Geodesic
Sign Regularity Conditions for Discontinuous Boundary-Value Problems
Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 643-655

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For a multipoint discontinuous boundary-value problem with nonoscillating differential operator, we present an analog of the Kalafati conditions that ensure the sign regularity property, i.e., the property that the number of sign changes of the solution does not exceed the number of sign changes of the function on the right-hand side. The sign regularity property allows one to verify whether the spectrum of the corresponding spectral problem exhibits the Sturm properties (i.e., the reality, positiveness, and simplicity of eigenvalues, the alternation of zeros of eigenfunctions, etc.). 
@article{MZM_2003_74_5_a0,
     author = {A. V. Borovskikh},
     title = {Sign {Regularity} {Conditions} for {Discontinuous} {Boundary-Value} {Problems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--655},
     publisher = {mathdoc},
     volume = {74},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a0/}
}
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A. V. Borovskikh. Sign Regularity Conditions for Discontinuous Boundary-Value Problems. Matematičeskie zametki, Tome 74 (2003) no. 5, pp. 643-655. http://geodesic.mathdoc.fr/item/MZM_2003_74_5_a0/