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11.—Limit-n Criteria of Integral Type

Published online by Cambridge University Press:  14 February 2012

F. V. Atkinson
F. V. Atkinson
Affiliation:
University of Toronto.
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Synopsis

Conditions are obtained which ensure that a 2nth order ordinary differential equation on a halfaxis have at most n linearly independent square-integrable solutions. The emphasis is on bounds on the integrals of the coefficients, and on conditions imposed over a sequence of intervals.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 167 - 198
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

1Ahiezer, N. I. and Glazman, I. M., Theory of linear operators in Hilbert space,2nd edn.Moscow: Nauka, 1966.Google Scholar
2Atkinson, F. V., Discrete and continuous boundary problems. New York: Academic Press, 1964.Google Scholar
3Atkinson, F. V. and Evans, W. D., On solutions of a differential equation which are not of integrable square. Math. Z., 127, 323–332, 1972.CrossRefGoogle Scholar
4Brinck, I., Selfadjointness and spectra of Sturm-Liouville operators. Math. Scand., 7, 219–239, 1959.CrossRefGoogle Scholar
5Brown, B. M. and Evans, W. D., On the limit-point and strong limit-point classification of 2n-th order differential expressions with wildly oscillating coefficients. Math. Z., 134, 351–368, 1973.Google Scholar
6Coddington, E. A. and Levinson, N., Theory of ordinary differential equations. New York: McGraw-Hill, 1955.Google Scholar
7Devinatz, A., The deficiency index problem for ordinary self-adjoint differential operators Bull. Amer. Math. Soc, 79, 1109–1127, 1973.CrossRefGoogle Scholar
8Devinatz, A., The deficiency index of certain fourth-order ordinary self-adjoint differentialoperators. Quart. J. Math. Oxford, Ser. 23, 267–286, 1972.Google Scholar
9Devinatz, A., On limit-2 fourth-order differential operators. J.London Math. Soc, 7, 135–146, 1973.Google Scholar
10Devinatz, A., Positive definite fourth-order differential operators. J. London Math. Soc, 6, 412–416, 1973.Google Scholar
11Dunford, N. and Schwartz, J., Linear operators, II: Spectral theory. New York:Interscience, 1963.Google Scholar
12Eastham, M. S. P., The limit-2 case of fourth-order differential equations. Quart.J. Math. Oxford Ser., 22, 131–134, 1971.CrossRefGoogle Scholar
13Eastham, M. S. P., On the L2-classification of fourth-order differential equations. J.London. Math. Soc, 3, 297–300, 1971.CrossRefGoogle Scholar
14Eastham, M. S. P., On a limit-point method of Hartman. Bull. London Math. Soc, 4, 340–344, 1972.Google Scholar
15Eastham, M. S. P., The spectral theory of periodic differential equations. Edinburgh and London: Scottish Academic Press, 1973.Google Scholar
16Eastham, M. S. P., Limit-circle differential expressions of the second order with an oscillating coefficient. Quart. J. Math. Oxford Ser., 24, 257–263, 1973.CrossRefGoogle Scholar
17Eastham, M. S. P. and Thompson, M. L., On the limit-point, limit-circle classification of second-order ordinary differential equations. Quart. J. Math. Oxford Ser., 2A, 531–535, 1973.CrossRefGoogle Scholar
18Evans, W. D., On the limit-point, limit-circle classification of a second-order differential equation with a complex coefficient. J.London Math. Soc, 4, 245–256, 1971.Google Scholar
19Evans, W. D., On non-integrable-square solutions of a fourth-order differential equation andthe limit-2 classification. J. London Math. Soc, 7, 343–354, 1973.Google Scholar
20Everitt, W. N., On the limit-point classification of second-order differential operators. J.London Math. Soc, 41, 531–534, 1966.CrossRefGoogle Scholar
