Vol 230 (2023)
Статьи
On the existence and uniqueness of a positive solution to a boundary-value problem for one nonlinear fractional functional differential equation
Abstract
In this paper, using the Krasnoselsky fixed-point theorem, we establish sufficient conditions for the existence of a positive solution to the boundary-value problem for one nonlinear fractional functional differential equation. To prove the uniqueness of a positive solution, we use the Banach fixed-point theorem. The results presented continue the author’s research on this topic.
3-7
Inequalities for the best “angular” approximation and the smoothness modulus of a function in the Lorentz space
Abstract
In this paper, we consider the Lorentz space of -periodic functions of several variables, the best “angular” approximation of such functions by trigonometric polynomials, and the mixed smoothness modulus of functions from this space. The properties of the mixed smoothness modulus are given and strengthened versions of the direct and inverse theorems on the “angular” approximations are proved.
8-24
Optimal boundary control for a distributed inhomogeneous oscillatory system with given intermediate conditions
Abstract
In this paper, we develop a constructive approach to the problem of optimal boundary control for a distributed inhomogeneous oscillatory system whose dynamics is modeled by a one-dimensional wave equation with piecewise constant characteristics. Using the approach proposed, one may satisfy multi-point intermediate conditions. The results obtained are illustrated by a specific example.
25-40
On the algebra of integral operators with involution
Abstract
In this paper, we consider integral operators with kernels depending on the sum and difference of arguments in the space , . We prove that such operators form a subalgebra of the algebra of bounded linear operators. The study of operators with kernels depending on the difference of arguments was carried out using Banach -modules. The differences and similarities between the subalgebra of integral operators and the corresponding subalgebra of difference operators with involution are noted.
41-49
Uniqueness theorem for one class of pseudodifferential equations
Abstract
The uniqueness of solutions for homogeneous equations in the class of analytic functionals with pseudodifferential operators commuting under shifts is discussed. We establish conditions for the symbols of operators that allow one to partition this class of operators into equivalence classes in such a way that within each class, any condition of the regularity of solutions at infinity that guarantees the uniqueness of a solution for an equation with some representative of this class, also guarantees the uniqueness of a solution for equations with all representatives of the same class.
50-55
On the proximate growth function relative to the model growth function
Abstract
The concept of proximate order is widely used in the theories of integer, meromorphic, subharmonic, and plurisubharmonic functions. In this paper, we provide a general interpretation of this concept as a proximate growth function relative to the model growth function. The classical proximate order is the proximate order in the sense of Valiron. Our definition uses only one condition. This form of definition is new for the classical proximate order. In this review, we show that for any function defined on a positive ray whose growth is determined by a model growth function, there is a proximate growth function relative to the model growth function.
56-74
The influence of delay and spatial factors on the dynamics of solutions in the mathematical model “supply-demand”
Abstract
A generalized version of the macroeconomic model “supply-demand” is considered. The main version of this model possesses a single attractor, namely, the state of economic equilibrium. We analyze a nonlinear boundary-value problem for a partial differential equation with delay on the right-hand side. The analysis of solutions in a neighborhood of the equilibrium state is reduced to the study of local bifurcations of the complex Ginzburg–Landau equation. For the basic boundary-value problem, the existence of cycles is proved, including cycles depending on the spatial variable.
75-87
Optimal control of external loads in the equilibrium problem for a composite body contacting with a rigid inclusion with a sharp edge
Abstract
In this paper, we consider a nonclassical mathematical model that describes the mechanical point contact of a composite body with an obstacle of special geometry. The nonlinearity of the model is due to inequality-type conditions within the framework of the corresponding variational problem. An optimal control problem is formulated in which the controls are functions of external loads, and the cost functional is specified using a weakly upper semi-continuous functional defined on the Sobolev space. The solvability of the optimal control problem is proved. For the sequence of solutions corresponding to the maximizing sequence, the strong convergence in the corresponding Sobolev space is proved.
88-95
Tensor invariants of geodesic, potential and dissipative systems. IV. Systems on tangents bundles of n-dimensional manifolds
Abstract
In this paper, we present tensor invariants (first integrals and differential forms) for dynamical systems on the tangent bundles of smooth -dimensional manifolds separately for , , , , and for any finite . We demonstrate the connection between the existence of these invariants and the presence of a full set of first integrals that are necessary for integrating geodesic, potential, and dissipative systems. The force fields acting in systems considered make them dissipative (with alternating dissipation).
The first part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 227 (2023), pp. 100–128. The second part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 228 (2023), pp. 92–118. The third part of the paper: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 229 (2023), pp. 90–119.
96-130