On a nonlocal inverse problem for a Benney-Luke type integro-differential equation with degenerate kernel

Are considered the questions of generalized solvability of the nonlocal direct and inverse boundary value problems with integral conditions for a restore the resource and boundary regimes for a nonlinear Benney-Luke type integro-differential equation of the fourth order with degenerate kernel. The method of degenerate kernel is applied and developed for the case of considering Benney-Luke type integro-differential equation of the fourth order. The required solutions of the problem are decomposed into series by the aid of Fourier method of separation of variables. Is obtained a system of countable system of algebraic equations. Solving this system is derived a system of two countable system of nonlinear functional integral equations with respect to first and second unknowing variables and the formula for calculate the third unknowing variable. Is proved the one value solvability of the considered problem. Is used the method of compressing mapping.