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Inverse nodal problem for \(p\)-Laplacian Dirac system. (English) Zbl 1385.34019

Consider the following boundary value problem for the \(p\)-Laplacian Dirac system: \[ (y_{2}^{p-1})'=(p-1)(\lambda-q(x))y_{1}^{p-1}, \]
\[ (y_{1}^{p-1})'=(p-1)(-\lambda+r(x))y_{2}^{p-1} \] and \[ y_{1}(0)=y_{2}(0)=1, \]
\[ \lambda^{\frac{4}{p}}y_{1}(1)+\alpha \lambda^{3}y_{2}(1)=0. \] The authors solve an inverse nodal problem with boundary conditions depending on the spectral parameters. The asymptotic estimates of eigenvalues and nodal points are obtained by a modified Prüfer substitution. The asymptotic estimates of nodal length are also obtained. Formulae for the reconstructions of \(r\) and \(q\) follow in the same way.

MSC:

34A55 Inverse problems involving ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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