3 ОПОВІДІ НАЦІОНАЛЬНОЇ АКАДЕМІЇ НАУК УКРАЇНИ ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2022. № 2: 3—11 Ц и т у в а н н я: Gutlyanskiĭ V.Ya., Nesmelova O.V., Ryazanov V.I., Yefimushkin A.S. Hilbert problem with measurable data for semilinear equations of the Vekua type. Допов. Нац. акад. наук Укр. 2022. № 2. С. 3—11. https://doi.org/10.15407/dopovidi2022.02.003 1. On completely continuous Hilbert operators. The basic part of definitions and historic comments can be found in papers [1] and [2]. However, let us recall here that a completely continuous mapping from a metric space M 1 into a metric space M 2 is defined as a continuous mapping on M 1 which takes bounded subsets of M 1 into relatively compact ones of M 2 , i.e., with compact closures in M 2 . When a continuous mapping takes M 1 into a relatively compact subset of M 1 , it is nowadays said to be compact on M 1 . The notion of completely continuous (compact) operators is due essentially, in the simplest partial cases, to Hilbert and F. Riesz, see the corresponding comments of Section VI.12 in [3], and to Leray and Schauder in the general case (see paper [4]). https://doi.org/10.15407/dopovidi2022.02.003 UDC 517.5 V.Ya. Gutlyanskiĭ 1 , https://orcid.org/0000-0002-8691-4617 O.V. Nesmelova 1, 2 , https://orcid.org/0000-0003-2542-5980 V.I. Ryazanov 1, 3 , https://orcid.org/0000-0002-4503-4939 A.S. Yefimushkin 1 1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk 2 Donbas State Pedagogical University, Slov’yansk 3 Bogdan Khmelnytsky National University of Cherkasy E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com, a.yefimushkin@gmail.com Hilbert problem with measurable data for semilinear equations of the Vekua type Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type () () ( ( )) z fz hzqfz   . The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form () () ( ( )) U z H zQU z   with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media. Keywords: Hilbert boundary-value problem, measurable boundary data, logarithmic capacity, semilinear equations of the Vekua type, nonlinear sources, angular limits, nontangent paths. МАТЕМАТИКА MATHEMATICS