposed on the operator coefficient of the equation and boundary elements.

A system of equations modeling the dynamic stretching of a homogeneous circular layer of incompressible ideally rigid-plastic transversely isotropic material obeying the Mises–Hencky criterion is studied. The upper and lower bases are stress-free, the radial velocity is set at the lateral boundary, and the possibility of thickening or thinning of the layer is taken into account, which simulates neck formation and further development of the neck. Using the method of asymptotic integration, two characteristic stretching modes are identified, that is, the relations of dimensionless parameters are determined, in which consideration of inertial terms is necessary. When considering the regime associated with the achievement of acceleration on the side face of its critical values, an approximate solution of the problem was constructed.

For a Cauchy problem associated with a controlled semilinear evolutionary equation with optionally bounded operator in a Hilbert space we obtain sufficient conditions for exact controllability to a given final state (and also to given intermediate states at intermediate time moments) on a arbitrarily fixed (without additional conditions) time interval. Here we use the Minty–Browder’s theorem and also a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples we consider a semilinear pseudoparabolic equation and a semilinear wave equation.