Two-fold completeness of root vectors of a system of quadratic pencils.

Abstract: <br /> <br /> We consider a spectral problem for a system of second order (in the <br /> spectral parameter) abstract pencils in a Hilbert space and <br /> 1) prove the completeness of a system of eigenvectors and associated <br /> vectors, <br /> 2) obtain, in some special cases, the expansion of vectors with respect <br /> to eigenvectors, <br /> 3) consider a relevant application of these abstract results to <br /> boundary-value problems for second order ordinary differential equations with a <br /> quadratic spectral parameter in both the equation and boundary conditions. <br /> We also <br /> 4) study an initial boundary-value problem for hyperbolic <br /> differential-operator equations and <br /> 5) give a relevant application to hyperbolic PDEs with the same <br /> (second) order of differentiation with respect to the time in both the equation and <br /> boundary conditions.