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Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane. [ABSTRACT FROM AUTHOR]

We consider integro-differential equations (IDEs) with a rapidly oscillating inhomogeneity and with a Volterra-type integral operator whose kernels can contain both a classical rapidly decreasing exponential (the simplest case) and fundamental solutions of differential systems (the general case). Difficulty in constructing a regularized (according to S.A. Lomov) asymptotics in the general case is due to the complex asymptotic structure of the fundamental solution matrix (Cauchy matrix) of the homogeneous differential system. In the present paper, we first construct a regularized asymptotics of the Cauchy matrix, which is then used to construct a regularized asymptotics of the solution of the IDE. [ABSTRACT FROM AUTHOR]

The paper studies the properties of a linear operator that is the generator of a semigroup associated with Volterra integro-differential equations in Hilbert spaces. These integro-differential equations can be realized as partial integro-differential equations arising in the theory of viscoelasticity and the theory of heat propagation in media with memory and have a number of other important applications. The results obtained in the paper can be used to study the behavior of and obtain estimates for the solutions of Volterra integro-differential equations. [ABSTRACT FROM AUTHOR]

Solutions of singular differential equations with the Bessel operator of negative order are studied. In this regard, of great interest are the solutions of the singular differential Bessel equation , which are presented in the paper as linearly independent functions and , . Some properties of the functions expressed in terms of the properties of the Bessel–Levitan -function are considered. The direct and inverse -Bessel transforms are introduced, and the -pseudoshift operator that commutes with the Bessel operator is defined. The fundamental solution of the ordinary singular differential operator is found. A representation of the fundamental solution of the Kipriyanov -operator with a singularity at the point and on the cone in the Euclidean -dimensional half-space is given. [ABSTRACT FROM AUTHOR]

We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations. [ABSTRACT FROM AUTHOR]

The spectral properties of a linear operator that is the generator of a semigroup generated by a Volterra integro-differential equation in a Hilbert space are studied. Such integro-differential equations can be implemented as partial integro-differential equations arising in the theory of viscoelasticity and the theory of heat propagation in media with memory and also have many other important applications.The established results on the Riesz basis property of the root vectors of the semigroup generator can be used in studying the properties of solutions of integro-differential equations. [ABSTRACT FROM AUTHOR]

We consider the first boundary value problem in a rectangular domain for an inhomogeneous fourth-order differential equation with lower-order terms. The uniqueness of a solution of the stated problem is proved. The solution is obtained explicitly using the Green's function constructed. [ABSTRACT FROM AUTHOR]

We discuss the well-posed solvability and exponential stability of solutions of abstract integro-differential equations where the kernels of integral operators are of general type and lie in the space of functions integrable on the positive half-line. The abstract integro-differential equations studied in the present paper are operator models of viscoelasticity theory problems. The proposed approach to the study of these integro-differential equations is related to an application of semigroup theory and can also be used to study other integro-differential equations containing integral terms of the Volterra convolution type. [ABSTRACT FROM AUTHOR]

The paper is devoted to constructing a numerical algorithm for calculating the blood flow in a volume vessel. A system of differential equations describing the dynamics of fluid in a single vessel with moving walls in cylindrical coordinates is derived assuming axial symmetry in arbitrary Eulerian-Lagrangian variables. A balance-characteristic scheme based on the Cabaret methodology is constructed for the obtained system of equations. The results of calculations of test problems are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the direct scattering problem on the half-line for the Dirac system of differential equations in the case of finite density with the boundary condition . The spectrum is studied, the resolvent is constructed, and the spectral expansion in the eigenfunctions of the Dirac operator is obtained. [ABSTRACT FROM AUTHOR]

The paper discusses the development of a method for constructing asymptotic formulas as for the fundamental solution system of two-term singular symmetric differential equations of odd order with coefficients in a broad class of functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation , the asymptotics of solutions in the case of various behavior of the coefficients and is studied. New asymptotic formulas are obtained for the case in which . [ABSTRACT FROM AUTHOR]

