Eigenvalue problem associated with the fourth order differential-operator equation.

*(English)*Zbl 1404.34075Summary: In this paper, we investigate the boundary value problem for fourth order differential operator equations with unbounded operator coefficients and one \(\lambda\)-dependent boundary condition. We obtain an asymptotic formula for eigenvalues and a trace formula for the corresponding self-adjoint operator.

##### MSC:

34G10 | Linear differential equations in abstract spaces |

34L05 | General spectral theory of ordinary differential operators |

34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

34B09 | Boundary eigenvalue problems for ordinary differential equations |

34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |

##### Keywords:

Hilbert space; differential operator equation; spectrum; eigenvalues; trace class operators; regularized trace##### References:

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