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Том 54, № 10 (2018)
- Год: 2018
- Статей: 12
- URL: https://journals.rcsi.science/0012-2661/issue/view/9354
Ordinary Differential Equations
Matrix Linearization of Functional-Differential Equations of Point Type and Existence and Uniqueness of Periodic Solutions
Аннотация
We study ω-periodic solutions of a functional-differential equation of point type that is ω-periodic in the independent variable. In terms of the right-hand side of the equation, we state easy-to-verify sufficient conditions for the existence and uniqueness of an ω-periodic solution and describe an iteration process for constructing the solution. In contrast to the previously considered scalar linearization, we use a more complicated matrix linearization, which permits extending the class of equations for which one can establish the existence and uniqueness of an ω-periodic solution.
1271-1284
Stable Relaxation Cycle in a Bilocal Neuron Model
Аннотация
We consider the so-called bilocal neuron model, which is a special system of two nonlinear delay differential equations coupled by linear diffusion terms. The system is invariant under the interchange of phase variables. We prove that, under an appropriate choice of parameters, the system under study has a stable relaxation cycle whose components turn into each other under a certain phase shift.
1285-1309
Traces of Higher Negative Orders for a String with a Singular Weight
Аннотация
We study the linear operator pencil A(λ) = L−λV, λ ∈ ℂ, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W2−1[0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λn, n ∈ ℕ, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L−1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.
1310-1320
Strong Solutions of Stochastic Differential Inclusions with Unbounded Right-Hand Side in a Hilbert Space
Аннотация
In a separable Hilbert space, a stochastic differential inclusion with coefficients whose values are closed not necessarily convex sets is considered. Two existence theorems for strong solutions are proved. In the first theorem, the proof is based on the use of Euler polygonal lines; in the second, on the successive approximation method. Instead of the assumption that the coefficients of the inclusion are globally Lipschitz, which is traditional in such cases, some conditions that are less restrictive for the problems in question are used.
1321-1337
Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I
Аннотация
The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L1(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L2(0, π).
1338-1353
Inverse Problem for a Fourth-Order Differential Operator with Nonseparated Boundary Conditions
Аннотация
Uniqueness theorems are proved for two inverse problems for a fourth-order differential operator with nonseparated boundary conditions. The first of the problems, which has technical applications, is the problem of identification of a differential equation and two boundary conditions, and the second problem is the problem of identification of a differential equation and four boundary conditions. One of two data sets is used as the spectral data of the problem. The first data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectral data of a system of three problems, and the second data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectra of ten boundary value problems.
1354-1362
Partial Differential Equations
Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions
Аннотация
The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.
1363-1370
Mixed Oblique Derivative Problem for the Helmholtz Equation in a Half-Disk
Аннотация
A mixed oblique derivative boundary value problem is considered for the Helmholtz equation in a half-disk. We prove the unique solvability of this problem for sufficiently large values of the parameter occurring in the equation, the leading part of the inverse operator being constructed explicitly.
1371-1383
Solvability of a Boundary Value Problem for a Differential Equation of the Boussinesq Type
Аннотация
We study the solvability and the construction of the solution of a boundary value problem with a nonlocal integral boundary condition for a three-dimensional analog of the fourth-order homogeneous Boussinesq type differential equation. Separation of variables is used to derive a criterion for the unique solvability of this nonlocal problem. The problem is also considered in the case of violation of the unique solvability criterion.
1384-1393
Short Communications
Variational Statement of the Schrödinger Equation with a Nonstationary Nonlinearity and Its Integrals of Motion
Аннотация
The inverse variational problem is solved for the nonlocal nonlinear Schrödinger equation modeling filamentation processes in various nonlinear media. The corresponding integral relations generalizing conservation laws to the nonconservative case are obtained.
1394-1398
Basis Property of the System of Root Functions of a Classical Spectral Problem with a Multiple Eigenvalue
Аннотация
The problem of describing the biorthogonal system of a classical system of root functions is considered for a loaded string problem with a multiple eigenvalue. The basis property of selected complete and minimal subsystems of the system of root functions in the presence of one or two associated functions is studied.
1399-1402
Solvability of Nonlocal Boundary Value Problems for an Equation of Mixed Type with Various Boundary Conditions
Аннотация
Two boundary value problems in which one of the conditions is nonlocal and contains a real parameter are studied for an equation of mixed type in a half-strip. Sufficient conditions for the unique solvability of these problems are obtained under some restrictions on the parameter.
1403-1407