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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On boundedness of solutions of second order differential equations in the limit circle case
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by Man Kam Kwong PDF
Proc. Amer. Math. Soc. 52 (1975), 242-246 Request permission

Abstract:

A differential equation of the form $x''(t) + a(t)x(t) = 0,t \geqslant 0$, is said to be in the limit circle case if all its solutions are square integrable on $[0,\infty )$. It has been conjectured in [1] that all its solutions are bounded. J. Walter recently gave a counterexample. This paper gives a method of modifying any given equation in the limit circle case with bounded solutions to produce one with unbounded solutions.
References
  • W. T. Patula and J. S. W. Wong, An $L^{p}$-analogue of the Weyl alternative, Math. Ann. 197 (1972), 9–28. MR 299865, DOI 10.1007/BF01427949
  • Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, DOI 10.1007/BF01474161
  • James S. W. Wong, Square integrable solutions of $L^{p}$ perturbations of second order linear differential equations, Ordinary and partial differential equations (Proc. Conf., Univ. Dundee, Dundee, 1974) Lecture Notes in Math., Vol. 415, Springer, Berlin, 1974, pp. 282–292. MR 0422742
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 52 (1975), 242-246
  • MSC: Primary 34B20
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0387710-5
  • MathSciNet review: 0387710