Generalized solvability of the mixed value problem for a nonlinear integro-differential equation of higher order with a degenerate kernel

This paper is concerned with the generalized solvability of a mixed problem for a nonlinear integro-differential equation with a pseudo-parabolic operator of an arbitrary natural degree and with a degenerate kernel. V. A. Il'in's approach to the definition of a weak generalized solution of the problem posed with initial and boundary conditions is used. A Fourier series method based on the separation of variables is applied. A countable system of algebraic equations is obtained using the degeneracy of the kernel and by integration under initial conditions. The well-known Cramer method is modified to solve the countable system of algebraic equations and to derive the desired function from the sign of the determinant. This makes it possible to obtain a countable system of nonlinear integral equations for regular values of the spectral parameter. First, a lemma on the unique solvability of this countable system of nonlinear integral equations in a Banach space is proved by the method of contracting mappings. Next, a theorem on the convergence of the Fourier series obtained as a formal solution of the given mixed problem is proved. In the proofs of the lemma and the theorem, the Holder, Minkowski and Bessel inequalities are repeatedly applied.