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10.—The Area-function for Non-linear Second-order Oscillations

Published online by Cambridge University Press:  14 February 2012

F. V. Atkinson
F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto.
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Extract

This paper is concerned with the rate of growth or decline as t → ∞ of oscillatory solutions of equations of the form

we have in mind equations which resemble in some degree the Lane-Emden-Fowler equation

where n ≧ is integral, and σ is either positive, or if negative is not too large, so that oscillatory solutions exist.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 72 , Issue 2 , 1974 , pp. 135 - 147
Copyright
Copyright © Royal Society of Edinburgh 1974

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References

References to Literature

[1] Atkinson, F. V., 1954. The asymptotic solution of second-order differential equations. Annali Mat. Pura Appl., 37, 347–378.CrossRefGoogle Scholar
[2] Atkinson, F. V., 1954. On linear perturbation of non-linear differential equations. Can. J. Math., 6, 561–571.CrossRefGoogle Scholar
[3] Atkinson, F. V., 1955. On asymptotically linear second-order oscillations. J. Rat. Mech. Analysis, 4, 769–793.Google Scholar
[4]Atkinson, F. V., 1960. A constant-area principle for steady oscillations. I. J. Math. Analysis Applic., 1, 133–144.CrossRefGoogle Scholar
[5] Bellman, R., 1953. Stability Theory of Differential Equations. New York: McGraw-Hill.Google Scholar
[6] Burton, T. and Grimmer, R. C., 1971. On the asymptotic behaviour of solutions of x″ + a(t)f(x) = 0. Proc. Camb. Phil. Soc. Math. Phys. Sci., 70 77–88.CrossRefGoogle Scholar
[7] Chandrasekhar, S., 1939. Introduction to the Theory of Stellar Structure. Chicago Univ. Press.Google Scholar
[8] Coffman, C. V. and Wong, J. S. W., 1972. Oscillation and non-oscillation of solutions of generalised Emden-Fowler equations. Trans. Am. Math. Soc., 167, 399–434.CrossRefGoogle Scholar
[9] Ehrenfest, P., 1913. A mechanical theorem of Boltzmann and its relation to the theory of energy quanta. K. Ned. Akad. Wet., 16 591–597. (See also Collected Scientific Papers, 1959. Amsterdam: North-Holland.)Google Scholar
[10] Fowler, R. H., 1931. On Emden's and similar differential equations. Q. Jl Math., 2, 259–288.CrossRefGoogle Scholar
[11] Münster, A., 1959. In Enzykl. Phys., 3/2, 244–245. Berlin-Göttingen-Heidelberg: Springer.Google Scholar
[12] Nehari, Z., 1969. A nonlinear oscillation problem. J. Diff. Equations, 5, 452–460.CrossRefGoogle Scholar