Contemporary Mathematics. Fundamental Directions

We consider an ordinary 2m-th order differential operator with purely integral conditions. In this case, the domain of definition of the corresponding operator is not dense in L₂(0,1). Under certain conditions for the weight functions included in the integral conditions, the Abel summability of the system of root functions of the corresponding differential operator is proved.

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This work is a continuation of the research conducted in [2,3]. When forming the capital structure of enterprises (determining the shares of borrowed and equity funds), the balance of interests of owners and managers plays an important role. In this case, the owner is interested in obtaining the maximum return on his capital (maximizing its rentability ROE), and the manager — in increasing the financial stability of the company by reducing the weighted average cost of capital WACC. Resolving the conflict situation is possible, as shown by the authors, within the framework of nonantagonistic game theory, namely in the context of a bimatrix game and a cooperative game of two people. The owners of enterprises Π_i were chosen as player A, and the ROE values of enterprises from a representative sample, observed in a certain period T, were chosen as the components of his game matrix. The role of player B was played by the managers of enterprises Π_i, and the components of his game matrix were values of the form 1/(1+ WACC) determined in the same period T. The algorithms for finding the Nash equilibrium capital structure for both models were tested using real data for five enterprises in the metallurgical segment observed over a period of 5 years. As a result, it was possible not only to obtain new methods for determining the balanced capital structure, but also to analyze the stability of the obtained solutions with respect to perturbations of the initial data (reduced game matrices), which made it possible to calculate the corridors of possible changes in ROE and WACC for the reference set of enterprises Π_α in the selected (shortened) observation period, as well as to outline paths for further development of this topic.

In this paper, we study an incomplete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to one equation describing a number of applications from the mechanics of viscoelastic systems.

The oscillatory properties of the spectrum of a problem on the natural vibrations of a cross-shaped rod system are investigated. The model is reduced to a fourth-order boundary value problem on a graph with rigid joint conditions for the rods. A method is proposed for reducing the original problem to a multipoint boundary value problem on a selected route, allowing the system to be interpreted as an elastically supported rod. A justification for this method is provided, and a condition for the oscillatory nature of the spectrum is formulated.

We study the initial-boundary value problem for a second-order nonhomogeneous hyperbolic equation in a plane half-strip with constant coefficients, containing a mixed derivative, and with zero and nonzero potentials. This equation is the equation for the transverse oscillations of a moving finite string. We consider the case of Dirichlet-Neumann boundary conditions: the left end is fixed, and the right end is free. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on opposite sides of the origin. We seek a classical solution (or a solution almost everywhere, sometimes called a “strong solution”) to this problem. A spectral problem associated with the original initial-boundary value problem, generated by an ordinary differential operator-valued function (pencil) of second order, is investigated. The asymptotic behavior of the eigenvalues and resolvent is determined, the operator-valued function is linearized in the corresponding space of vector functions, and a theorem on the expansion of the first component of the vector function in root functions of the spectral problem is proved. A theorem on the uniqueness of the classical solution is formulated and proved, and a formula for the classical solution is derived in the form of a series of contour integrals. Then, using these formulas in the case of zero potential, theorems on finite formulas for the classical solution in special cases are proved, and based on these, a finite formula for the classical solution in the general case is obtained.

This study examines the small oscillations of an ideal stratified fluid within bounded domains. In classical formulations of these problems, the lateral boundaries of the container are typically assumed to align with the gravity vector g and the direction of density stratification. This research investigates non-classical cases where the container walls and the stratification direction form a specific angle. This geometric discrepancy results in a qualitative transformation of the internal wave spectrum. Analysis of a tilted rectangular vessel demonstrates that the angle between the domain boundary and the vector g significantly affects the spectral formation. Specifically, the spectrum of the small-oscillation operator is no longer purely discrete but includes regions of a continuous spectrum. Identifying the boundaries of the continuous spectrum is essential for the accurate resolution of non-homogeneous evolution problems. The tilt angle and the geometric parameters of the vessel determine both these boundaries and the transition points between the discrete and continuous spectra. The results indicate that the orientation of the cavity relative to the gravitational field is a primary factor determining the properties of internal waves in a bounded volume of stratified fluid.