Abstract We give a statement of a multipoint boundary value problem on a geometric graph. The characteristics of the graph admitting such a statement are described. The form of the boundary value problem adjoint to the problem posed is found. An analog of the Keldysh residue theorem is obtained for the Green’s function of the spectral multipoint boundary value problem on the graph.

Abstract Some theorems on the uniqueness of the solution of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with a spectral parameter \(\lambda \) with data on parallel and external characteristics are obtained.

Abstract We consider a nonlinear Schrödinger equation arising in a number of physical problems. It is shown that when the real part is separated in this equation, there arises a nonlinear differential equation that has at least two types of solutions, a multiwave solution and a solution in the form of standing waves. Numerical examples of the multiwave solution and its transition to the standing wave

Abstract We consider problems of optimal control of processes described by the Dirichlet boundary value problem for elliptic equations with mixed derivatives and unbounded nonlinearity. The controls are contained in the coefficients multiplying the highest derivatives of the state equation. Finite-difference approximations to nonlinear optimization models are constructed and investigated, and to find

Abstract For the equation \((\mathrm {sign}\thinspace y)|y|^{m}u_{{xx}}+u_{{yy}}-(m/2y)u_{y}=0\) considered in a mixed domain, we prove theorems on the uniqueness and existence of a solution of a problem with a missing Tricomi condition on the boundary characteristic and with an analog of the Frankl condition on an internal characteristic and on the degeneracy segment.

Abstract We consider inverse problems of determining the initial conditions and a time-invariant inhomogeneity in boundary value problems for the Burgers equation. A transformation is used permitting one to reduce the Burgers equation to an equation with a variable coefficient for a function available for measurement in tomographic observations. We prove theorems on the unique reconstruction of the

Abstract For abstract nonlinear difference schemes with operators acting in finite-dimensional Banach spaces, a stability criterion is stated and proved; namely, for a consistent finite-difference approximation to a well-posed differential problem, the solution of the difference scheme converges if and only if the scheme is unconditionally stable. In a sense, this criterion generalizes Lax’s equivalence

Abstract We find the asymptotics of the eigenpairs of the Dirichlet and Neumann spectral problems for the Laplace operator in a domain separated by several partitions with holes of small diameters and splitting into several independent cells in the limit as the diameters tend to zero. Using asymptotic methods for singularly perturbed domains, we study the splitting of a multiple eigenvalue of the limit

Abstract We consider a mathematical model of scattering of \(H \)-polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based

Abstract We consider a nonlinear integro-differential equation with zero operator in the differential part whose integral operator contains several rapidly varying kernels. This paper continues the research carried out earlier for equations with only one rapidly varying kernel. We prove that the conditions for the solvability of the corresponding iteration problems, as in the linear case, are not differential

Abstract We study a class of nonlinear integral equations with a noncompact Hammerstein– Nemytskii operator on the entire line. Some special cases of such equations have specific applications in various fields of natural science. The combination of a method for constructing invariant cone segments for the corresponding nonlinear monotone operator with methods of the theory of functions of a real variable

Abstract We consider the first initial–boundary value problem for a spatially one-dimensional second-order Petrovskii parabolic system with differentiable coefficients in a half-strip with nonsmooth lateral boundary. A theorem on the uniqueness of the classical solution of this problem is proved.

Abstract A first-order regularized trace formula has been obtained for the Sturm–Liouville operator with a point of \( \delta ^{\prime }\)-interaction. For large values of the spectral parameter, asymptotic representations have been found for solutions of the Sturm–Liouville equation with discontinuity conditions. The asymptotics of the eigenvalues of the operator under study has been derived. It is

Abstract The global solvability of boundary value problems for the reaction–diffusion–convection equation is proved for the case in which the reaction coefficient in the equation and the mass transfer coefficient in the boundary condition nonlinearly depend on the substance concentration. The minimum and maximum principle for the concentration is established. The solvability of multiplicative control

Abstract We obtain sufficient as well as necessary and sufficient conditions for linear one-dimensional homogeneous stochastic differential equations with independent standard and fractional Brownian motions to possess some types of stability.

