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Higher-order differential equations with a slowly decaying coefficient containing a periodic factor

Published online by Cambridge University Press:  24 October 2008

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
13 Abbotswood, Speen, Princes Risborough, Bucks. HP27 0SR
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Extract

Let p(x) be periodic with mean value c and let α > 0. Then the investigation of the asymptotic solution of the second-order equation

can be split into four cases: (i) α > 2, (ii) α > 1 and c = 0, (iii) 0 < α ≤ 1 and c = 0, (iv) 0 < α ≤ 2 and c ≠ 0. The first two cases are straightforward because then there are solutions y1 and y2 of (1·1) such that y1(x) ~ x and y2(x) → 1 as x → ∞ [5, theorems 2·7·2 and 2·7·5; 7, p. 375; 8]. Thus the coefficient of y in (1·1) represents a perturbation which has no effect on the dominant asymptotic behaviour of solutions. Case (iii) is less simple because here the nature of p(x) does influence the asymptotic form of solutions and, indeed, when α < 1 the solutions of (1·1) are oscillatory. This case has been fully covered by a number of authors: Atkinson, Eastham and McLeod[2, §3], Cassell[3, 4], Eastham[5, §4·13], Mahoney[10], Willett[13] and Wong[14, 15]. In addition, corresponding results for a higher-order self-adjoint analogue of (1·1) have been obtained recently by Al-Hammadi and Eastham [1, 6] as an application of general asymptotic methods for differential systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 175 - 184
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Al-Hammadi, A. S. A. and Eastham, M. S. P.. Higher-order differential equations with small oscillatory coefficients. J. London Math. Soc. (2) 40 (1989), 507–518.CrossRefGoogle Scholar
[2]Atkinson, F. V., Eastham, M. S. P. and McLeod, J. B.. The limit-point, limit-circle nature of rapidly oscillating potentials. Proc. Roy. Soc. Edinburgh 76A (1977), 183–196.CrossRefGoogle Scholar
[3]Cassell, J. S.. The asymptotic behaviour of a class of linear oscillators. Quart. J. Math. (2) 32 (1981), 287–302.CrossRefGoogle Scholar
[4]Cassell, J. S.. The asymptotic integration of some oscillatory differential equations. Quart. J. Math. (2) 33 (1982), 281–296.CrossRefGoogle Scholar
[5]Eastham, M. S. P.. The asymptotic solution of linear differential systems (London Math. Soc. Monographs 4, Oxford, 1989).Google Scholar
[6]Eastham, M. S. P. and Al-Hammadi, A. S. A.. Asymptotic solution of differential equations with small oscillatory coefficients. Results in Mathematics 19 (1991), 57–73.CrossRefGoogle Scholar
[7]Hartman, P.. Ordinary differential equations (2nd ed., Birkhäuser, 1982).Google Scholar
[8]Hartman, P. and Wintner, A.. On nonoscillatory linear differential equations. Amer. J. Math. 75 (1953), 717–730.CrossRefGoogle Scholar
[9]Howard, H. and Marić, V.. Asymptotics of nonocillatory solutions of some second-order linear differential equations. Bull. London Math. Soc. (to appear).Google Scholar
[10]Mahoney, J. J.. Asymptotic results for the solution of a certain differential equation. J. Austral. Math. Soc. 13 (1972), 147–158.CrossRefGoogle Scholar
[11]Šimša, J.. Asymptotic integration of a second order ordinary differential equation. Proc. Amer. Math. Soc. 101 (1987), 96–100.CrossRefGoogle Scholar
[12]Trench, W. F.. Linear perturbations of a nonoscillatory second order equation. Proc. Amer. Math. Soc. 97 (1986), 423–428.CrossRefGoogle Scholar
[13]Willett, D.. On the oscillatory behaviour of the solutions of second order linear differential equations. Ann. Polon. Math. 21 (1969), 175–194.CrossRefGoogle Scholar
[14]Wong, J. S. W.. Second order linear oscillation with integrable coefficients. Bull. Amer. Math. Soc. 74 (1968), 909–911.CrossRefGoogle Scholar
[15]Wong, J. S. W.. Oscillation and non-oscillation of solutions of second order linear differential equations with integrable coefficients. Trans. Amer. Math. Soc. 44 (1969), 197–215.CrossRefGoogle Scholar