Sturm--Liouville oscillation theory for impulsive problems

Yu. V. Pokornyi ; M. B. Zvereva ; S. A. Shabrov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 1, pp. 109-153

Résumé

This paper extends the Sturm–Liouville oscillation theory on the distribution of zeros of eigenfunctions to the case of problems with strong singularities of the coefficients (of $\delta$-function type). For instance, these are problems arising in the study of eigenoscillations of an elastic continuum with concentrated masses and localized interactions with the surrounding medium. The extension of the standard description of the problem is carried out by replacing the usual form of the ordinary differential equation $$ -(pu')'+qu=\lambda mu $$ by the substantially more general form $$ -(pu')(x)+(pu')(0)+\int_0^xu\,dQ=\lambda\int_0^xu\,dM $$ with absolutely continuous solutions whose derivatives, as well as the coefficients $p$, $Q$, $M$, belong to $\operatorname{BV}[0,l]$. The integral is understood in the Stieltjes sense.