Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University868410.18287/2541-7525-2020-26-3-7-16Research ArticleON THE SOLVABILITY OF SOME BOUNDARY VALUE PROBLEMS
WITH INVOLUTIONNazarovaK. Zh.<p>Candidate of Physical and Mathematical Sciences, associate professor of the Department of Mathematics</p>gjnazarova@mail.ruhttps://orcid.org/0000-0002-2093-1879TurmetovB. Kh.<p>Doctor of Physical and Mathematical Sciences, professor of the Department of Mathematics</p>turmetovbh@mail.ruhttps://orcid.org/0000-0001-7735-6484UsmanovK. I.<p>Candidate of Physical and Mathematical Sciences, associate professor of the<br />Department of Mathematics</p>y_kairat@mail.ruhttps://orcid.org/0000-0002-1377-4633Akhmet Yassawi International Kazakh-Turkish University250920202637163004202130042021Copyright © 2020, Nazarova K.Z., Turmetov B.K., Usmanov K.I.2020<p>This article is devoted to the study of the solvability of some boundary value problems with involution.<br>In the space Rn, the map Sx=−x is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.</p>инволюциянелокальное уравнениенелокальная задачанаклонная производнаяуравнение Пуассонагладкостьсуществованиеединственностьinvolutionnonlocal equationnonlocal problemoblique derivativePoisson equationsmoothnessexistenceuniqueness[[1] Carleman T. Sur la theorie des equations integrales et ses applications. Verhandlungen des Internationalen Mathematiker-Kongresse Zurich, 1932, vol. 1, pp. 132–151. 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