On solvability of nonlocal boundary value problems for integro-differential equations
We study the solvability of the initial-boundary value problem for linear integro-differential equations with a lateral boundary condition correlating values of the solution or its conormal derivative with values of some integral operator on the solution. We prove existence and uniqueness theorems for regular solutions. Recently, nonlocal boundary value problems for parabolic and hyperbolic equations with integral conditions on the lateral boundary are intensively studied, primarily in the classical case of second- and fourth-order equations. The systematic study of nonlocal boundary value problems, the problems of finding periodic solutions to elliptic equations, began in the article by A. V. Bitsadze and A. A. Samarskii (1969). A great contribution to the development of the theory of nonlocal problems for differential equations of various classes was made by A. L. Skubachevsky (1997) and A. M. Nakhushev(2006, 2012).
 Kozhanov A. I., “Problems with integral-type conditions for some classes of nonstationary equations,” Dokl. Math., 90, No. 1, 440–443 (2014).
 Lykov A. V., “The application of methods of thermodynamics of irreversible processes to the study of heat-mass transfer,” Eng. Phys. J., 9, No. 3, 287–304 (1965).
 Nakhushev A. M., Problems with Displacement for Equations in Partial Derivatives [in Russian], Nauka, Moscow (2006).
 Nakhushev A. M., Loaded Equations and Their Application [in Russian], Nauka, Moscow (2012).
 Kozhanov A. I., “A nonlinear loaded parabolic equation and a related inverse problem,” Math. Notes, 76, No. 6, 784–795 (2004).
 Telesheva L. A., “On the solvability of the linear inverse problem for a higher order parabolic equation,” Mat. Zametki YAGU, 20, No. 2, 186–196 (2013).
 Fridman A., “Monotone decay of solutions of parabolic equations with nonlocal boundary conditions,” Q. Appl. Math., 44, No. 3, 401–407 (1986).
 Kozhanov A. I., “About solvability of a boundary problem with nonlocal boundary condition for linear parabolic equations,” Vestn. Samar. Gos. Tekhn. Univ., No. 30, 63–69 (2004).
 Abdrahmanov A. M. and Kozhanov A. I., “A problem with a nonlocal boundary condition for one class of odd-order equations,” Russ. Math., 51, No. 5, 1–10 (2007).
 Kozhanov A. I. and Pulkina L. S., “On the solvability of boundary value problems with a nonlocal boundary condition of an integral form for the multidimensional hyperbolic equations,” Differ. Equ., 42, No. 9, 1233–1246 (2006).
 Popov N. S., “On the solvability of boundary value problems for multidimensional pseudoparabolic equations with nonlocal boundary condition of integral type,” Mat. Zametki YaGU, 19, No. 1, 82–95 (2012).
 Popov N. S., “On the solvability of boundary value problems for multidimensional pseudo hyperbolic equations with a nonlocal boundary condition of integral type,” Mat. Zametki SVFU, 21, No. 2, 69–80 (2014).
 Kozhanov A. I., “Boundary value problems for a class of nonlocal integro-differential equations with degeneracy,” Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 23, No. 4, 19–24 (2017).
 Trenogin V. A., Functional Analysis [in Russian], Nauka, Moscow (1980).
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