A. I. Inozemtsev

Russian State Agrarian University -- Moscow Timiryazev Agricultural Academy (Moscow, Russian Federation)

Abstract. The paper contains sufficient conditions for existence and uniqueness of a solution to the Romanovsky equation with a generalized partial integral operator in anisotropic Lebesgue spaces. The solution obtained by the method of successive approximations is represented in the form of a Neumann operator series. Conditions for its convergence are indicated. Also conditions for boundedness of partial integral operators, their iterations, and estimates of their norms in the corresponding anisotropic Lebesgue spaces are presented. A formula is established for the iterated kernels of the generalized Romanovsky partial integral operators, and their relationships with the iterations of partial integral operators are obtained, too. For the norm of the iterated kernel, its boundedness is proven, and dependence of this norm on the norm of the kernel of the generalized Romanovsky partial integral operator is shown. The necessity of considering generalized Romanovsky partial integral operators and their iterated kernels in spaces of essentially bounded functions is indicated. A formula for the resolvent of the generalized Romanovsky partial integral equation and its belonging to the space of functions essentially bounded with respect to some of their variables is established. The obtained conditions for the existence and uniqueness of a solution to an equation with a generalized partial integral Romanovsky operator in anisotropic Lebesgue spaces are applied to the study of linear generalized partial integral equations of the Romanovsky type, to which the problem of $n$-connected Markov chains is reduced.

Key Words: partial integral operator, Romanovsky operator, Romanovsky-type equation, anisotropic spaces, operator iterations, resolvent

For citation: A. I. Inozemtsev. On Romanovsky equations with a generalized partial-integral operator in anisotropic Lebesgue spaces. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:3(2026), 179–192. DOI: https://doi.org/10.15507/2079-6900.28.202603.179-192

Information about the author:

Aleksey I. Inozemtsev, Ph.D. (Phys. and Math.), Associate Professor, Department of Higher Mathematics, Russian State Agrarian University -- Moscow Timiryazev Agricultural Academy (49, Timiryazevskay St., Moscow 127434, Russian Federation), ORCID: http://orcid.org/0000-0002-7662-8991, a.inozemcev@rgau-msha.ru