Vol 68, No 2 (2022)


Stochastic Equations and Inclusions with Mean Derivatives and Their Applications

Gliklikh Y.E.


This work is a detailed presentation of the results, mainly obtained in recent years by the author and his school of the research of mean derivatives of random processes, stochastic equations and inclusions with mean derivatives, as well as their applications in various mathematical disciplines, mainly in mathematical physics. In addition, the work contains introductory material on mean derivatives by E. Nelson, who introduced this concept in the 60s of the XXs century, the results of other researchers on this topic, and preliminary concepts from various areas of mathematics used in this work.

Contemporary Mathematics. Fundamental Directions. 2022;68(2):191-337
pages 191-337 views

On the Completeness of Eigenfunctions of One 5th-Order Differential Operator

Rykhlov V.S.


In this paper, we fully solve the problem of the completeness of the eigenfunctions of an ordinary 5th-order differential operator in the space of square-summable functions on the segment [0, 1] generated by the simplest differential expression \( y^{(5)} \) and two-point two-term boundary conditions \(\alpha_v y^{(v−1)}(0) + \beta_v y^{(v−1)}(1) = 0\) and \(v = \overline{1, 5}\), under the main assumption \(\alpha_v \ne 0\), \(v = \overline{1, 5}\) or \(\beta_v \ne 0\), \(v = \overline{1, 5}\) (in this case, without loss of generality, we can assume that all \(\alpha_v\) or all \(\beta_v\) , respectively, are equal to one). The classical methods of studying completeness, which go back to well-known articles by M. V. Keldysh, A. P. Khromov, A. A. Shkalikov, and many others, are not applicable to the operator under consideration. These methods are based on “good” estimates for the spectral parameter of the used generating functions (“classical”) for the system of eigenfunctions and associated functions. In the case of a strong irregularity of the operator under consideration, these «classical» generating functions have too large rate of grows in the spectral parameter. To solve the problem of multiple completeness, we propose a new approach that uses a special parametric solution that generalizes «classical» generating functions. The main idea of this approach is to select the parameters of this special solution to construct generating functions that are no longer «classical» with suitable estimates in terms of the spectral parameter. Such a selection for the operator under consideration turned out to be possible, although rather nontrivial, which allowed us to follow the traditional scheme of proving the completeness of the system of eigenfunctions in the space of square-summable functions on the segment [0, 1].

Contemporary Mathematics. Fundamental Directions. 2022;68(2):338-375
pages 338-375 views

Asymptotic Behavior of the Solution for One Class of Nonlinear Integral Equations of Hammerstein Type on the Whole Axis

Khachatryan K.A., Petrosyan H.S.


A class of nonlinear integral equations on the whole axis with a noncompact integral operator of Hammerstein type is investigated. This class of equations has applications in various fields of natural science. In particular, such equations are found in mathematical biology, in the kinetic theory of gases, in the theory of radiation transfer, etc. The existence of a nonnegative nontrivial and bounded solution is proved. The asymptotic behavior of the constructed solution on ±∞ is studied. In one important special case, the uniqueness of the constructed solution in a certain weighted space is established. At the end of the work, specific applied examples of the equations under study are given.

Contemporary Mathematics. Fundamental Directions. 2022;68(2):376-391
pages 376-391 views

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies