On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

In this article the problems of the unique classical solvability and the construction of the solution of a nonlinear boundary value problem for a fifth order partial integro-differential equations with degenerate kernel are studied. Dirichlet boundary conditions are specified with respect to the spatial variable. So, the Fourier series method, based on the separation of variables is used. A countable system of the second order ordinary integro-differential equations with degenerate kernel is obtained. The method of degenerate kernel is applied to this countable system of ordinary integro-differential equations. A system of countable systems of algebraic equations is derived. Then the countable system of nonlinear Fredholm integral equations is obtained. Iteration process of solving this integral equation is constructed. Sufficient coefficient conditions of the unique solvability of the countable system of nonlinear integral equations are established for the regular values of parameter. In proof of unique solvability of the obtained countable system of nonlinear integral equations the method of successive approximations in combination with the contraction mapping method is used. In the proof of the convergence of Fourier series the Cauchy–Schwarz and Bessel inequalities are applied. The smoothness of solution of the boundary value problem is also proved.