A linear integro-differential equation with a singular differential operator in the principal part is studied. Special versions of the generalized collocation method are proposed and justified to find its approximate solution in the space of generalized functions. [ABSTRACT FROM AUTHOR]

We consider the differential inclusion with the constraint , , on the derivative of the unknown function, where and are set-valued mappings, is superpositionally measurable, and is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form , , , . [ABSTRACT FROM AUTHOR]

We study integrally bounded solutions of the differential equation , where is a linear differential operator of order defined on functions () and is a finite-dimensional Euclidean space). The right-hand side is an integrally bounded function on ranging in and satisfying the inequality , , . Conditions are given on the operator and the function that guarantee an inverse inequality of the following form for the solutions under consideration: where the constant is independent of the choice of a real number and function . [ABSTRACT FROM AUTHOR]

A first-order regularized trace formula has been obtained for the Sturm–Liouville operator with a point of -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 shown that there appears an additional term in the regularized trace formula that takes into account the jump in the charge distribution function in the middle of the interval. Note that regularized traces are used to approximately calculate the first eigenvalues of the operator under consideration. These traces are also useful when solving inverse spectral analysis problems for differential equations. [ABSTRACT FROM AUTHOR]

We consider an initial–boundary value problem for a nonlinear Sobolev-type second-order differential equation with a noncoercive source describing the behavior of charges in a semiconductor plasma in an external field. The local solvability of this problem is proved, and upper bounds are obtained for the solution blow-up time. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

For the Kipriyanov operator written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator on the sphere and the corresponding spherical functions ( -harmonics) are introduced. The main properties of the operator on the sphere and the differential equation of -harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in is given. The solution is obtained by the Fourier method in the form of Laplace series in -harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in belong to the interval , , . [ABSTRACT FROM AUTHOR]

For the differential equation on the interval , we consider the three-point eigenvalue problem , , where is the spectral parameter, the point is fixed, and and , , are some complex numbers. Necessary and sufficient conditions that the coefficients and must satisfy for the indicated three-point problem to have degenerate boundary conditions are obtained. [ABSTRACT FROM AUTHOR]

A linear integro-differential equation with a singular differential operator in the main part is studied. To find its approximate solution in the space of generalized functions, a special version of the generalized collocation method is proposed and justified. [ABSTRACT FROM AUTHOR]

A nonlocal problem with an integral matching condition is studied for a parabolic-hyperbolic equation with two lines of type change containing a nonlinear loaded term. The uniqueness of the solution of the problem is proved by the method of energy integrals and the existence, using the theory of integral equations. Classes and sufficient conditions are determined for given functions that ensure the unique solvability of the problem under study. [ABSTRACT FROM AUTHOR]

The present paper deals with finding an optimal control of vibrations of a spherical layer, where the control is given by conditions of the third kind on the boundary. We continue and develop the results obtained by V.A. Il’in, E.I. Moiseev, and their students. We consider a more general control optimality criterion than in the previous papers. [ABSTRACT FROM AUTHOR]

We consider a linear homogeneous th-order differential equation, , with constant bounded operator coefficients in a Banach space. Under some conditions on the (real and complex) roots of the corresponding characteristic equation, we obtain a formula expressing the general solution of the differential equation via the operator functions given by the exponential, sine, and cosine of the roots. The case in which is even and the characteristic equation has pairs of complex-conjugate pure imaginary roots is investigated in detail. The case of a second-order differential equation is considered separately. [ABSTRACT FROM AUTHOR]

We consider the problem of dynamic reconstruction of unknown boundary disturbances in a parabolic equation. The disturbances are assumed to occur in the Neumann boundary condition. Based on the Osipov-Kryazhimskii dynamic regularization method, the problem is reduced to that of constructing a feedback control law for some auxiliary differential equation. We indicate an information-noise- and roundoff-error-robust algorithm for solving this problem. [ABSTRACT FROM AUTHOR]

The article reviews "On the Ill-Posedness of Structural Identification," a research paper, that was published in the 2006 issue of Differential Equations.

