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Vol 52, No 10 (2016)

Ordinary Differential Equations

Remark on the theory of Sergeev frequencies of zeros, signs, and roots for solutions of linear differential equations: I

Barabanov E.A., Voidelevich A.S.

Abstract

For linear differential equations with continuous coefficients, we compute the Borel types of Lebesgue sets of their lower and upper characteristic frequencies of zeros, signs, and roots, which are treated as functions on the direct product of the space of equations with the compact-open topology and the space of initial vectors of solutions.

Differential Equations. 2016;52(10):1249-1267
pages 1249-1267 views

Asymptotics of the solution of a system with constant delay

Grebenshchikov B.G.

Abstract

We study a complicated linear homogeneous system with constant delay consisting of two subsystems, one of which contains an exponential factor multiplying the right-hand side. Under some conditions, we prove the existence of an asymptotically periodic solution of this system.

Differential Equations. 2016;52(10):1268-1273
pages 1268-1273 views

Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

Knyazhishche L.B.

Abstract

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.

Differential Equations. 2016;52(10):1274-1281
pages 1274-1281 views

Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

Kornev S.V., Obukhovskii V.V., Zecca P.

Abstract

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x′(t) ∈ F(t, xt) under the assumption that the multimapping F has convex compact values and satisfies the upper Carathéodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Carathéodory conditions. In both cases, a generalized integral guiding function is used to study the problem.

Differential Equations. 2016;52(10):1282-1292
pages 1282-1292 views

On Γ-limit classes of perturbations defined by integral conditions

Makarov E.K.

Abstract

For the coefficients of linear differential systems, we consider classes of piecewise continuous perturbations that are infinitesimal in mean on the positive half-line with some positive piecewise continuous weight belonging to a given set. We obtain sufficient conditions for such a class to be Γ-limit, i.e., to admit the computation of a reachable upper bound of the exponents of linear differential systems with perturbations in that class by a formula similar to the well-known formulas for the central and exponential exponents.

Differential Equations. 2016;52(10):1293-1299
pages 1293-1299 views

Existence of a linear Pfaff system with arbitrary bounded disconnected lower characteristic set of positive Lebesgue m-measure

Platonov A.S., Krasovskii S.G.

Abstract

We give a constructive proof of the existence of a completely integrable Pfaff system with infinitely differentiable bounded coefficient matrices and with an arbitrary disconnected lower characteristic set of positive Lebesgue m-measure.

Differential Equations. 2016;52(10):1300-1311
pages 1300-1311 views

Partial Differential Equations

Tricomi problem for an advance–delay equation of mixed type with variable deviation of the argument

Zarubin A.N., Kholomeeva A.A.

Abstract

We study a boundary value problem for an equation of mixed type with the Lavrent’ev–Bitsadze operator in the leading part and with variable deviation of the argument in lower-order terms. The general solution of the equation is constructed. We prove a uniqueness theorem without any conditions on the value of the deviation. The problem is uniquely solvable. We derive integral representations of the solutions in closed form in the elliptic and hyperbolic domains.

Differential Equations. 2016;52(10):1312-1322
pages 1312-1322 views

Eigenfunctions of the Tricomi problem with an inclined type change line

Moiseev E.I., Likhomanenko T.N.

Abstract

We construct the eigenfunctions of the Tricomi problem for the case in which the type change line of the elliptic-hyperbolic equation is inclined and forms an arbitrary angle α with the x-axis. These eigenfunctions form a basis in the elliptic domain. In addition, we find an integral constraint on the inclined type change line.

Differential Equations. 2016;52(10):1323-1330
pages 1323-1330 views

Elliptic dilation–contraction problems on manifolds with boundary. C*-theory

Savin A.Y., Sternin B.Y.

Abstract

We study boundary value problems with dilations and contractions on manifolds with boundary. We construct a C*- algebra of such problems generated by zero-order operators. We compute the trajectory symbols of elements of this algebra, obtain an analog of the Shapiro–Lopatinskii condition for such problems, and prove the corresponding finiteness theorem.

Differential Equations. 2016;52(10):1331-1340
pages 1331-1340 views

Numerical Methods

On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation

Beshtokov M.K.

Abstract

We study a nonlocal boundary value problem for a degenerating pseudoparabolic third-order equation of the general form. For the solution of the problem, we obtain a priori estimates in differential and difference form, which imply the stability of the solution with respect to the initial data and right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.

Differential Equations. 2016;52(10):1341-1354
pages 1341-1354 views

Estimates for the difference between exact and approximate solutions of parabolic equations on the basis of Poincaré inequalities for traces of functions on the boundary

Matculevich S.V., Repin S.I.

Abstract

We study a method for the derivation of majorants for the distance between the exact solution of an initial–boundary value reaction–convection–diffusion problem of the parabolic type and an arbitrary function in the corresponding energy class. We obtain an estimate (for the deviation from the exact solution) of a new type with the use of a maximally broad set of admissible fluxes. In the definition of this set, the requirement of pointwise continuity of normal components of the dual variable (which was a necessary condition in earlier-obtained estimates) is replaced by the requirement of continuity in the weak (integral) sense. This result can be achieved with the use of the domain decomposition and special embedding inequalities for functions with zero mean on part of the boundary or for functions with the zero mean over the entire domain.

Differential Equations. 2016;52(10):1355-1365
pages 1355-1365 views

Short Communications

Some properties of solutions of a scalar Riccati equation with complex coefficients

Grigoryan G.A.

Abstract

We give the definition of normal and limit solutions of the Riccati equation with complex coefficients and study some properties of these solutions. We prove the minimality property of a solution of a system of two linear first-order ordinary differential equations.

Differential Equations. 2016;52(10):1366-1370
pages 1366-1370 views

On an inhomogeneous problem for parabolic-hyperbolic equation

Gulyaev D.A.

Abstract

We study a boundary value problem for an inhomogeneous parabolic-hyperbolic equation with a noncharacteristic type change line. Boundary conditions of the first kind are posed on characteristics in the parabolic and hyperbolic parts of the domain where the equation is given, and a condition of the third kind is posed on the noncharacteristic part of the boundary in the parabolic part. First, we study the solvability of an inhomogeneous initial–boundary value problem for a parabolic equation.

Differential Equations. 2016;52(10):1371-1373
pages 1371-1373 views

On the multiple spectrum of a problem for the Bessel equation with spectral parameter in the boundary condition

Kapustin N.Y., Moiseev T.E.

Abstract

A problem for the Bessel equation of zero order with complex physical and spectral parameters in the boundary condition is considered. We study whether the system of eigenfunctions has the basis property.

Differential Equations. 2016;52(10):1374-1378
pages 1374-1378 views

On solutions of a boundary value problem for the biharmonic equation

Matevosyan O.A.

Abstract

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.

Differential Equations. 2016;52(10):1379-1383
pages 1379-1383 views

On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

Repin O.A., Frolov A.A.

Abstract

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.

Differential Equations. 2016;52(10):1384-1388
pages 1384-1388 views