Vol 72, No 3 (2026): On the centenary of the birth of Georgii Alexandrovich Kamenskii

This paper addresses the eigenvalue problem for the Laplace operator in a bounded planar domain. The general theory of compact self-adjoint operators implies that the spectrum of the problem is discrete and tends to infinity. While finding the spectrum for an arbitrary domain is a highly complex problem, well-developed methods exist for estimating the first eigenvalue in certain types of domains. One such method involves constructing an ``effectless cut'' --- a topological cut that does not alter the first eigenvalue of the operator --- as proposed by Hersch and Weinberger. Additionally, when the domain is a ring, a method exists for approximating the first eigenfunction using a Morse--Smale function. In this paper, assuming the first eigenfunction of the Laplace operator on a ring to be a Morse--Smale function, we establish the existence of an effectless cut homeomorphic to a circle. For a domain consisting of a 2-disk with more than one hole, we also demonstrate the existence of an effectless cut that separates one boundary component from the others. We present examples of cuts that are not homeomorphic to a circle.

A criterion for the asymptotic stability of the zero solution for systems of differential equations with almost-periodic coefficients in the linear terms is established in terms of the solvability of the Lyapunov differential equation. A uniform estimate for the solutions of such systems is derived, making it possible to determine the rate of decay as $t \to +\infty.$ The attraction domain is characterized, and an estimate for the rate of convergence of the zero solution for systems of nonlinear differential equations with almost-periodic coefficients in the linear terms is obtained. Robust stability under almost-periodic perturbations is investigated.

This paper establishes local and global existence theorems for integral solutions to the Cauchy problem for a class of semilinear differential inclusions of mixed type with deviating argument in Banach spaces. The proofs employ topological methods of nonlinear analysis for set-valued condensing mappings.

This work addresses the stability (uniform, asymptotic, and exponential) of linear autonomous differential equations of neutral type. These types of stability are interpreted as properties of an integral functional defined on Lebesgue spaces of measurable functions. This approach enables a refined analysis of the most complex form of stability --- asymptotic stability --- and allows for distinguishing between cases where it coincides with exponential stability and cases where it does not. The latter case arises when the operator acting on the derivative is noninvertible, causing the equation to exhibit properties that distinguish it from differential equations of retarded type. A dedicated section examines uniform stability, for which an easily verifiable criterion has been established.

A nonlinear system of three differential equations with distributed delay and nonnegative initial data is studied. The first two equations describe the number of two interacting populations, and the third equation describes the number of complexes formed as a result of contact interactions between pairs of individuals. The duration of the complexes' existence is determined by a distribution function on a finite period of time. The resulting Cauchy problem has a unique solution on the semiaxis, and the components of the solution are nonnegative. The system of differential equations has one equilibrium position with nonnegative components, and this equilibrium position is locally asymptotically stable. Similar results have been established for systems of equations containing a concentrated delay and without delay.

In this paper we examine the initial-value problem for parabolic-type differential-difference equations with a single spatial shift in two cases. In the first case, the equation contains, in addition to a differential operator, a difference operator corresponding to shifts in the higher-order terms; in the second case, the shift appears in the lower-order term. We construct the solutions of the equations and then prove why these solutions are fundamental. In both cases, the solution to the initial-value problem is constructed in a form of a convolution of the fundamental solution with the initial-value function, which is assumed to be summable. The behavior of the constructed solutions and their derivatives at large time values are examined and the convergence of the resulting solutions to the initial function is also established. A visual comparison of the behavior and temporal dynamics of the fundamental solutions of the equations under different values of the parameters involved is implemented using the numerical left rectangle rule.