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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.

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