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Proceedings of the American Mathematical Society

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On boundedness of solutions of second order differential equations in the limit circle case

Author: Man Kam Kwong
Journal: Proc. Amer. Math. Soc. 52 (1975), 242-246
MSC: Primary 34B20
MathSciNet review: 0387710
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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.

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  • [1] W. T. Patula and J. S. W. Wong, An 𝐿^{𝑝}-analogue of the Weyl alternative, Math. Ann. 197 (1972), 9–28. MR 0299865
  • [2] Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, 10.1007/BF01474161
  • [3] 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) Springer, Berlin, 1974, pp. 282–292. Lecture Notes in Math., Vol. 415. MR 0422742

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Keywords: Limit circle case, unbounded solutions
Article copyright: © Copyright 1975 American Mathematical Society