Abstract We consider the analytic continuation problem for the solution of the system of thermoelasticity equations in a spatial domain based on its values and the values of its stresses known on part of the domain boundary, i.e., the Cauchy problem.

Abstract We consider unsteady triaxial tension–compression of a moving parallelepiped filled with a Newtonian viscous fluid and changing its linear dimensions (with a constant volume) during motion. The statement of the linearized problem is given in terms of three-dimensional perturbations imposed on the main process. To study this problem, we apply the method of integral relations based on the use

Abstract Various tests for the exponential stability of a linear system of ordinary differential equations are obtained by the method of frozen coefficients. To this end, we prove and use an improved Gelfand–Shilov estimate for the matrix exponential. The cases in which the coefficient matrix of the system satisfies the Lipschitz condition or the Hölder condition on the positive real line are considered

Abstract Two- and one-dimensional Kirchhoff models of thin isotropic plates and rods are combined into a single problem describing the deformation of the joint of these elastic objects. The conjugation conditions at the points of attachment of the rods to the plate are assigned using the technique of self-adjoint extensions of fourth-order differential operators in a two-dimensional domain and second-order

Abstract We consider a boundary value problem for an elliptic differential equation with analytic coefficients that is degenerate in one of the variables in a rectangle. Using the method of spectral separation of singularities, a solution of this problem is constructed in the form of a Poisson series—a series in the eigenfunctions of the second-order limit linear ordinary differential operator with

Abstract We consider biquaternionic wave (biwave) equations. They are biquaternionic generalizations of the Maxwell and Dirac equations and are equivalent to a system of eight differential equations of hyperbolic type. Using the theory of generalized functions, we construct fundamental and generalized solutions of such equations, including discontinuous ones, describing shock waves and obtain conditions

Abstract Green’s functions for the Navier and Riquier–Neumann problems for the biharmonic equation in the unit ball are constructed, and integral representations of the solution of these problems are given.

Abstract We prove that for any real-valued function \(\theta (\cdot ) \) defined on the half-line \([0,+\infty ) \), monotone increasing to \(+\infty \), and such that \(\theta (t)/t\to 0 \) as \(t\to {+\infty }\) and for any positive integer \(n\geq 2\) there exists an \(n \)-dimensional Lyapunov regular linear differential system whose Lyapunov exponents change under some perturbation of its coefficient

Abstract Some theorems on the uniqueness of the solution of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with a spectral parameter \(\lambda \) are obtained.

Abstract On a finite time horizon, we consider a control system described by a vector differential equation with right-hand side that changes its structure at some times spaced by a distance that cannot be less than a certain given value. In between two adjacent structure change instants, the right-hand side is a function that is Lipschitz in state variables, continuous in time, and linear in the control

Abstract We prove the existence of a classical solution of the contact problem for Petrovskii parabolic systems of the second order with Dini continuous coefficients in a strip separated into two domains by a nonsmooth curve.

Abstract We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with distinct powers of the small parameter multiplying the second derivatives. The problem has the specific feature that one of the two equations in the degenerate system has three disjoint roots, one being double and the other two being simple. We prove that for sufficiently

Abstract We study the solvability of a nonlinear boundary value problem for a system of five second-order partial differential equations under given boundary conditions. The system describes the equilibrium state of elastic shallow inhomogeneous anisotropic shells with free edges in the framework of the Timoshenko shear model. The boundary value problem is reduced to a nonlinear operator equation for

Abstract We consider a linear control system with an almost periodic coefficient matrix whose mean is a diagonal matrix and with a linear state feedback control. The control matrix is a constant square matrix and has zero rows, and its nonzero rows are linearly independent. The feedback gain is assumed to be almost periodic and such that its frequency module is contained in the frequency module of

Abstract We consider abstract linear inhomogeneous second-order integro-differential equations in a Hilbert space that are defined on the positive half-line and have unbounded coefficients and integral terms of the Volterra convolution type with kernels represented by the Stieltjes integral of a decaying exponential. The equations under study represent an abstract form of partial integro-differential