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1-7]. The present paper belongs to the same series. We analyze the possibility of constructing approximate solutions of high-order accuracy on a uniform or almost uniform grid for weakly singular integro-differential equations of Volterra type. [ABSTRACT FROM AUTHOR]

We study the uniqueness of the solution of a time-regular problem for the operator-differential equation with the Tricomi operator . The order of the differential expression is considered to be an arbitrary positive integer , and the regular boundary conditions are given with respect to the time variable . The operator is generated by the Tricomi equation . The boundary conditions for the Tricomi operator are given by the Dirichlet condition on the elliptic part and by the fractional derivative traces of the solution along the characteristics. It is indicated that this operator is a self-adjoint operator in the space . The self-adjointness of the operator guarantees the existence of a complete system of eigenfunctions orthonormal in if is a domain bounded by a Lyapunov curve and by characteristics of the wave equation. [ABSTRACT FROM AUTHOR]

The asymptotics of the eigenvalues and vector eigenfunctions of the Lamé system in a thin cylinder with the Dirichlet boundary conditions on the entire boundary is constructed and justified. The plane problem of elasticity theory with new elastic constants play the role of the limit problem. It is shown that when setting the Neumann conditions on the lateral surface of the cylinder, any boundary value problem for a system of differential equations on a section cannot provide the asymptotics of the eigenvalues of the three-dimensional problem, since the vector eigenfunctions are localized near the edge of a thin cylinder. Open questions are stated. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for general linear systems of complex partial differential equations in Banach spaces of entire vector functions with integral metrics. The type of exponential growth of the solution of the problem and the right-hand side of the system of differential equations with respect to the variable may depend in a certain way on the other variable . Necessary and sufficient conditions are obtained under which this problem is well-posed solvable in a given scale of function spaces. Therefore, a description of the structure of systems of differential equations for which the Cauchy problem is well posed is given. [ABSTRACT FROM AUTHOR]

The existence of vector functions of equilibrium prices in market models described by differential equations is studied. In the models under study, supply and demand depend not only on the commodity prices but also on the price change rate. Sufficient conditions for the existence of equilibrium in such models are obtained as corollaries of theorems on the existence of coincidence points for mappings acting from a -symmetric -quasimetric price space into the metric space of purchased sets of goods. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem in three-dimensional space for a first-order almost linear differential equation with discontinuous coefficients of the derivatives. Two special cases related to the behavior of characteristics are singled out and studied. [ABSTRACT FROM AUTHOR]

We study the properties of eigenvalues of a boundary value problem for a fourth-order differential equation on a geometric graph modeling the elastic deformations of a system of rods elastically hinged at the joints. We establish the simplicity conditions for the points of the spectrum of the corresponding differential operator. Estimates for the multiplicity of eigenvalues are derived. These estimates are given in terms of the topological characteristics of the graph. The estimates are sharp within the framework of these concepts. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions of an ordinary singular differential equation of arbitrary odd order. The potential in the equation can be either a rapidly oscillating function or a distribution. With the help of special quasiderivatives, the equation is reduced to a system of first-order differential equations, which is then reduced to -diagonal form by a successive application of Hausdorff transformations. [ABSTRACT FROM AUTHOR]

The aim of the paper is to show, using the one-dimensional problem as an example, what is to be expected and what should be pursued when studying the multidimensional case. The one-dimensional case has been chosen as a model, because here the problem admits an explicit solution permitting one to follow the phase transformation process. [ABSTRACT FROM AUTHOR]

