We obtain sufficient conditions for weighted integrability of a generalized Fourier–Bessel transform of functions from generalized integral Lipschitz classes. These conditions are analogues of the well known Moricz conditions for classical Fourier transform. Also a Boas type result connecting the behavior of a function and the smoothness of its generalized Fourier–Bessel transform is proved.

Let $T$ be a contraction on a complex Hilbert space $\mathcal{H}$, and for $f\in \mathcal{H}$ define $$A_n(T)f=\frac{1}{n}\sum_{j=1}^nT^jf.$$ Let $(n_k)$ be an increasing sequence and $M$ be any sequence. We prove that there exists a positive constant $C$ such that $$\ang(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in M}}\|A_m(T)f-A_{n_k}(T)f\|_{\mathcal{H}}^2\ang)^{1/2}\leq C\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.

In this paper we consider an initial boundary value problem (direct problem) for a fourth order equation with the fractional Caputo derivative. Two inverse problems of determining the right-hand side of the equation by a given solution of the direct problem at some point are studied. The unknown of the first problem is a one-dimensional function depending on a spatial variable, while in the second problem a function depending on a time variable is found. Using eigenvalues and eigenfunctions, a solution of the direct problem is found in the form of Fourier series. Sufficient conditions are established for the given functions, under which the solution to this problem is classical. Using the results obtained for the direct problem and applying the method of integral equations, we study the inverse problems. Thus the uniqueness and existence theorems of the direct and inverse problems are proved.

We consider a simply connected domain of strip type with the symmetry of transfer. The boundary of the domain consists of circular arcs (circular numerable polygon). We write the Schwarz derivative of the mapping of a strip onto a circular numerable polygon in terms of elliptic functions. We obtain a generalization of the Schwarz--Christoffel formula for mapping of a strip onto a numerable polygon with the boundary consisting of straight line segments. One special case of a numerable polygon with additional symmetry with respect to a vertical line is considered.

In this research, we consider the fractional semilinear problem in a sequentially compact Banach space $X$: $x^{\alpha}(t)=A(t)x(t)+f(t,x(t))$, $t\in \mathbb R^{+} $, with the initial condition $x(0)=x_{0}$, $ x_{0} \in X $, where $A$ is the generator of an evolution system $({U(t,s)})_{t\leq s \leq {0}}$ and $f$ is a given function satisfying some assumptions. We study this fractional semilinear integro-differential equation and examine when it has an asymptotically almost periodic solution.

Rational approximations of the conjugate function on the segment $[-1,~1]$ by Abel--Poisson sums of conjugate rational integral Fourier-Chebyshev operators with restrictions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established.

Rational approximations on the segment $[-1,~1]$ of the conjugate function with density $(1-x)^\gamma,$ $\gamma\in (1/2,~1),$ by Abel-Poisson sums are studied. An integral representation of approximations and estimates of approximations taking into account the position of a point on the segment $[-1,~1]$ are obtained. An asymptotic expression as $r\to 1$ for the majorant of approximations, depending on the parameters of the approximating function is established. In the final part, the optimal values of parameters which provide the highest rate of decrease of this majorant are found. As a corollary we give some asymptotic estimates of approximations on the segment $[-1,~1]$ of the conjugate function by Abel-Poisson sums of conjugate polynomial Fourier-Chebyshev series.

In this paper we consider a mathematical model of control system in the form differential inclusion. The problem of controllability of this system under the condition of mobility of the terminal set $M=M(t)$ is researched. For this model of a dynamic system we define a notion of the $M$-controllability set. Using the methods of the theory of differential inclusions and multi-valued maps, the structural and topological properties of the $M$-controllability set are studied.

In the paper, we study direct and inverse problems for fractional partial differential equations of the Benney–Luke type. The conditions for the existence and uniqueness of solutions to the Cauchy problems for a Benney–Luke type equation of fractional order are derived. In addition, the inverse problem of finding the right-hand side of the equation is investigated.

For plane domains we define a new metric close to the Poincar\'{e} metric with the Gaussian curvature $k=-4$. For this quasi-hyperbolic metric we study inequalities of isoperimetric type. It is proved that the constant of the linear quasi-hyperbolic isoperimetric inequality for admissible subdomains of a given domain is finite if and only if the domain does not contain the point at infinity and has a uniformly perfect boundary. Also, we give estimates of these constants using some known numerical characteristics of domains.

\selectlanguage{english} We give the description of filtered Lie algebras over a perfect field of characteristic~2, to which graded non-alternating Hamiltonian Lie algebras are associated. Derivations and automorphisms of filtered Lie algebras are found.