We give some necessary and sufficient conditions for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray and C\v .V. Stanojevi´c and extend some results of F. M´oricz proved for Walsh–Fourier series

We construct the asymptotics of the eigenvalues for a quasidifferential Sturm-Liouville boundary value problem on eigenvalues and eigenfunctions considered on a segment J=[a,b], with the boundary conditions of type I on the left - type I on the right, i.e., for a problem of the form (in the explicit form of record) p22(t)(p11(t)(p00(t)x(t))′+p10(t)(p00(t)x(t)))′+p21(t)(p11(t)(p00(t)x(t))′+p10(t)(p00(t)x(t)))+ +p20(t)(p00(t)x(t))=−λ(p00(t)x(t))(t∈J=[a,b]), p00(a)x(a)=p00(b)x(b)=0. The requirements for smoothness of the coefficients (i.e., functions pik(⋅):J→R,k∈0:i,i∈0:2) in the equation are minimal, namely, these are: functions pik(⋅):J→R are such that functions p00(⋅) and p22(⋅) are measurable, nonnegative, almost everywhere finite and almost everywhere nonzero, functions p11(⋅) and p21(⋅) are also nonnegative on segment J, and in addition, functions p11(⋅) and p22(⋅) are essentially bounded on J, functions 1p11(⋅),p10(⋅)p11(⋅), p20(⋅)p22(⋅),p21(⋅)p22(⋅),1min{p11(t)p22(t),1} are summable on segment J. Function p20(⋅) acts as a potential. It is proved that under the condition of nonoscillation of a homogeneous quasidifferential equation of the second order on J, the asymptotics of the eigenvalues of the boundary value problem under consideration has the form \lambda_k=\big(\pi k\big)^2 \Big(D+O\big({1}\big{/}{k^2}\big)\Big) as k→∞, where D is a real positive constant defined in some way.

In this paper, we study the direct and two inverse problems for a model equation of mixed parabolic-hyperbolic type. In the direct problem, the Tricomi problem for this equation with a non-characteristic line of type change is considered. The unknown of the inverse problem is the variable coefficient at the lowest derivative in the parabolic equation. To determine it, two inverse problems are studied: with respect to the solution defined in the parabolic part of the domain, the integral overdetermination condition (inverse problem 1) and one simple observation at a fixed point (inverse problem 2) are given. Theorems on the unique solvability of the formulated problems in the sense of classical solution are proved.

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray-Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.

A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly time-oscillating right-hand side is considered on a positive time semi-axis. For this problem, which depends on a large parameter (high oscillation frequency), a limiting (averaged) multipoint boundary value problem is constructed and a limiting transition in the Hölder space of bounded vector functions defined on the considered semi-axis is justified. Thus, for normal systems of differential equations in the case of a multipoint boundary value problem, the Krylov-Bogolyubov averaging method on the semi-axis is justified.

In this paper, the inverse spectral problem method is used to integrate a nonlinear sine-Gordon type equation with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies sine-Gordon-type equations with an additional term.

In this paper we study the locally uniform convergence of homeomorphisms with bounded (1,σ)-weighted (q,p)-distortion to a limit homeomorphism. Under some additional conditions we prove that the limit homeomorphism is a mapping with bounded (1,σ)-weighted (q,p)-distortion. Moreover, we obtain the property of lower semicontinuity of distortion characteristics of homeomorphisms.

We consider a matrix model A, related to a system describing two identical fermions and one particle of another nature on a lattice, interacting via annihilation and creation operators. The problem of the study of the spectrum of a block operator matrix A is reduced to the investigation of the spectrum of block operator matrices of order three with a discrete variable, and relations for the spectrum, essential spectrum and point spectrum are established. Two-particle and three-particle branches of the essential spectrum of the block operator matrix A are singled out.