Vol 61, No 1 (2025)

The paper investigates the question of the existence of a classical solution to the initial value problem with incomplete initial data on the boundary of the strip for a hyperbolic differential-difference equation. The equation contains a superposition of a differential operator and a translation operator with respect to a spatial variable that varies along the entire real axis. Using the Gelfand–Shilov operational scheme, a solution to the problem was obtained in explicit form.

In this paper the boundary value problem (BVP) for diffusion equation with piecewise constant arguments is studied. By using the separation of variables method, the considered BVP is reduced to the investigation of the existence conditions of solutions of initial value problems for differential equation with piecewise constant arguments. Existence conditions of infinitely many solutions or emptiness for considered differential equation are established and explicit formula for these solutions are obtained. Several examples are given to illustrate the obtained results.

In this work, the first boundary value problem is studied for a two-dimensional wave equation in a cylindrical domain. A uniqueness criterion has been established. The solution is constructed as the sum of an orthogonal series. When justifying the convergence of a series, the problem of small denominators from two natural arguments arose for the first time. An estimate for separation from zero with the corresponding asymptotics was established, which made it possible to prove the convergence of the series in the class of regular solutions and the stability of the solution.

The work is devoted to the study of questions of existence and uniqueness of a continuous bounded and positive solution to one system of nonlinear multidimensional integral equations. The scalar analogue of the indicated system of integral equations, with different representations of the corresponding matrix kernel and nonlinearities, has important applied significance in a number of areas of physics and biology. This article proposes a special iterative approach for constructing a positive continuous and bounded solution to the system under study. It is possible to prove that the corresponding iterations uniformly converge to a continuous solution of the specified system. Using some a priori estimates for strictly concave functions, we also prove the uniqueness of the solution in a fairly wide subclass of continuous bounded and coordinately nonnegative vector functions. In the case when the integral of the matrix kernel has a unit spectral radius, it is proved that in a certain subclass of continuous bounded and coordinate-wise non-negative vector functions, this system has only a trivial solution, which is an eigenvector of the kernel integral matrix.

The problem of dynamic reconstruction of input actions in a system of ordinary differential equations and the problem of tracking a trajectory of a system by some trajectory of another one influenced by an unknown disturbance are under consideration. An input action is assumed to be an unbounded function, namely, an element of the space of square integrable functions. Two solving algorithms, which are stable with respect to informational noises and computational errors and oriented to program realization, are designed. Upper estimates of their convergence rates are established. The algorithms are based on constructions from feedback control theory. They operate under conditions of (inaccurate) measuring the phase states of the given systems at discrete times.

In this paper we solve the problem of assigning the desired characteristic polynomial of a linear stationary dynamic system with one input and output dynamic feedback in the form of a first-order dynamic compensator. Necessary and sufficient conditions for the existence of the solution of the problem are considered. An explicit formula for the compensator parameters, analogous to the Bass–Gura formula for a state feedback system, is derived.