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

Kwong, Man Kam

doi:10.1090/S0002-9939-1975-0387710-5

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

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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