In the work boundary value problems have been formulated and investigated for a fourth-order equations degenerating on the whole boundary of the domain. The existence, uniqueness and stability of the solution of the problem have been proved. At the same time, by applying the method of separation of variables to the considered problem, a spectral problem for an ordinary differential equation has been obtained. Next, the Green's function of the spectral problem was constructed, with the help of which it is equivalently reduced to an the second kind Fredholm integral equation with a symmetric kernel. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for solution the problem was obtained, from which follows its continuous dependence on the given functions.