Hostname: page-component-7bb8b95d7b-pwrkn Total loading time: 0 Render date: 2024-09-29T22:57:13.431Z Has data issue: false hasContentIssue false
  • English
  • Français

A stability problem with algebraic aspects

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson
F. V. Atkinson
Affiliation:
University of Toronto
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the set S(f) of λ-values for which there exists a function g ∊ L(t0, ∞) such that

has a solution not tending to 0; here f is a fixed function which is positive, non-decreasing and tends to ∞ with t. It is shown that if the jumps of logf(t) at its discontinuities are uniformly bounded, then S(f) is an additive group. This group is determined in some cases; some related groups are noted, which may coincide with S(f).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 3-4 , 1978 , pp. 299 - 314
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

1 Atkinson, F. V.. Asymptotic formulae for linear oscillations. Proc. Glasgow Math. Assoc. 3 (1957), 105–111.CrossRefGoogle Scholar
2 Burton, T. and Grimmer, R.. On the asymptotic behaviour of solutions of x” + a(t)f{x) = 0. Proc. Cambridge Philos. Soc. 70 (1971), 77–88.CrossRefGoogle Scholar
3 Cesari, L.. Asymptotic behaviour and stability problems in ordinary differential equations (Berlin: Springer, 1963).CrossRefGoogle Scholar
4 Kleine, H. De. A counter-example to a conjecture in second order linear equations. Michigan Math. J. 17 (1970), 29–32.Google Scholar
5 Hartman, P.. On a theorem of Milloux. Amer. J. Math. 70 (1948), 395–399.CrossRefGoogle Scholar
6 Hartman, P.. The existence of large or small solutions of linear ordinary differential equations. Duke Math. J. 28 (1961), 421–430.CrossRefGoogle Scholar
7 Lazer, A. C.. A stability condition for the differential equation y” +p2 (x)y = 0. Michigan Math. J. 12 (1965), 193–196.CrossRefGoogle Scholar
8 McShane, E. J.. On the solutions of the differential equation y” +p2y = 0. Proc. Amer. Math. Soc. 17 (1966), 55–61.Google Scholar
9 Meir, A., Willett, D. and Wong, J. S. W.. On the asymptotic behaviour of the solutions of x”+a(t)x = 0. Michigan Math. J. 14 (1967), 47–52.CrossRefGoogle Scholar
10 Milloux, H.. Sur l'équation différentielle x”+A(t)x = 0. Prace Nauk. Inst. Mat. Fiz. Teoret Politechn. Wroclaw. Ser. Stadia i Materialy 41 (1934), 39–54.Google Scholar
11 Muldowney, J. S. and Macki, J. W.. The asymptotic behaviour of solutions to linear systems of ordinary differential equations. Pacific J. Math. 33 (1970), 693–706.Google Scholar
12 Sansone, G.. Equazioni differenziali nel campo reale (Bologna: Zanichelli, 1948).Google Scholar
13 Teufel, H. Jr. The stability of solutions of generalized Emden-Fowler equations. Canad. Math. Bull. 17 (1974), 397–401.CrossRefGoogle Scholar
14 Willett, D.. On an example in second-order linear ordinary differential equations. Proc. Amer. Math. Soc. 17 (1966), 1263–1266.CrossRefGoogle Scholar