Hostname: page-component-84b7d79bbc-dwq4g Total loading time: 0 Render date: 2024-07-30T09:01:03.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit circle classification and boundedness of solutions

Published online by Cambridge University Press:  14 November 2011

F. Neuman
F. Neuman
Affiliation:
Mathematical Institute, Czechoslovak Academy of Sciences (Brno Branch)
Get access Rights & Permissions [Opens in a new window]

Synopsis

Using an algebraic approach to the nth order linear differential equations (see [3] for n = 2 and [13 and 14] for n ≧ 2) it is shown the essence of the relation between the limit circle classification and boundedness of solutions of y″ = q(t)y. On this example it is demonstrated that if a problem can be formulated in the rank of the approach, then it is only a technical matter to answer it, e.g. the relationship studied here is based on a standard fact from the theory of functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 31 - 34
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bartuŝek, M.Connection between asymptotic properties of zeros of solutions of y' = q(t)y. Arch. Math. (Brno) 3 (1972), 113–124.Google Scholar
2Borůvka, O.Sur les intégrates oscilationes des equations differentielles linearies du second ordre. Czechoslovak Math. J. 78 (1953), 199–251.CrossRefGoogle Scholar
3Borůvka, O.Linear Differential Transformations of the Second Order (London: English Univ. Press, 1971).Google Scholar
4Burton, T. A. and Patula, W. T.Limit circle results for second order equations. Monatsh. Math. 81 (1976), 185–194.CrossRefGoogle Scholar
5Everitt, W. N.A note on the Dirichlet conditions for second-order differential expressions. Canad. J. Math. 28 (1976), 312–320.CrossRefGoogle Scholar
6Kwong, M. K.On boundedness of solutions of second order differential equations in the limit circle case. Proc. Amer. Math. Soc. 52 (1975), 242–246.CrossRefGoogle Scholar
7Naimark, M. A.Linear Differential Operators, 2nd edn (Moscow: Nauka, 1969).Google Scholar
8Neuman, F.Relation between the distribution of the zeros of the solutions of a 2nd order linear differential equation and the boundedness of these solutions. Ada Math. Acad. Sci. Hungar. 19 (1968), 1–6.CrossRefGoogle Scholar
9Neuman, F.L2-solutions of y'=q(t)y and a functional equation. Aequationes Math. 6 (1971), 66–70.CrossRefGoogle Scholar
10Neuman, F.Distribution of zeros of solutions of y' = q(t)y in relation to their behaviour in large. Studia Set Math. Hungar. 8 (1973), 177–185.Google Scholar
11Neuman, F.On a problem of transformations between limit-circle and limit-point differential equations. Proc Roy. Soc Edinburgh Sect A 72 (1973), 187–193.CrossRefGoogle Scholar
12Neuman, F.Geometrical approach to linear differential equations of the n-th order. Rend. Mat. 5 (1972), 579–602.Google Scholar
13Neuman, F.Global transformations of linear differential equations of the n-th order. Kniznice Odbom. Vêd. Spisů Vysoké. Uĉení Tech. v Bmê B 56 (1975), 165–171.Google Scholar
14Neuman, F.Categorial approach to global transformations of the n-th order linear differential equations. Casopis Pest. Mat. 102 (1977), 350–355.CrossRefGoogle Scholar
15Patula, W. T. and Wong, J. S. W.An Lp-analogue of the Weyl alternative. Math. Ann. 197 (1972), 9–28.CrossRefGoogle Scholar
16Weyl, H.Über gewohnliche Differentialgleichungen mit Singularitäten und die zugehörige Entwicklung willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269.CrossRefGoogle Scholar