Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T07:28:45.793Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic solution of linear differential systems

Part of: Asymptotic theory

Published online by Cambridge University Press:  26 February 2010

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
Chelsea College (University of London), 552, King's Road, London, SW10 0UA
Get access

Extract

In the paper which forms the basis for the modern asymptotic theory of linear differential systems, Levinson [22] considered the system

on an interval [a, ∞), where

(1) A is a constant nÜn matrix with n distinct eigenvalues and either

(2a) R(x) is L(a, ∞)

or

(2b) R(x)→0 as x→∞ and R'(x) is L(a, ∞).

MSC classification

Secondary: 34E99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 131 - 138
Copyright
Copyright © University College London 1985

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

1.Atkinson, F. V.. Asymptotic formulae for linear oscillations. Proc. Glasgow Math. Assoc, 3 (1956–1958), 105–111.CrossRefGoogle Scholar
2.Becker, R. I.. Asymptotic expansions of second-order linear differential equations having conditionally integrable coefficients. J. London Math. Soc. (2), 20 (1979), 472–484.CrossRefGoogle Scholar
3.Becker, R. I.. Asymptotic expansions of formally self-adjoint differential equations with conditionally integrable coefficients. I, Quart. J. Math. (Oxford) (2), 31 (1980), 49–64; II, I, Quart. J. Math. (Oxford), 33 (1982), 11–25.CrossRefGoogle Scholar
4.Bellman, R.. Stability theory of differential equations (McGraw-Hill, 1953).Google Scholar
5.Cassell, J. S.. The asymptotic behaviour of a class of linear oscillators. Quart. J. Math. (Oxford) (2), 32 (1981), 287–302.CrossRefGoogle Scholar
6.Cassell, J. S.. The asymptotic integration of some oscillatory differential equations. Quart. J. Math. (Oxford) (2), 33 (1982), 281–296.CrossRefGoogle Scholar
7.Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (McGraw-Hill, 1955).Google Scholar
8.Coppel, W. A.. Stability and Asymptotic Behaviour of Differential Equations (Heath, Boston, 1965).Google Scholar
9.Eastham, M. S. P.. The deficiency index of a second-order differential system. j. London Math. Soc. (2), 23 (1981), 311–320.CrossRefGoogle Scholar
10.Eastham, M. S. P.. Asymptotic theory and deficiency indices for differential equations of odd order. Proc. Roy. Soc. Edinburgh, 90A (1981), 263–279.CrossRefGoogle Scholar
11.Eastham, M. S. P.. Asymptotic theory for a critical class of fourth-order differential equations. Proc. Roy. Soc. London, 383A (1982), 465–476.Google Scholar
12.Eastham, M. S. P.. Asymptotic formulae of Liouville-Green type for higher-order differential equations. J. London Math. Soc. (2), 28 (1983), 507–518.CrossRefGoogle Scholar
13.Eastham, M. S. P.. Transformations of differential equations, the Levinson asymptotic theorem and deficiency indices. Differential Equations, North-Holland Mathematics Studies 92. Edited by Knowles, I. W. and Lewis, R. T.. (North-Holland, 1984), 155–164.Google Scholar
14.Eastham, M. S. P. and Grudniewicz, C. G. M.. Asymptotic theory and deficiency indices for higher-order self-adjoint differential equations. J. London Math. Soc. (2), 24 (1981), 255–271.CrossRefGoogle Scholar
15.Fedorjuk, M. V.. Asymptotic behaviour of eigenvalues and eigenfunctions of one-dimensional singular differential operators. Soviet Math., 7 (1966), 929–932.Google Scholar
16.Grudniewicz, C. G. M.. Asymptotic methods for fourth-order differential equations. Proc. Roy. Soc. Edinburgh, 87A (1980), 53–63.CrossRefGoogle Scholar
17.Harris, W. A. and Lutz, D. A.. On the asymptotic integration of linear differential systems. J. Math. Anal. AppL, 48 (1974), 1–16.CrossRefGoogle Scholar
18.Harris, W. A. and Lutz, D. A.. Asymptotic integration of adiabatic oscillators. j. Math. Anal Appl., 51 (1975), 76–93.CrossRefGoogle Scholar
19.Harris, W. A. and Lutz, D. A.. A unified theory of asymptotic integration. J. Math. Anal. AppL, 57 (1977), 571–586.CrossRefGoogle Scholar
20.Hinton, D. B.. Asymptotic behaviour of solutions of (ry (m))(k) ±qy = 0. J. Diff. Equations, 4 (1968), 590–596.CrossRefGoogle Scholar
21.Hinton, D. B.. Asymptotic formulae for linear equations. Glasgow Math. J., 13 (1972), 147–152.CrossRefGoogle Scholar
22.Levinson, N.. The asymptotic nature of solutions of linear differential equations. Duke Math. J., 15 (1948), 111–126.CrossRefGoogle Scholar
24.Naimark, M. A.. Linear Differential Operators, Part 2 (Harrap, 1968).Google Scholar