Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T05:19:01.969Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit-point criteria for systems of differential equations

Published online by Cambridge University Press:  14 November 2011

Hilbert Frentzen
Hilbert Frentzen
Affiliation:
Department of Mathematics, University of Essen, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

For a certain class of first order systems of differential equations several theorems are derived which give sufficient conditions for an appropriate sesquilinear form to be identically zero on suitable spaces of solutions of the system. As a consequence for second order systems limit-point criteria are obtained which include rather general criteria in the case of second order equations. The method used involves sequences of auxiliary functions and is most expedient for the proof of interval limit-point criteria. The theory is also applicable to second order equations with complex coefficients yielding sufficient conditions for the existence of solutions which are not of integrable square.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 3-4 , 1980 , pp. 233 - 245
Copyright
Copyright © Royal Society of Edinburgh 1980

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

1Atkinson, F. V.. Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 167–198.CrossRefGoogle Scholar
2Atkinson, F. V., Eastham, M. S. P. and McLeod, J. B.. The limit-point, limit-circle nature of rapidly oscillating potentials. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 183–196.CrossRefGoogle Scholar
3Atkinson, F. V. and Evans, W. D.. On solutions of a differential equation which are not of integrable square. Math. Z. 127 (1972), 323–332.CrossRefGoogle Scholar
4Evans, W. D.. On limit-point and Dirichlet-type results for second-order differential expressions. Proceedings of the 1976 Dundee Conference on Differential Equations. Lecture Notes in Mathematics 564, 78–92 (Berlin: Springer, 1976).Google Scholar
5Evans, W. D. and Zettl, A.. Interval limit-point criteria for differential expressions and their powers. J. London Math. Soc. 15 (1977), 119–133.CrossRefGoogle Scholar
6Hinton, D.. Limit point criteria for differential equations. Canad. J. Math. 24 (1972), 293–305.CrossRefGoogle Scholar
7Kogan, V. I. and Rofe-Beketov, F. S.. On square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 74 (1976), 5–40.CrossRefGoogle Scholar
8Read, T. T.. A limit-point criterion for expressions with oscillatory coefficients. Pacific J. Math. 66 (1976), 243–255.CrossRefGoogle Scholar
9Read, T. T.. A limit-point criterion for −(py')' + qy. Proceedings of the 1976 Dundee Conference on Differential Equations. Lecture Notes in Mathematics 564, 383–390 (Berlin: Springer 1976).Google Scholar
10Read, T. T.. A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. Soc. 15 (1977), 271–276.CrossRefGoogle Scholar
11Rofe-Beketov, F. S.. Square-integrable solutions, self adjoint extensions and spectrum of differential systems. Differential Equations. Acta Univ. Uppsala, Symp. Univ. Uppsala 7 (1977), 169–178.Google Scholar