In a finite-dimensional Euclidean space, we consider a differential game of two persons—a pursuer and an evader—described by a nonlinear autonomous controlled system of differential equations in normal form the right-hand side of which is the sum of two functions, one of which depends only on the state variable and the pursuer's control and the other, only on the state variable and the evader's control. The set of values of the pursuer's control is finite, and the set of values of the evader's control is compact. The goal of the pursuer is to bring the trajectory of the system from the initial position to any predetermined neighborhood of zero in finite time. The pursuer strategy is constructed as a piecewise constant function with values in a given finite set. To construct the pursuer control, it is allowed to use only information about the value of the current state coordinates. The evader's control is a measurable function for the construction of which there are no constraints on available information. It is shown that, to transfer the system to any predetermined neighborhood of zero, it is sufficient for the pursuer to use a strategy with a constant step of partitioning the time interval. The value of the fixed partitioning step is found in closed form. A class of systems is singled out for which an estimate of the transfer time from an arbitrary initial position to a given neighborhood of zero is obtained. The estimate is sharp in some well-defined sense. The solution essentially uses the notion of a positive basis in a vector space. [ABSTRACT FROM AUTHOR]

Here nonsmooth solutions of a differential equation are treated as solutions for which the compatibility conditions are not required to hold at the corner points of the domain and hence corner singularities can occur. In the present paper, we drop the compatibility conditions at three of the four vertices of a rectangle. At the remaining vertex, from which a characteristic (inclined) of the reduced equation issues, we impose compatibility conditions providing the C3, λ-smoothness of the desired solution in a neighborhood of that vertex as well as additional conditions leading to the smoothness of solutions of the reduced equation occurring in the regular component of the solution of the considered problem. Under our assumptions and for a sufficient smoothness of the coefficients of the equation and its right-hand side, we show that the classical five-point upwind approximation on a Shishkin piecewise uniform mesh preserves the accuracy specific for the smooth case; i.e., the mesh solution uniformly (with respect to a small parameter) converges in the L-norm to the exact solution at the rate O( N−1 ln2 N), where N is the number of mesh nodes in each of the coordinate directions. [ABSTRACT FROM AUTHOR]

In the present paper, in terms of a generalized solution of the wave equation, we perform an exhaustive study of the problem on the boundary control by an elastic force u x(0, t) = µ( t) at one endpoint x = 0 of a string in the presence of a model nonlocal boundary condition of one of four types relating (with the sign “+” or “−”) the values of the displacement u( x, t) or its derivative u x( x, t) at the boundary point x = l of the string to their values at some interior point $$ \mathop x\limits^ \circ $$ of the string (0 < $$ \mathop x\limits^ \circ $$ < l). We prove necessary and sufficient conditions for the existence of such boundary controls. Under these conditions, we optimize the controls by minimizing the boundary energy integral and then write out the optimal boundary controls in closed analytic form. [ABSTRACT FROM AUTHOR]

The article discusses the career and achievements of Anatolii Mikhailovich Samoilenko, academician of the National Academy of Sciences of Ukraine. Samoilenko graduated he graduated from the Faculty of Mechanics and Mathematics of Kyiv University in 1960 and defended his thesis for postgraduate and doctoral studies. He published several scientific papers and became one of the leading specialists in the field of ordinary differential equations and the theory of nonlinear oscillations.

The article reviews "On the Smoothness of the Solution of the Two-Endpoint Boundary Control Problem for the String Equation on an Arbitrary Time Interval," a research paper that was published in the 2006 issue of Differential Equations.

The present paper is a continuation of [1] and uses the notation and definitions introduced there. In [1], the authors also presented the properties of transformation operators, studied new function spaces, introduced the notion of s-trace, and proved the direct and inverse trace theorems. In the present paper, we pose a singular boundary value problem in a domain with a single corner (singular) point. We prove some auxiliary results. The main result is given by the theorem on the unique solvability of the corresponding boundary value problem. [ABSTRACT FROM AUTHOR]

The present paper gives a comparative analysis of spectral properties of two hyperbolic systems in a bounded plane domain. The basis property of systems of eigenelements is widely used in the construction of explicit solutions of boundary value problems in the form of biorthogonal series in the corresponding systems; thus special attention is paid in our investigations to this problem of spectral theory. The main results of the present paper are the following: (1.) the study of the spectrum structure and (2.) the analysis of properties of vector eigenfunctions: completeness, minimality, basis property, and conditional basis property. It is noteworthy that the systems of equations that we study have different spectral characteristics. [ABSTRACT FROM AUTHOR]

