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).