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Estimation of the solutions of the Sturm-Liouville equation

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Abstract

Exact estimates are presented for the solutions of the problem\(\ddot y + \lambda ^2 p(t)y = 0, y(0) = 0, \dot y(0) = 1\) withp(t) satisfying one of the following conditions:

$$(i) |p(t)| \leqslant M< \infty ; (ii) 0< \omega _1 \leqslant p(t) \leqslant \omega _2< \infty ; (iii) \mathop {sup}\limits_x \int_x^{x + T} {p(t)dt = P_T /T.} $$

The extremal solutions are found.

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Authors and Affiliations

  1. Physico-Technical Institute of Low Temperatures, Ukrainian Academy of Sciences, Kharkov

    B. Ya. Levin

  2. Kharkov Polytechnical Institute, Kharkov

    L. Ya. Mirochnik

Authors
  1. B. Ya. Levin
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  2. L. Ya. Mirochnik
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Additional information

Deceased.

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 244–278, March, 1994.

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Levin, B.Y., Mirochnik, L.Y. Estimation of the solutions of the Sturm-Liouville equation. Ukr Math J 46, 251–289 (1994). https://doi.org/10.1007/BF01062239

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  • DOI: https://doi.org/10.1007/BF01062239

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