On a boundary value problem for Benney–Luke type differential equation with nonlinear function of redefinition and integral conditions

In three-dimensional domain a Benney–Luke type partial differential equation of the even order with integral form conditions, spectral parameter and small positive parameters in mixed derivatives is considered. The solution of this partial differential equation is studied in the class of regular functions. The Fourier method of separation of variables (Fourier series method) and the method of successive approximation in combination with the method of compressing mapping are used. Using the method of Fourier series, we obtain countable system of ordinary differential equations. So, the nonlocal boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the spectral parameter. The problem is reduced to solving countable system of linear algebraic equations. Using the given additional condition, we obtained the nonlinear countable system of functional equation with respect to redefinition function. Using the Cauchy-Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series.