The present paper continues the study of the Neumann boundary value problem for the Laplace equation with a generalized boundary condition. This study was initiated in [1-4], where a screen problem was considered: the solution is sought outside a thin surface (screen) and the boundary condition is posed on both sides of the surface; moreover, the surface was assumed to be a part of the plane. The present paper deals with the case of interior and exterior problems in domains bounded by a smooth closed surface. The essential new feature is that the surface is not plane. We assume that the right-hand side of the boundary condition is a distribution treated as a continuous linear functional on the space of some test functions. We introduce the notion of generalized boundary values and generalized normal derivative of a function on the boundary of the domain. We prove the uniqueness of the solution of the problem. The existence of the solution is proved for the case in which the right-hand side of the boundary conditions is a continuous linear functional on the space of continuous functions on the boundary of the domain; moreover, the solution is constructed as a superposition of elements of a system of fundamental solutions. The present paper uses the notation introduced in [4]. [ABSTRACT FROM AUTHOR]

The problem of electromagnetic wave diffraction by bodies with ideally conducting surface or magnetodielectric bodies was considered by numerous authors. A general method for proving the solvability of the diffraction problem for elliptic equations and systems was developed by Vainberg. At the same time, integral, integro-differential, and singular integral equation methods, which admit a straightforward numerical implementation that can be useful in applications, were widely used in analysing the solvability of the diffraction problem for the Maxwell equations. The Galerkin method and a combined method based on the simultaneous consideration of the Helmholtz equation and the surface integral equation were also applied to this end. The cited papers establish the existence and uniqueness of a solution of the diffraction problem for certain characteristics of the bodies, for example, if the permittivity and permeability of the magnetodielectric scatterer are continuous or piecewise constant functions or if the magnetodielectric scatterer has some absorption. In the present paper, we use a different approach based on proving that an auxiliary interior boundary value problem has a weak solution, which determines a classical solution of the diffraction problem in the infinite domain. For the statement of the interior boundary value problem, we use the partial radiation conditions introduced in for a similar problem in a cylinder. The proof of the existence of a weak solution is based on the embedding theorem for vector fields and the introduction of a function space that takes into account the partial radiation conditions. [ABSTRACT FROM AUTHOR]

In this article, the author shows that eigenvalue distribution is necessary for the Bessel property of systems of normalized root functions of a differential operator, obtain a sharp estimate for the number of eigenvalues, and refine the Bessel property criterion. The Bessel property of systems of root functions of ordinary differential operators was studied in a number of papers, in all of these papers, the condition that the rank of eigenvalues is uniformly bounded or the "sum of units" was imposed for higher-order operators.

The main methods for integration of differential equations with the use of an admissible transformation group Gr are based on the theorem on the invariant representation of equations (e.g., see [1, 2]). By this theorem, the equations to be studied can be represented in an equivalent form determined by invariant functions of the group Gr. In the present paper, we prove a similar assertion for approximate groups of transformations introduced in [3]. (For a bibliographical survey and references, see [4].) By way of application, we show that the construction of regular approximately invariant solutions of differential equations with a small parameter can be reduced to the integration of differential equations with fewer independent variables. The exposition is carried out with first-order accuracy with respect to the small parameter. The generalization of results to the case of an arbitrary accuracy order leads to more cumbersome formulas with essentially the same meaning. In the present paper, we use the notation in [3, 4]. In particular, the approximate equality f ≅ g means that f = g + o(ε). [ABSTRACT FROM AUTHOR]

Complete quasi-hyperbolic operator-differential equations of even order with cons-tant domains were considered in [1, 2]. Quasi-hyperbolic operator-differential equa-tions of even order with variable domains in the case of a two-term leading part were analyzed in [3]. Complete hyperbolic operator-differential equations of the second order with variable domains were investigated in [4, 5]. In the present paper, we generalize and improve the results of all above-mentioned papers and consider complete quasi-hyperbolic operator-differential equations of even order with variable domains. In applications, such equations include hyperbolic equations such that the coefficients in the equations and in the boundary conditions [3] smoothly depend on time, singular hyperbolic equations [4], and "hyperbolic" equations of higher-order in the space variables, represented in the second part of the present paper. [ABSTRACT FROM AUTHOR]

