We study a connection between $\rho$ - trigonometrically convex functions and the class of subharmonic functions. The established connection is used to prove new inequalities characterizing $\rho$ - trigonometrically convex functions and to find integral equations of the first kind for $\rho$ - trigonometric functions. Under a detailed development of this issue, there appears the convolution integral equation $$ h(\theta)=\int\limits_{-\infty}^{\infty}h(\theta-u)d\sigma(u), $$ where $\sigma$ is a finite compactly supported measure. The results on the theory of this equation are exposed following A.F. Leontiev, who studied this equation in relation with the theory of Dirichlet series. Using the Leontiev interpolating function, we propose additional conditions ensuring that a continuous solution to the equation \begin{equation*} h(\theta)=\int\limits_{-\infty}^{\infty}a_R(u)h(\theta-u)du \end{equation*} for a fixed $R$ is a $\rho$ - trigonometric function.

In this paper we consider parabolic equation with nonlinear memory and absorption \begin{equation*} u_t= \Delta u + a \int\limits_0^t u^q (x,\tau) d\tau - b u^m, x \in \Omega, t>0, \end{equation*} under nonlinear nonlocal boundary condition \begin{equation*} u(x,t) = \int\limits_{\Omega}{k(x,y,t)u^l(y,t)} dy, x\in\partial\Omega, t > 0, \end{equation*} and nonnegative continuous initial data. Here $a,$ $b,$ $q,$ $m,$ $l$ are positive numbers, $\Omega$ is a bounded domain in $\mathbb{R}^N,$ $N\geq1,$ with smooth boundary $\partial\Omega,$ $k(x,y,t)$ is a nonnegative continuous function defined for $x \in \partial \Omega$, $y \in \overline\Omega$ and $ t \ge 0.$ We prove that each solution of the problem is global if $\max (q,l) \leq 1$ or $\max (q,l) > 1$ and $ l < (m + 1)/2,$ $q \leq m.$ If $l>\max\{1, (p+1)/2\}$ and the function $k(x,y,t)$ is positive for small $t,$ the solutions blow up in finite time for large enough initial data. The obtained results improve previously established conditions for the existence and absence of global solutions.

This paper is devoted to studying the reaction - diffusion systems with rapidly oscillating coefficients in the equations and in boundary conditions in domains with locally periodic oscillating boundary; on this boundary a Robin boundary condition is imposed. We consider the supercritical case, when the homogenization changes the Robin boundary condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition in the limit as the small parameter, which characterizes oscillations of the boundary, tends to zero. In this case, we prove that the trajectory attractors of these systems converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction - diffusion systems in the domain independent of the small parameter. For this aim we use the homogenization theory, asymptotic analysis and the approach of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of dissipative evolution equations. The homogenization method and asymptotic analysis are used to derive the homogenized reaction - diffusion system and to prove the convergence of solutions. First we define the appropriate auxiliary functional spaces with weak topology, then, we prove the existence of trajectory attractors for these systems and formulate the main Theorem. Finally, we prove the main convergence result with the help of auxiliary lemmas.

We describe the commutant of system of integration operators in the Fréchet space $H(\Omega)$ of all functions holomorphic in a domain $\Omega$ in $\mathbb C^N,$ which is polystar with respect to the origin. In particular, among such domains, there are the products of domains $\mathbb C$ being star with respect to the origin and complete Reinhardt domains with center at the origin. As in the one - dimensional case, the operators in the commutant are the Duhamel operators. We show that $H(\Omega)$ with the Duhamel product $\ast$ is an associative and commutative topological algebra. It is topologically isomorphic to the commutant with the product, which is the composition of operators, and with the topology of bounded convergence. We obtain a similar to one - dimensional representation of the product $f\ast g$ as a sum containing one term being a multiple of $f$ and terms with the integrals at least in one variable of the function independent of the derivatives of $f$. By means of this representation we prove the criterion of $\ast$ - invertibility of a function in $H(\Omega)$ and the corresponding convolution operator. We establish that the algebra $(H(\Omega), \ast)$ is local. In the case when the domain $\Omega$ is in addition convex, in the dual situation we obtain the criterion for the invertibility of operator from the commutant of system of operators of partial backward shift.

