On completeness of a part of eigen and associated vectors of a quadratic operator pencil for a double-point boundary value problem. (English) Zbl 1481.47016
Summary: In the paper we study some spectral properties of a quadratic operator pencil, solvability of one type of double-point boundary value problem for elliptic type operator-differential equation. Here, at first analytic properties of the resolvent of a quadratic pencil, structure of the spectrum of the given operator pensil are studied. Then the completeness of a part of the system of eigen and associated vectors of the space of traces of regular solutions and also completeness of descending elementary solutions in the space of all regular solutions of a homogeneous equation, are proved. All obtained results are expressed in terms of the properties of the coefficients of the given quadratic pencil.
MSC:
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 39B42 | Matrix and operator functional equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
Keywords:
operator pencil; completeness; system of eigenvectors; regular solution; elementary solutions; resolventReferences:
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