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On a boundary value problem with a spectral parameter in the boundary conditions. (English) Zbl 0942.34067

The authors study the boundary value problem with a spectral parameter in the equation and the boundary conditions:

-u '' +q(x)u=λ 2 u,0<x<1,
(α 0 +α 1 λ+α 2 λ 2 )u(0)+u ' (0)=0,(β 0 +β 1 λ+β 2 λ 2 )u(1)+u ' (1)=0·

Here, λ is a spectral parameter, q(x) is a nonnegative continuous function on the interval [0,1], and α i and β i are real constants.

The authors study properties of the eigenvalues and eigenfunctions of the boundary value problem. The main result is a theorem on the number of zeros of eigenfunctions under their natural enumeration and a theorem on asymptotic formulas for eigenvalues and eigenfunctions. It was Shkalikov who drew the authors’ attention to the nontriviality of this problem in the case when the boundary conditions involve λ and λ 2 .

MSC:
34L05General spectral theory for OD operators
34B24Sturm-Liouville theory
34L20Asymptotic distribution of eigenvalues for OD operators
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory