On solvability of a Boussinesq type integro-differential equation with nonlocal integral conditions

This article considers the questions of one value solvability of the nonlocal boundary value problem for a nonlinear Boussinesq type fourth-order integro-differential equation. The Fourier method of separation of variables is employed. The countable system of nonlinear integral equations (CSNIE) is obtained. To prove the theorem of one-value solvability of CSNIE the method of successive approximations is used. The convergence of the Fourier series to an unknown function of considering nonlocal boundary value problem is shown. This paper advances the theory of Boussinesq type differential equations.