Abstract
We consider a spectral problem for a system of second order (in the spectral parameter) abstract pencils in a Hilbert space and prove the completeness and the Abel basis property of a system of eigenvectors and associated vectors. In some special cases, we obtain the expansion of vectors with respect to eigenvectors. Further, it is considered a relevant application of these abstract results to boundary-value problems for second and fourth order ordinary differential equations with a quadratic spectral parameter both in the equation and in boundary-value conditions.
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