Abstract We consider linear time-invariant differential equations with the Hukuhara derivative. We prove that if a solution of such an equation has a nonempty interior at the initial time, then the Lyapunov exponents of the radii of the inscribed and circumscribed spheres of this solution are strict and are equal to the minimum and maximum absolute values, respectively, of the eigenvalues of the coefficient

Abstract We consider the Showalter–Sidorov problem for the stochastic version of the linear Ginzburg–Landau equation in Hilbert spaces of smooth differential forms defined on a compact oriented Riemannian manifold without boundary with stochastic processes serving as coefficients. This equation is reduced to an abstract Sobolev type stochastic equation with a relatively radial operator on the right-hand

Abstract We consider the class of linear parametric differential systems \(\dot {x}=\mu A(t)x \) defined on the half-line \(t\geq 0 \), where \(\mu \in \mathbb {R} \) is a parameter, \(x\in \mathbb {R}^n \), and the matrix \(A(\cdot )\colon [0,+\infty )\to \mathbb {R}^{n\times n}\) is piecewise continuous and bounded. It is proved that for each \(n\geq 2\) some set is the irregularity set of one of

Abstract The connections between stochastic differential equations in which continuous and discontinuous random processes serve as sources of randomness and deterministic equations for the probabilistic characteristics of solutions of these stochastic equations are studied. In the study, we use various approaches based on the stochastic change of variables formula (Itô’s formula), on the analysis of

Abstract Initial value problems with various boundary conditions at the ends are studied for the beam vibration equation taking into account the beam rotational motion under bending. An energy inequality is established, which implies the uniqueness of the solution of the four initial–boundary value problems posed. In the case of hinged ends, theorems on the existence and stability of the solution of

Abstract We study a mixed problem with nonlinear dissipative boundary conditions for systems of one-dimensional semilinear wave equations with a focusing nonlinear source that has a variable growth exponent. Theorems on the blow-up of solutions in finite time are proved.

Abstract Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow

Abstract We prove the existence and uniqueness of the solution of the Cauchy problem for an integro-differential equation related to the peridynamic model in mechanics of solids.

Abstract We prove that a necessary and sufficient condition for the existence of a classical solution of the inhomogeneous biharmonic equation in a bounded plane domain is the requirement of stronger continuity of the function on the right-hand side in the equation.

Abstract We prove the existence and uniqueness of a solution of the Darboux problem for the fourth-order Bianchi equation. The Riemann–Hadamard function is determined for the Darboux problem. The solution of the problem is constructed in terms of this function.

Abstract We extend an analog of the Chetaev–Malkin–Massera theorem on stability by the first approximation for differential systems to discrete-time systems in which the linear first approximation system is subjected to linear and nonlinear perturbations. The result is stated in terms of the characteristic exponents and the Lyapunov irregularity coefficient of the linear first approximation system

Abstract For an integro-differential equation of the convolution type with a power nonlinearity and a variable coefficient defined on the half-line \([0,\infty )\), we use the method of weight metrics in the cone of the space \(C^1(0,\infty ) \) formed by functions positive on \((0,\infty ) \) and vanishing at the origin to prove a global theorem on the existence and uniqueness of a solution belonging

Abstract We study the Tricomi problem for an advance–delay mixed type equation. Theorems on the uniqueness and existence of a twice continuously differentiable solution are proved.

Abstract We establish estimates (interior and up to the boundary) of solutions of the inequality \(\mathscr {L}(u)\geq 0\), where \(\mathscr {L} \) is a linear second-order differential operator with impulse singularities. The results in particular apply to the Neumann boundary value problem. The relationship between this inequality and the class of concave functions of one variable is studied.

Abstract We use the Lyapunov vector function method and the guiding function method to obtain sufficient conditions for the existence of Poisson bounded and partially Poisson bounded solutions of systems of differential equations.

