Geodesic
Spectral and Oscillatory Properties of a Linear Pencil of Fourth-Order Differential Operators
Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 55-67.

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The paper deals with the spectral and oscillatory properties of a linear operator pencil $A-\lambda B$, where the coefficient $A$ corresponds to the differential expression $(py'')''$ and the coefficient $B$ corresponds to the differential expression $-y''+cry$. In particular, it is shown that all negative eigenvalues of the pencil are simple and, under some additional conditions, the number of zeros of the corresponding eigenfunctions is related to the serial number of the corresponding eigenvalue.
Keywords: linear differential operator, initial boundary-value problem, pencil of operators, number of zeros of eigenfunctions. 
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J. Ben Amara; A. A. Vladimirov; A. A. Shkalikov. Spectral and Oscillatory Properties of a Linear Pencil of Fourth-Order Differential Operators. Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 55-67. http://geodesic.mathdoc.fr/item/MZM_2013_94_1_a4/

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