Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author's paper in 2009, where the abstract theory of fold completeness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may contain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary conditions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach L_q-framework, in contrast to previously known results in the Hilbert L₂-framework. Some concrete mechanical problems are also presented.