Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel

We consider the questions of unique solvability of the inverse problem for a nonlinear partial Benney–Luke type integro-differential equation of the fourth order with degenerate kernel. The method of degenerate kernel designed for Fredholm integral equations of the second kind is modified to the case of considered partial Benney–Luke type integro-differential equation of the fourth order. We use the Fourier method of separation of variables. After designation, the Benney–Luke type integro-differential equation is reduced to a system of algebraic equations. With the help of additional condition we obtain the countable system of nonlinear integral equations with respect to main unknown function. We use the method of successive approximations combined with the method of compressing maps. Further is defined the restore function.