Vol 59, No 2 (2023)
Articles
Viktor Valentinovich Vlasov (18.11.1956-04.01.2023)
Abstract
On One Type of Oscillating Solutions of a Second-Order Ordinary Differential Equation with a Three-Position Hysteresis Relay and a Perturbation
Abstract
A second-order ordinary differential equation with a three-position hysteresis relay characteristic and a periodic perturbation function is considered. The existence theorem is proved for an oscillatory solution with a complete traversal of the characteristic with a possible exit into its saturation zones in some finite time and with a closed phase trajectory of an arbitrary shape. Sufficient conditions for the existence of periodic solutions with arbitrary and symmetric phase trajectories are established, as well as conditions for the nonexistence of a periodic solution with a symmetric phase trajectory. Numerical examples are given.
On the Spectral Properties of a Fourth-Order Self-Adjoint Operator
Abstract
We consider a spectral problem for a fourth-order differential operator with real periodic coefficients and Neumann-type boundary conditions. For this operator, the eigenvalue asymptotics and a regularized trace formula are obtained.
On the Gröbner Basis of the Ideal of Lyapunov Quantities of the Kukles System
Abstract
The center problem and the cyclicity of singular points of the Kukles system are studied. The necessary conditions for the center at the origin are obtained as the variety of the ideal of Lyapunov quantities calculated by direct solution of the polynomial system whose left-hand sides form the Gröbner basis of the ideal. This ideal is also used to calculate the cyclicity of the centers and foci of the system. A theorem is proved that allows one to find the cyclicity of the centers of polynomial systems by using its Gröbner basis instead of the ideal of Lyapunov quantities.
Boundary Value Problem for an Inhomogeneous Fourth-Order Equation with Lower-Order Terms
Abstract
We consider the first boundary value problem in a rectangular domain for an inhomogeneous fourth-order differential equation with lower-order terms. The uniqueness of a solution of the stated problem is proved. The solution is obtained explicitly using the Green’s function constructed.
On the Fundamental Solution of a Parabolic Equation with Dini Continuous Coefficients
Abstract
We consider a parabolic equation in one spatial variable with Dini continuous coefficients. For this equation, the existence of a classical fundamental solution is proved and estimates are given. The condition on the nature of the continuity of the leading coefficient of the equation for the existence of a fundamental solution is sharp.
Finding the Two-Dimensional Relaxation Kernel of an Integro-Differential Wave Equation
Abstract
We consider the multidimensional inverse problem of determining the kernel of the integral term in an integro-differential wave equation. In the direct problem, it is required to find the displacement function from an initial–boundary value problem, and in the inverse one, to determine the kernel of the integral term depending on both time and one of the spatial variables. The local unique solvability of the problem in the class of functions continuous in one of the variables and analytic in the other one is proved on the basis of the method of scales of Banach spaces of real analytic functions.
Well-Posedness of the Generalized Samarskii–Ionkin Problem for Elliptic Equations in a Cylindrical Domain
Abstract
We study the well-posedness of some analogs of the nonlocal Samarskii–Ionkin problem for second-order elliptic equations in Sobolev spaces. For the problems in question, existence and uniqueness theorems are proved for regular solutions, i.e., solutions that have all generalized Sobolev derivatives occurring in the corresponding equation. Some spectral problems for elliptic equations with the nonlocal Samarskii–Ionkin condition are studied.
Stability Criterion and Sharp Estimates for the “Super-Twisting” Algorithm
Abstract
A new method for proving necessary and sufficient conditions for the global asymptotic stability of the “super-twisting” algorithm is given. The new method is based on obtaining a complete analytical solution of the system for the “worst-case” perturbation and permits one to obtain a criterion in a simpler, completely real form as well as to find estimates for the worst-case (majorizing) trajectory.
On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator
Abstract
For the Cauchy problem associated with a controlled semilinear evolution equation with an unbounded maximal monotone operator in a Hilbert space, sufficient conditions are obtained for exact controllability to a given final state. Here a generalization of the Browder–Minty theorem and results on the total global solvability of this equation obtained by the author earlier are used. As an example, a semilinear wave equation is considered.
On the Bifurcation of Thresholds of the Essential Spectrum with a Spectral Singularity
Abstract
We consider the Schrödinger operator on the plane with bounded potential, where is a real potential, and are compactly supported complex potentials, and
is a small parameter, assuming that the lower part of the spectrum of the one-dimensional Schrödinger operator consists of a pair of isolated eigenvalues and the essential spectrum of the operator has a virtual level at its lower edge and a spectral singularity inside.
Additionally, we assume that there is a certain superposition of eigenvalues of the operator with the virtual level and spectral singularity of the operator; this leads to the emergence of a special threshold in the essential spectrum of the perturbed operator, with the perturbation leading to a bifurcation of this threshold into eigenvalues and resonances with multiplicity doubling. The bifurcation scenario described in this paper is qualitatively different from the previously known ones.
Spectral Properties of the Generator of a Semigroup Generated by the Volterra Integro-Differential Equation
Abstract
The spectral properties of a linear operator that is the generator of a semigroup generated by a Volterra integro-differential equation in a Hilbert space are studied. Such integro-differential equations can be implemented as partial integro-differential equations arising in the theory of viscoelasticity and the theory of heat propagation in media with memory and also have many other important applications.The established results on the Riesz basis property of the root vectors of the semigroup generator can be used in studying the properties of solutions of integro-differential equations.
On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity
Abstract
We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.