Abstract This work is devoted to study the existence and uniqueness of solution of an analogue of the Gellerstedt problem with nonlocal assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation with nonlinear terms and Gerasimov–Caputo operator of differentiation. Using the method of integral energy, the uniqueness of solution have been proved. Existence

Abstract We consider the first boundary value problem for quasilinear parabolic differential–difference equations. The uniqueness and existence of generalized solutions are proved. The proofs are based on the theory of linear differential–difference equations and the theory of monotone (pseudomonotone) operators.

Abstract We consider a class of gradient-like flows on three-dimensional closed manifolds whose attractors and repellers belongs to a finite union of embedded surfaces and find conditions when the ambient manifold is Seifert.

Abstract In this article we calculate the order projection in a space \(\mathcal{OA}_{r}(E,F)\) of all regular orthogonally additive operators from a vector lattice \(E\) to a Dedekind complete vector lattice \(F\), onto the band \(\{T\}^{\perp\perp}\) of \(\mathcal{OA}_{r}(E,F)\) which is generated by a disjointness preserving orthogonally additive operator \(T\colon E\to F\).

Abstract This paper is devoted to small motions problem of composite hydrodynamic systems containing a viscoelastic fluid. Hydrodynamic systems of two or more viscoelastic fluids are considered, as well as partially dissipative systems containing ideal fluids or a barotropic gas (in addition to a viscoelastic fluid). The listed initial-boundary-value problems are investigated using the operator approach

Abstract Let \(\mathbf{E}=\mathbf{E}(\Omega,\mathcal{F},\mu)\) be a symmetric Banach space of measurable functions on a measure space \((\Omega,\mathcal{F},\mu)\). We prove a version of Mean (Statistical) Ergodic Theorem for Cesáro averages \(A_{n,T}f=1/n\sum_{k=1}^{n}T^{k-1}f\), \(f\in\mathbf{E}\), while operators on \(\mathbf{E}\) are induced by positive absolute contraction in \(\mathbf{L}_{1}+

Abstract Problem review of existence and stability nonlinear parabolic functionally differential equations structures with transformation of spatial variables. Such equations model optical systems that contain a thin layer of a nonlinear Kerr-type medium. Review of Initial boundary value problems for circumference, a circle, and a ring with an involution operator, corresponding to transformation of

Abstract The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral

Abstract In this paper, we investigate a problem on small motions of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. The initial boundary value problem is reduced to the Cauchy problem for a first-order differential operator equation on a Hilbert space. Properties of the resulting operator matrices, which are coefficients of the equation, are studied

Abstract A class of quadratic irregular pencils of the second order ordinary differential operators with constant coefficients is considered in this paper. It is assumed that the boundary conditions are splitting and the characteristic equation has positive roots. Both amounts of two-fold expansions in the series in terms of eigenfunctions of such pencils and necessary and sufficient conditions for

Abstract The purpose of this article is to demonstrate with a simple example the complexities that arise when considering boundary-value problems for differential-difference equations. In this paper we discuss in detail smoothness issues of solutions to the Dirichlet problem for linear differential-difference equation with integer shifts. After analysis of this example we consider more general statement

Abstract In previous works ([4–8]), the author studied the general approach to solving mixed boundary, spectral, and initial-boundary transmission problems. In this paper, this approach is applied to the mixed boundary transmission problem of the linear theory of elasticity. On the basis of the corresponding Green formulas the solution of the problem can be represented as the sum of the solutions of

Abstract An initial-boundary value problem posed on a bounded interval is considered for a class of odd-order (more than one) quasilinear evolution equations with general nonlinearity. Assumptions on the equations do not provide global a priori estimates for solutions of an arbitrary size. For small initial and boundary data, small right-hand side function results on global existence and uniqueness

Abstract The paper deals with the Sturm–Liouville operator generated on the finite interval \([0,\pi]\) by the differential expression \(-y^{\prime\prime}+q(x)y\), where \(q=u^{\prime}\), \(u\in L_{\varkappa}[0,\pi]\) for some \(\varkappa\geq 2\), and arbitrary regular boundary conditions. Consider two such operators with different potentials but the same boundary conditions. We prove that the difference

