Vol 88, No 4 (2024)

We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where, for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.Bibliography: 13 titles.

The first boundary value problem is studied for a mixed type equation with the Lavrentiev-Bitsadze operator in a rectangular domain. It is shown that the correctness of the problem statement essentially depends on the ratio of the sides of the rectangle from the hyperbolic part of the mixed type. The criterion of the uniqueness of the solution is established. The solution is constructed as a sum of Fourier series. At substantiation of uniform convergence of the series the problem of small denominators arises. In this connection the estimates of small denominators about separability from zero with the corresponding asymptotics are established. These estimates allowed us to prove the convergence of the series in the class of regular solutions of this equation. Estimates on the stability of the solution from given boundary functions and right-hand side are proved.

A class of nonlinear integral equations with a monotone Nemytskii operator on the positive half-line is considered. This class of integral equations occurs in many areas of modern natural science. In particular, such equations, under various restrictions on the nonlinearity and the corresponding kernel, arise in the dynamical theory $p$-adic strings for the scalar field of tachyons, in the kinetic theory of gases and plasmas in the framework of the conventional and modified nonlinear Bhatnagar-Gross-Krook model for the Boltzmann kinetic equation. Equations of a similar nature are also found in the theory of nonlinear radiative transfer in inhomogeneous media and even in the mathematical theory of the spread of epidemic diseases in the framework of the modified Diekmann-Kaper model. A constructive existence theorem for a bounded positive and continuous solution is proved. A uniform estimate of the difference between the previous and next iterations is obtained, and these successive approximations uniformly converge to a bounded continuous solution of the considered equation. We also study the asymptotic behavior of the constructed solution at infinity. In particular, it is proved that the solution at infinity has a positive limit, which is uniquely determined from some characteristic equation. It is also proved that the difference between the limit and the solution is an integrable function on the positive semi-axis. Using certain geometric inequalities for convex and concave functions, as well as relying on the proven integral asymptotics theorem, it is possible to prove the uniqueness of the solution in a certain subclass of non-negative non-trivial continuous and bounded functions. With the help of the results obtained, it is also possible to study a special class of nonlinear integral equations of the Urysohn type on the positive half-line. In particular, the existence of a positive and bounded solution of this class of equations is proved, and some qualitative properties of the constructed solution are studied. At the end of the paper, concrete examples of the applied nature of the corresponding kernel and nonlinearity are given to illustrate the importance of the results obtained.