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Partial inverse nodal problems for differential pencils on a star-shaped graph. (English) Zbl 1460.34030

The paper deals with the boundary value problem \(B:=B(q_{0},q_{1},h)\) consisting of the differential equation \[ -y''_{l}+(q_{l0}(x)+2\lambda q_{l1}(x)) y_{l}=\lambda^{2} y_{l}, \tag{\(\ast\)} \] on the finite interval \((0,1)\), with Robin conditions on the boundary vertices \[ y'_{l}(0)-h_{l}y_{l}(0)=0,\,\, l=\overline{1,m_{0}}, \] and the following continuity and Kirchhoff’s conditions \[ y_{1}(1,\lambda)=y_{l}(1,\lambda),\,\, l=\overline{2,m_{0}}, \] \[ \sum^{m_{0}}_{l=1}\,\,y'_{l}(1,\lambda)=0, \] respectively.
The authors study partial inverse nodal problems for differential pencils on a star-shaped graph, and propose a novel scheme for these problems. First, they obtain the asymptotic form of the solutions of \((\ast)\), the asymptotic formulae of the eigenvalues and the nodal points of the boundary value problem \(B\). Then, by the analysis of Weyl’s \(m\)-functions, they prove that the coefficients of the problem \(B\) can be uniquely determined by the twin-dense nodal subsets on the interior intervals \([a_{l},1]\) for any \(a_{l}\in[0,\frac{1}{2})\) under some conditions. Finally, the results are generalized on arbitrary intervals.

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
34B45 Boundary value problems on graphs and networks for ordinary differential equations
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