Abstract We study the Cauchy problem for nonlinear degenerate parabolic equations with almost periodic initial data. Existence and uniqueness (in the Besicovitch space) of entropy solutions are established. It is demonstrated that the entropy solution remains to be spatially almost periodic and that its spectrum (more precisely, the additive group generated by the spectrum) does not increase in the

Abstract An electric auto-generator on two coupled Van der Pol circuits has been considered. Conditions for the existence of asymptotically stable two-frequency oscillations are obtained. Conditions are equalities and inequalities expressed through the numerical characteristics of the parameters of the oscillator. An algorithm is given for finding the numerical characteristics of the oscillator, at

Abstract This review paper is devoted to the description of the results of N.D. Kopachevsky and S. Krein (form 1964 till nowadays) related to spectral properties of strong dissipative hydrodynamical systems and corresponding operator-differential equations. Such spectral problems usually lead to eigenvalue problem for the so-called operator pencil of S. Krein with some possible modifications. We provide

Abstract In this paper, we consider the hyperbolic thermoelastic system of memory type and the corresponding system of abstract integro-differential equations. We study forced motions of this system of abstract integro-differential equations and investigate the asymptotic behavior of solutions to this system as time goes to infinity.

Abstract The Lamè system on the plane is considered for the displacement vector in a general anisotropic medium. Cases of explicit calculation of the roots of its characteristic equation are indicated. It is proved that this system cannot be written in the complex form of a single equation. An integral representation of general solution of the Lamè system is received in the class of continuous functions

Abstract In this article we explore orthogonally additive (nonlinear) operators in vector lattices. First we investigate the lateral order on vector lattices and show that with every element \(e\) of a \(C\)-complete vector lattice \(E\) is associated a lateral-to-order continuous orthogonally additive projection \(\mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}\). Then we prove that for an order bounded

Abstract Let \(\simeq\) be a binary relation between sets of integers, and \(\leq_{R}\) be a Post reducibility, i.e. a reflexive and transitive relation between sets of integers such that if \(A\leq_{R}B\) then the computational complexity of recognition of elements of \(A\) is easier than (or equal to) the recognition of elements of \(B\). Suppose that for a class \({\mathcal{A}}\) of arithmetical

Abstract Let \(D\) be a domain in the complex plane and \(M\) be an extended real-valued function on \(D\). If \(f\) is a non-zero holomorphic function on \(D\) such that \(|f|\leq\exp M\), then it is natural to expect that there should be some upper boundedness for the distribution of the zeros of \(f\) expressed exclusively in terms of the function \(M\) and the geometry of the domain \(D\). We have

Abstract We study in this paper negative \(\mathbb{A}\)-numberings where \(\mathbb{A}\) are admissible structures. We establish that a series of classical assumptions is remained to hold for negative \(\mathbb{A}\)-numberings in the case of the application of certain limits as for numberings as for admissible structures. We find examples of admissible structures \(\mathbb{A}\) whose families of all

Abstract An \(\alpha\)-coloring \(\xi\) of a structure \(\mathcal{S}\) is distinguishing if there are no nontrivial automorphisms of \(\mathcal{S}\) respecting \(\xi\). In this note we prove several results illustrating that computing the distinguishing number of a structure can be very hard in general. In contrast, we show that every computable Boolean algebra has a \(0^{\prime\prime}\)-computable

Abstract Two max- and min-approximation problems on solarity of sets in dual spaces are considered. It is shown that if the metric projection onto a set \(M\subset X^{*}\) is \(w^{*}\)-upper semicontinuous and has nonempty \(w^{*}\)-closed acyclic values, then \(M\) is a sun. In particular, a Chebyshev set with \(w^{*}\)-continuous metric projection is a sun. In the max-approximation setting, a set

Abstract We study \(P^{*}\)-combinations of almost \(\omega\)-categorical weakly o-minimal theories. One of natural questions in the study of such a combination is the question of preserving or weakening certain properties of initial theories. Earlier, criteria for Ehrenfeuchtness of \(P^{*}\)-combinations of countably many \(\omega\)-categorical ordered theories were obtained. Here we prove that if

Abstract The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings. Under the assumption of Projective Determinacy, we prove that for every non-zero natural number \(n\), there are infinitely many pairwise elementarily non-equivalent Rogers semilattices for \(\Sigma^{1}_{n}\)-computable families.

