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Vol 53, No 1 (2017)
- Year: 2017
- Articles: 14
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9295
Ordinary Differential Equations
Generalized-periodic motions of nonautonomous systems
Abstract
The definition and existence criterion are given for the generalized-periodic motions of a certain wide class of systems. The class contains all the systems that can be characterized by the classical periodic operator of displacement, the systems generated by the Volterra integral equations, and some others. A relationship is established between generalized-periodic motions and integral invariant sets.
1-7
Linear differential operators and operator matrices of the second order
Abstract
Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.
8-17
Stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables
Abstract
The method of model equations is used to study the moment stability of solutions to linear impulsive systems of Itô differential equations with aftereffect with respect to initial data in part of variables. Sufficient conditions of stability are obtained in terms of parameters of these systems.
18-32
Riesz basis property of system of root functions of second-order differential operator with involution
Abstract
The properties of the root functions are studied for an arbitrary operator generated in L2(−1, 1) by the operation with involution of the form Lu = −u″(x)+αu″(−x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (−1, 1), ν(x) is an absolutely continuous involution of the segment [−1, 1] and the coefficients q(x) and qν(x) are summable functions on (−1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L2(−1, 1) for the system of the root functions of the operator.
33-46
Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions
Abstract
Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.
47-55
Partial Differential Equations
Nonlocal time-multipoint problem for a certain class of evolutionary pseudodifferential equations with variable symbols: I
Abstract
It has been proved that in generalized spaces of the type S, a pseudodifferential operator constructed based on a variable symbol can be treated as the operator of infiniteorder differentiation if the operator symbol satisfies certain conditions. The properties of the fundamental solution to a nonlocal time-multipoint problem for the evolutionary equation with this operator have been studied.
56-73
Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients
Abstract
The classical solution of the first mixed problem for a second-order hyperbolic equation with variable coefficients in the case of two independent variables in a curvilinear half-strip is considered. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. A method is proposed for constructing the solution using the method of sequential approximations for a system of integral equations of the second kind.
74-85
86-98
Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel
Abstract
A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.
99-108
Control Theory
109-114
115-121
Optimal control for the coefficients of quasilinear parabolic equation with a goal functional on domain boundary
Abstract
The optimal control problem for a quasilinear parabolic equation with controls in the coefficients and a goal functional on the domain boundary is considered. The correctness of the statement of the problem is studied and a necessary condition of optimality is derived.
122-132
Short Communications
133-138
Representation of solutions of integro-differential equations with kernels depending on the parameter
Abstract
Integro-differential equations with unbounded operator coefficients in a separable Hilbert space are studied. These equations are an abstract form of the Gurtin–Pipkin-type equation, which describes finite-speed propagation of heat in media with memory. A representation of strong solutions of these equations is derived in the form of the sums of series in exponents that correspond to the spectral points of operator-functions that are the symbols of these equations.
139-143