One important property of Painlevé equations is their representability in the form of equivalent Hamiltonian systems with polynomial Hamiltonians. This property, originally discovered in [1] and later used in a number of papers [2-8], is especially important for the analysis of τ-functions [9], direct constructions of analogs of Painlevé equations from Hamiltonian systems [10], and isomonodromic deformation of linear systems described by Painlevé equations [11, 12]. In the present paper, we construct equivalent Hamiltonian systems for the first few equations in the series of higher-order Painlevé equations obtained by reduction from higher-order Korteweg-de Vries equations [2], construct analogs of τ-functions, and study polynomial Hamiltonians. [ABSTRACT FROM AUTHOR]

The investigation of asymptotic properties of some singularly perturbed boundary value problems and the choice of an appropriate method for the construction of solutions or approximations to solutions often prove to be very difficult. An analysis shows that such problems include boundary value problems leading to infinitely large values of the solution at a discontinuity point as the small parameter tends to zero. Typically, there is an initial jump in such problems [1-3]. The most general results in this direction were obtained in [4-7]. However, the asymptotics of the solution of a boundary value problem is constructed in these papers under the a priori assumption that the derivative of the desired solution or the fast variable is infinitely large as the small parameter tends to zero. We naturally encounter the problem of finding a class of boundary value problems for which the initial jump phenomenon occurs. This is the aim of the present paper. [ABSTRACT FROM AUTHOR]

The notion of stability of a solution of a differential equation is well known. Strong (or coefficient) stability is the stability of the solution under perturbations of the coefficients of the equation. On the abstract level, a perturbation of the coefficients can be treated as a perturbation of the operator of the problem. Here the main difficulty is that, as a rule, differential operators are unbounded and become bounded only for a special choice of function spaces. Strong stability was considered in a number of papers. For example, the strong stability of stationary boundary value problems was studied in [1-4]. The strong stability of operator-differential equations and operator-difference schemes in Hilbert spaces was considered in [5-10]. Note also the paper [11], where a new definition of strong stability was introduced. The present paper deals with the construction of a priori strong stability estimates for the Cauchy problem for an abstract first-order linear differential equation in Hilbert spaces under perturbations of the initial condition, right-hand side, and operators of the problem. We show that a priori strong stability estimates can be obtained directly on the basis of various a priori estimates for the solution of the Cauchy problem. The perturbations of the operators of the problem are estimated in the corresponding operator norms. [ABSTRACT FROM AUTHOR]

The theory of ordinary regular linear differential operators of order n on a finite interval was developed in the classical papers [1, 2] and further generalized in [3-7] and many other publications. Original, complete results for wide classes of differential operators, including partial differential operators, were obtained by Il'in and his students [8-14]. Other classes of differential operators (irregular differential operators and pencils of ordinary linear differential operators) were also analyzed starting from [15-28]. However, there is no complete theory of regular pencils of general ordinary differential operators [see formulas (10) and (11) below] in the literature. In particular, (1) definitive smoothness conditions for the coefficients of a pencil have not been found; (2) the notion of regularity, aptly stated by Birkhoff in a special case, has not received clear development; instead, it has acquired an intricate form involving a lot of parameters and useless in practice; (3) rigorously speaking, there is no paper specifically dealing with the pencil (10), (11), which is an important generalization of the case considered by Birkhoff. We suggest new asymptotic formulas with respect to a parameter for solutions of ordinary linear differential equations under very weak smoothness conditions on the coefficients of the equation. We give a definition of regularity in terms of simple algebraic parameters of the pencil and obtain necessary conditions for regularity in terms of "indices." [ABSTRACT FROM AUTHOR]