Abstract In this paper, we investigate the inverse problem of a simple layer potential where, by the known values of the potential gradient given as a finite series in a neighborhood of the point at infinity, we search for a simple closed contour with gravitating masses that generates potential outside the contour. The desired contour is an image of the unit circle under a mapping by polynomials of

Abstract The vertex 3-colourability problem is to verify whether it is possible to split the vertex set of a given graph into three subsets of pairwise nonadjacent vertices or not. This problem is known to be NP-complete for planar graphs of the maximum face length at most 4 (and, even, additionally, of the maximum vertex degree at most 5), and it can be solved in linear time for planar triangulations

Abstract The aim of this paper is to study the multiple interpolation problem in the spaces of analytical functions of finite order \(\rho>1\) in the half-plane. The necessary and sufficient conditions for solvability of interpolation problem are obtained. These conditions are formulated in terms of the Nevanlinna measure determined by the interpolation divisor.

Abstract We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle \(f\) means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with \(f\). We will prove that the punctual

Abstract We introduce a new class of torsion-free abelian groups which are special epimorphic images of finite rank completely decomposable groups. They belong to the well-known class of Butler groups by their definition. In comparison with the existing results the presented groups are not only of the maximal possible rank that is obviously less than the rank of the pre-image. The kernel of the above

Abstract This paper is a review of results on endomorphism-extendable modules, automorphism-extendable modules, \(\pi\)-injective modules, endomorphism-liftable modules, and automorphism-liftable modules over non-primitive hereditary Noetherian prime rings. A module \(M\) is said to be endomorphism-extendable (resp., automorphism-extendable) if every endomorphism (resp., automorphism) of any submodule

Abstract Let \(G=(V,E)\) be a simple graph. A set \(D\subseteq V\) is a \(2\)-dominating set of \(G\), if every vertex of \(V\setminus D\) has at least two neighbors in \(D\). The \(2\)-domination number of a graph \(G\), is denoted by \(\gamma_{2}(G)\) and is the minimum size of the \(2\)-dominating sets of \(G\). In this paper, we count the number of \(2\)-dominating sets of \(G\). To do this, we

Abstract The functional equation for the Riemann zeta function allows us to construct a kernel of a Toeplitz operator containing a modified zeta function. Hypergeometric functions are used for a description of subspaces approximating the kernel. Expressions for the limit functions are found.

Abstract A constructive method is proposed for the solution of the Riemann–Hilbert problem which is realized in the case of five singular points. The basic components of this method are the logarithmization of the product of matrix functions and the determination of the parameters of a related differential equation of Fuchsian type. The proposed method allows us to solve several related problems, namely

Abstract Punctual presentations of algebraic structures in several familiar classes are studied. It is proved that the punctual dimension of punctual structures from the following classes is either 1 or \(\infty\): equivalence structures, linear orders, torsion-free abelian groups, abelian \(p\)-groups, Boolean algebras.

Abstract The paper deals with a family of second-order partial differential equations on the complex plane. The coefficient of the unknown function depends on a natural parameter \(n\). Solutions of the equations are called generalized analytic functions of order \(n\). In a simply connected domain \(T^{+}\), we investigate Neumann boundary value problem for generalized analytic functions of order

Abstract In this paper, we are applied a new approach to prove that the solution of the linear Fredholm operator equation of the third kind given on a segment and having a finite number of multipoint singularities is equivalent to the solution of the linear Fredholm operator equation of the second kind with additional conditions. We are showed an example of solving the system of linear integral Fredholm

Abstract An initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary order elliptic differential operator is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding

Abstract In this paper, certain spectral properties related with the first order linear differential expression in the weighted Hilbert space at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential expression in the weighted Hilbert space have been determined. Then, the deficiency indices of the minimal operator have

Abstract A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded

Abstract Two-point boundary value problem for fourth order partial integro-differential equation is considered. By new unknown function the problem is reduced to an equivalent family of two-point boundary value problems for the Volterra integro-differential equations of second order. By the method of introduction functional parameters are constructed the algorithms for finding of solution to family