Abstract We consider the polynomial Liénard system \(\dot {x}=-y \), \(\dot {y}=x+A(x)-B(x)y \) under the assumption that the real polynomials \(A(x) \) and \(B(x) \) and the derivative \(A^{\prime }(x) \) satisfy the conditions \(A(0)=B(0)=A^{\prime }(0)=0 \). We prove that this system has an isochronous center at the singular point \(O(0,0)\) if and only if the polynomials \( A(x)\) and \(B(x) \)

Abstract We consider nonautonomous linear systems of differential equations admitting a time-dependent first integral that is a quadratic form. The duality between mutually adjoint linear systems with quadratic integrals is established. Conditions for the spectrum of such linear systems to be symmetric about zero are indicated. We prove that a linear system is stable if and only if it admits a first

Abstract We study the asymptotics of the spectrum of the Dirac operator on the real line with a potential in \(L_2 \). It is shown that the spectrum of such an operator lies in a domain of the complex plane symmetric about the real axis and bounded by the graph of some continuous real-valued square integrable function. To prove this, we use the \(L_1 \)-functional calculus for self-adjoint operators

Abstract We consider a system of \(n\)th-order ordinary differential equations whose right-hand side is the sum of a linear function of the solution with a constant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbing continuous periodic function. The matrix of the linear function has only real simple nonzero eigenvalues, of which at least one is positive. We study

Abstract We study how a boundary value problem with a nonlocal integral condition of the first kind for a mixed type equation with a singular coefficient in a rectangular domain depends on a numerical parameter occurring in the equation. A uniqueness criterion is established, and theorems on the existence and stability of a solution of the problem are proved. The solution is constructed in closed form

Abstract For a linear periodic system of differential equations, we indicate a method permitting one to find a matrix similar to the matrix of the Poincaré map in closed form.

Abstract We prove that the \(p\)th moments, \(p\ge 1 \), of strong solutions of a mixed-type stochastic differential equation driven by a standard Brownian motion and a fractional Brownian motion with Hurst index greater than \(1/2\) are finite provided that the coefficients of the equation, together with all of their partial derivatives of order \(\le 2 \), are continuous and bounded and the \(p \)th

Abstract We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla \) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated

Abstract We study the existence of periodic solutions for a class of systems of first-order nonlinear ordinary differential equations on the plane. Using an a priori estimate of periodic solutions, we establish the homotopy invariance of the solvability property of the periodic problem, give a description of the homotopy classes, and prove a theorem about a necessary and sufficient condition for the

Abstract We establish the property of completeness in the space \(L_2(D) \), where \(D \) is a domain in \(\mathbb {R}^n \), \(n\geq 3\), of the products of all possible regular solutions of the equation \(\Delta u-\kappa ^2 u=0 \), \(\kappa \in \mathbb {R}_{+}\cup i\mathbb {R}_{-} \), by the fundamental solutions of this equation with singularities on a straight line that does not meet \(\overline

Abstract We prove theorems on the Lyapunov stability, asymptotic stability, and instability of the equilibrium position of time-varying semilinear differential-algebraic equations (DAEs) as well as theorems about the asymptotic (complete) stability of these equations. As an example of applying the theorems on the global solvability of time-varying semilinear DAEs proved in the first part of this paper

Abstract We present methods for proving new sufficient stability and asymptotic stability tests for the zero solution of a time-varying differential equation using nonmonotone sign-indefinite Lyapunov functions. As an example of application of our statements, we establish new stability tests for a gradient system in which the equilibrium is not isolated and the function defining the right-hand side

Abstract For linear neutral type systems, we construct a finite time observer in the form of the output of a delayed type system with commensurable concentrated delays and finite spectrum. The observer permits one to obtain an exact estimate of the solution of the original system in finite time. The results are illustrated by an example.

Abstract We consider a linear second-order differential equation with coefficients depending on a spectral parameter and present conditions on the coefficients under which there exist no degenerate two-point boundary conditions for the equation.

Abstract We consider initial value problems for a number of hyperbolic equations with a power-law degeneracy and with operator coefficients in a Banach space and establish sufficient conditions for the unique solvability of these problems in terms of the coefficients of the equation, the degeneracy order, and the initial elements.