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On the non-existence of L2 solutions of a class of non-linear differential equations†

Published online by Cambridge University Press:  20 January 2009

J. Burlak
J. Burlak
Affiliation:
The University, Glasgow, W.2
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In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equation

does not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 4 , December 1965 , pp. 257 - 268
Copyright
Copyright © Edinburgh Mathematical Society 1965

References

REFERENCE

(1) Bieberbach, L.Einfiihrung in die Theorie der Differentialgleichungen im reellen Gebiet (Springer, Berlin, 1956).Google Scholar
(2) Cesari, L.Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, Berlin, 1959).Google Scholar
(3) Conti, R.Limitazioni “in ampiezza” Delle Soluzioni Di Un Sistema Di Equazioni differenziali e applicazioni, Boll. Un. Mat. Ital. (3) 11(1956), 344–349.Google Scholar
(4) Conti, R.Sulla prolungabilita delle soluzioni di un sistema di equazioni differenziali ordinarie, Boll. Un. Mat. Ital. (3) 11(1956), 510–514.Google Scholar
(5) Ince, E. L.Ordinary Differential Equations (Longmans, Green, London, 1927).Google Scholar
(6) Kestelman, H.Modern Theories of Integration (University Press, Oxford, 1937).Google Scholar
(7) Sansone, G.Equazioni differenziali nel campo reale, vol. 2 (Zanichelli, Bologna, 2nd ed., 1949).Google Scholar
(8) Titchmarsh, E. C.The Theory of Functions (University Press, Oxford, 2nd ed., 1939).Google Scholar
(9) Wintner, A.The non-local existence problem of ordinary differential equations, Amer. J. Math. 67 (1945), 277–284.Google Scholar
(10) Wintner, A.The infinities in the non-local existence problem of ordinary differential equations, Amer. J. Math. 68 (1946), 173–178.CrossRefGoogle Scholar
(11) Wintner, A.A criterion for the non-existence of (L2)-solutions of a nonoscillatory differential equation, J. London Math. Soc. 25 (1950), 347–351.Google Scholar