21Everitt, W. N., Some positive definite differential operators, J.London Math. Soc, 43, 456–473, 1968.Google Scholar
22Everitt, W. N., On the limit-point classification of fourth-order differential equations. J. London Math. Soc, 44, 273–281, 1969.Google Scholar
23Everitt, W. N. and Giertz, M., Some properties of the domains of certain differentialoperators. Proc. London Math. Soc, 23, 301–324, 1971.Google Scholar
24Everitt, W. N., Some inequalities associated with certain ordinary differential operators. Math. Z., 126, 308–326, 1972.Google Scholar
25Everitt, W. N., A Dirichlet type result for ordinary differential operators Math. Ann., 203, 119–128, 1973.CrossRefGoogle Scholar
26Everitt, W. N., Giertz, M. and Weidmann, J., Some remarks on a separation and limitpoint criterion of second-order, ordinary differential expressions. Math. Ann., 200, 335–346, 1973.CrossRefGoogle Scholar
27Everitt, W. N., Hinton, D. B. and Wong, J. S. W., On the strong limit-n classification of linear ordinary differentiale xpressions of order 2n. Proc. London Math. Soc, 29, 351–367, 1974.CrossRefGoogle Scholar
28Friedrichs, K., Über die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschrānkten gewöhnlichen Differentialoperatoren zweiter Ordnung. Math. Ann., 112, 1–23, 1935/1936.CrossRefGoogle Scholar
29Hallam, T. G., On the non-existence of Lp-solutions of certain non-linear differential equations. Glasgow Math. J., 8, 133–138, 1967.CrossRefGoogle Scholar
30Hartman, P., The number of L2-solutions of x”+q(t)x = 0. Amer. J. Math., 73, 635–645, 1951.Google Scholar
31Hartman, P. and Wintner, A., A criterion for non-degeneracy of the wave equation. Amer. J. Math., 71, 206–213, 1949.CrossRefGoogle Scholar
32Hinton, D. B., Limit-point criteria for differential equations. Canad. J. Math., 24, 293–305, 1972.Google Scholar
33Hinton, D. B., Solutions of (ryn)n +qy = 0 of class Lp [0, ∞). Proc. Amer. Math. Soc, 32, 134–138, 1972.Google Scholar
34Hinton, D. B., Limit-point criteria for differential equations, II (to be published), 1974.CrossRefGoogle Scholar
35Kalf, H., Remarks on some Dirichlet type results (preprint), 1973.Google Scholar
36Kurss, H., A limit point criterion for non-oscillatory Sturm-Liouville differential operators. Proc Amer. Math. Soc, 18, 445–449, 1967.Google Scholar
37Mcleod, J. B., Square-integrable solutions of a second-order differential equation with complex coefficients, Quart. J. Math. Oxford Ser., 13, 129–133, 1962.Google Scholar
38Naimark, M. A., Linear differential operators, Pt. II. New York: Ungar, 1968.Google Scholar
39Patula, W. T., A limit-point criterion for a fourth-order differential operator, J.London Math. Soc. 8, 217–225, 1974.Google Scholar
40Patula, W. T. and Waltman, P., Limit-point classification of second-order linear differential equations, J.London Math. Soc. 8, 209–216, 1974.Google Scholar
41Patula, W. T. and Wong, J. S. W., An Lp-analogue of the Weyl alternative. Math. Ann., 197, 9–28, 1972.CrossRefGoogle Scholar
42Walker, P., Deficiency indices of fourth-order differential operators. J.Differential Equations, 9, 133–140, 1971.CrossRefGoogle Scholar
43Walker, P., Asymptotics for a class of fourth-order differential equations, J. Differential Equations, 11, 321–324, 1972.Google Scholar
44Wong, J. S. W. and Zettl, A., On the limit-point classification of second-order differential equations. Math. Z, 132, 297–304, 1973.CrossRefGoogle Scholar
45Zettl, A., A note on square-integrable solutions of linear differential equations. Proc. Amer. Math. Soc, 21, 671–672, 1969.Google Scholar