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Equivalence of fourth order boundary value problems and matrix eigenvalue problems. (English) Zbl 1267.34036

The authors establish equivalent relations between a class of regular self-adjoint fourth-order linear boundary value problems and a class of matrix problems. More precisely, they show that
(i) for any boundary value problem in the first class, there exists a matrix problem in the second class with exactly the same spectrum;
(ii) for any matrix problem in the second class, there exists a boundary value problem in the first class with exactly the same spectrum.
This work is an extension of an earlier work by Kong, Wu, and Zettl in 2001 and that by Kong, Volkmer, and Zettl in 2009 for second-order problems, and is a follow-up of the authors’ work on fourth-order problems.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
15A18 Eigenvalues, singular values, and eigenvectors
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References:

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