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Prüfer angle and non-oscillation of linear equations with quasiperiodic data

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Abstract

We consider the Sturm–Liouville differential equations with a power of the independent variable and sums of periodic functions as coefficients (including the case when the periodic coefficients do not have any common period). Using known results, one can show that the studied equations are conditionally oscillatory, i.e., there exists a threshold value which can be expressed by the coefficients and which separates oscillatory equations from non-oscillatory ones. It is very complicated to specify the behaviour of the treated equations in the borderline case. In this paper, applying the method of the modified Prüfer angle, we answer this question and we prove that the considered equations are non-oscillatory in the critical borderline case.

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Acknowledgements

The both authors are supported by Grant GA17-03224S of the Czech Science Foundation.

Author information

Correspondence to Michal Veselý.

Additional information

Communicated by G. Teschl.

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Hasil, P., Veselý, M. Prüfer angle and non-oscillation of linear equations with quasiperiodic data. Monatsh Math 189, 101–124 (2019). https://doi.org/10.1007/s00605-018-1232-5

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Keywords

  • Sturm–Liouville equation
  • Prüfer angle
  • Oscillation theory
  • Periodic coefficient
  • Non-oscillation

Mathematics Subject Classification

  • 34A30
  • 34C10
  • 34C29