We show that the inverse scattering transform technique can be applied to obtain the time dependence of scattering data of the Zakharov - Shabat system, which is described by the loaded nonlinear Schrödinger equation in the class of fast decaying functions. In addition we prove that the Cauchy problem for the loaded nonlinear Schrödinger equation is uniquely solvable in the class of rapidly decreasing functions. We provide the explicit expression of a single soliton solution for the loaded nonlinear Schrödinger equation. As an example, we find the soliton solution of the considered problem for an arbitrary non - zero continuous function $\gamma(t).$

The Krause mean process serves as a comprehensive model for the dynamics of opinion exchange within multi - agent system wherein opinions are represented as vectors. In this paper, we propose a framework for opinion exchange dynamics by means of the Krause mean process that is generated by a cubic doubly stochastic matrix with positive influences. The primary objective is to establish a consensus within the multi - agent system.

We construct asymptotics for the eigenvalues and eigenfunctions of the Laplace operator in a thin polyhedron with parallel closely spaced bases and skewed narrow lateral faces. On the bases we impose the Dirichlet conditions, while on the lateral faces the Dirichlet or Neumann conditions are imposed. Their distribution over the faces, as well as the slope of the latter, significantly affect the behavior of eigenfunctions when the domain becomes thinner. We find situations, in which the eigenfunctions are distributed along the entire polyhedron and localized near its lateral faces or vertices. The results are based on the analysis of the spectrum (cut - off point, isolated eigenvalues, threshold resonances, etc.) of auxiliary problems in a half - strip and a quarter of a layer with skewed end and lateral sides, respectively. We formulate open questions concerning both spectral and asymptotic analysis.

In the work we obtain a necessary and sufficient condition for the zeros of analytic functions in area Privalov classes $\tilde{\Pi}_q$ $(0<q<1)$ in the unit circle $D=z\in \mathbb{C}: |z|<1\}$ located in the Stolz angles. We solve the free interpolation problem in these classes under the condition that the interpolation nodes are located in the Stolz angles. We also solve the interpolation problem in the area Privalov classes in circle on Carleson sets.

Among several approaches towards the classical Bernoulli polynomials $B_n(x)$, one is the definition by the generating function \begin{equation*} \frac{we^{xw}}{e^w-1}=\sum\limits_{n=0}^{\infty}B_{ n}(x)\frac{w^n}{n!} for |w|<2\pi. \end{equation*} As a generalization of $B_n(x)$, for any positive integer $N$, a new class of Bernoulli polynomials called Hypergeometric Bernoulli polynomials of order $N$, $B_n(N, x)$ was established. For the particular case $N=2$ these polynomials are given by \begin{equation*} \frac{1}{2}\frac{w^2e^{xw}}{e^w-1-w}=\sum\limits_{n=0}^{\infty}B_n(2,x) \frac{w^n}{n!} for |w|<2\pi. \end{equation*} Several asymptotic formulas for the Bernoulli and Euler polynomials inside different domains related to the roots of $\phi(w)=e^w-1$ were found. In this paper, we consider an integral representation for $B_n(2,x)$ and establish a zero attractor for the re-scaled polynomials $B_n(2,nx)$ for large values of $n$. We briefly discuss some analogous asymptotic formulas of $B_n(2,x)$ inside domains related to the roots of $\varphi(w)=e^w-1-w$.

We suggest an algorithm for deriving nonlinear integrable equations of the form $$u^{j+1}_{n,x}=F(u^{j}_{n,x},u^{j+1}_{n},u^{j}_{n+1},u^{j}_{n},u^{j+1}_{n-1})$$ with three independent variables; the algorithm uses the known list of Toda type integrable equations. The algorithm is based on the Darboux integrable finite field reductions, construction of a complete set of characteristic integrals and dicretization via integrals.