Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, II

A hierarchy of Sturm–Liouville problems

“…Combining them with the above method of descending and climbing the hierarchy (which gives explicit formulae for the transformed quantities at each stage), we extend this aspect of inverse problems as well. Finally the existence question (which asks for conditions on the sequences of eigenvalues and norming constants guaranteeing that they do arise from a Sturm-Liouville problem) can also be tackled this way: see Reference [17] for details.…”

“…References to related work will be given as appropriate, but we note now that proofs of most of the results here can be found in References [16,17].…”

A discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter

“…In [21], Binding et al considered the Sturm-Liouville problem with the right boundary condition rationally dependent on the eigenparameter and got the existence of eigenvalues and the oscillation theory of eigenfunctions. The authors in [22] also discussed the inverse problems in terms of given spectral data.…”

“…Most of authors in these literature works focus on the asymptotic formula of eigenvalues and eigenfunctions, oscillation theory, etc. Moreover, what calls for special attention is that the eigenparameter dependent boundary conditions are of rational fraction of parameter λ of order one except [21,22]. Inspired by the above-mentioned literature, we consider the following discontinuous Sturm-Liouville equation:…”

Discontinuous Sturm–Liouville Problems with Eigenparameter-Dependent Boundary and Transmissions Conditions

“…Particularly, in recent years, highly important results in this field have been obtained for the case when the eigenparameter appears not only in the differential equation but also in the boundary conditions. The literature on such results is voluminous and we refer to [1,2,11,17] and corresponding bibliography cited therein. In particular, [2,4,11,12,14] and [15] contains many references to problems in physics and mechanics.…”