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Oscillation theory for Sturm-Liouville problems with indefinite coefficients. (English) Zbl 1002.34018
The paper is concerned with the in the sense of A. Zettl [Hinton, Don (ed.) et al., Spectral theory and computational methods of Sturm-Liouville problems. Proceedings of the 1996 conference, Knoxville, TN, USA, in conjunction with the 26th Barrett memorial lecture series. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 191, 1-104 (1997; Zbl 0882.34032)] Sturm-Liouville equation \[ -(py')'+qy=\lambda ry \text{ a.e. on }I=[a_0,a_1],\tag{1} \] satisfying \[ (\cos\alpha_j)y(a_j)=(\sin\alpha_j)(py')(a_j). \tag{2} \] It is assumed that \(p(x)\neq 0\) a.e. and \(1/p,q,r\in L_1(I),\) with \(-\infty\leq a_0<a_1\leq\infty.\) The oscillation and related results are studied for two basic cases where one of \(p\) and \(r\) is indefinite, i.e., takes both signs on sets of positive measure, and the other is definite, i.e., either positive a.e. or negative a.e. In the case where \(p\) is indefinite and \(r\) is positive, equation (1) is interpreted in the sense of Carathéodory, so \(py'\) is absolutely continuous, but if \(p\) is discontinuous, \(y'\) can be also, and, in addition, the zeros of \(y\) and \(py'\) need not interlace each other. To overcome the difficulties, the authors use the absolutely continuous Prüfer angle \(\Theta=\Theta(.,\lambda),\) defined by \(\Theta(a_0)=\alpha_0\) and \[ \Theta'=\frac{1}{p}\cos^2\Theta+(\lambda r-q)\sin^2\Theta \] a.e. on \(I.\) They discuss oscillation theory for equation (1) and related results. An analogous theory is presented also for the case when \(p>0\) and \(r\) indefinite, as well as similarities and differences between the two cases are examined.

34B24 Sturm-Liouville theory
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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