Abstract In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has

Abstract We consider a one-dimensional two-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points \(\pm x_{0}\). The corresponding Schrödinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of

Abstract The Dirichlet and the Keldysh problems for a three-dimensional elliptic equation with three singular coefficients are investigated in a semi-infinite parallelepiped. The uniqueness and the existence theorems are proved using the spectral analysis method. For the problem posed, using the Fourier method, one dimensional spectral problems are obtained. Based on the completeness property of systems

Abstract In this paper we consider of a multidimensional loaded mixed type equation of the second order with some seminonlocal conditions on the coefficients. The existence and uniqueness of solution of the seminonlocal boundary value problem of second kind is proved in the Sobolev space \(W\ _{2}^{3}(Q)\). In proof of the theorems are used the methods of ‘‘\(\varepsilon\)-regularizations’’, a priori

Abstract In this work for a class of second order model and non-model partial integro-differential equations with singularity in the kernel by the aid of arbitrary functions obtained integral representation for solution manifold. In this given article it is entered a new class of functions that at point (a, b) is converted to zero with some asymptotic behavior and the solution of given equation is

Abstract In this article in a rectangular domain inverse boundary value problems with local and non-local boundary conditions for a fourth order partial differential equation were formulated and investigated. The solutions of the inverse problems were obtained in the form of Fourier series. It was proved the theorem on uniqueness and existence of the solution of the inverse problems under consideration

Abstract A two-parameter family of algorithms for finding an approximate solution to a linear two-point boundary value problem for a system of ordinary differential equations is offered. The convergence conditions for the algorithms are obtained. The necessary and sufficient coefficient conditions for the well-posedness of considered problem are established.

Abstract We considered the mixed parabolic-hyperbolic type equation with discontinuous coefficient in a rectangular domain.We established a criterion for the unique solvability of this problem,using this criterion constructed solution as the sum of Fourier series. Finally, the stability of the solution with respect to initial function is proved.

Abstract The nonlinear two-point boundary value problem with parameter for systems of ordinary differential equations is investigated by the parametrization method. The method consists in partition of the interval apart, the introduction of additional parameters and reducing the original problem to the equivalent multi-point boundary value problem with parameters. The algorithms for finding a solution

Abstract An inverse problem for determining the order of time-fractional derivative in a nonhomogeneous subdiffusion equation with an arbitrary elliptic differential operator with constant coefficients in \(N\)-dimensional torus is considered. Using the classical Fourier method it is proved, that the value of the solution at a fixed time instant as the observation data recovers uniquely the order of

Abstract Problems of optimal control of linear and nonlinear stochastic systems with quadratic criterion qualities are studied. For such problems the existence of optimal control in feedback control form is proved by method of dynamic programming. The work consists of two parts. The first part deals with the linear problem. In the first part, the existence of a solution to the Cauchy problem for the

Abstract The article investigates the Robin boundary-value problem for a linear inhomogeneous ordinary differential equation of the second order on a segment with a small parameter at the highest derivative of the unknown function. The aim of the study is to construct a uniform asymptotic expansion of the solution of the Roben problem with an arbitrary degree of accuracy on the considered segment,

Abstract In this article, we consider the scattering problem for the Sturm–Liouville operator on the positive half-line with boundary condition depending quadratically of spectral parameter.The scattering data is defined. The resolvent operator is constructed and the expansion formula is obtained.

Abstract In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation

Abstract In this article, a boundary value problem for a loaded partial differential equations of fourth order was considered on a rectangle domain. The solution of the problem is obtained in the form of Fourier series. Also, theorems about uniqueness and existence of the solution of the problem under consideration were proved by spectral method.

Abstract This article presents a brief on using a saddlepoint approximations technique to solve some branching process problems; this method of approximation provides a good understanding of branching process behaviors such as probability density, mass function, and cumulative distribution function based on a moment generating function for complicated models such as nuclear chain reaction, survival

Abstract It considered a non-homogeneous allocation scheme of distinguishable particles by different cells. We proved gaussian and Poisson limit theorems for the number of empty cells from the first \(K\) cells. These results we apply to study the type I error and the type II error in select analogs of the empty box test.