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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonlinear oscillations of second order
differential equations of Euler type


Authors: Jitsuro Sugie and Tadayuki Hara
Journal: Proc. Amer. Math. Soc. 124 (1996), 3173-3181
MSC (1991): Primary 34C10, 34C15; Secondary 34A12, 70K05
MathSciNet review: 1350965
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Abstract: We consider the nonlinear equation $t^{2}x'' + g(x) = 0$, where $g(x)$ satisfies $xg(x) > 0$ for $x \ne 0$, but is not assumed to be sublinear or superlinear. We discuss whether all nontrivial solutions of the equation are oscillatory or nonoscillatory. Our results can be applied even to the case $\frac {g(x)}{x} \to \frac {1}{4} \; \text {as} \; |x| \to \infty $, which is most difficult.


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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto 390, Japan
Address at time of publication: Department of Mathematics and Computer Science, Shimane University, Matsue 690, Japan
Email: jsugie@botan.shimane-u.ac.jp

Tadayuki Hara
Affiliation: Department of Mathematical Sciences, University of Osaka Prefecture, Sakai 593, Japan
Email: hara@ms.osakafu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03601-5
PII: S 0002-9939(96)03601-5
Keywords: Oscillation, nonlinear differential equations, Li\'{e}nard system, global phase portrait
Received by editor(s): April 6, 1995
Additional Notes: The first author was supported in part by Grant-in-Aid for Scientific Research 06804008.
Dedicated: Dedicated to Professor Junji Kato on the occasion of his 60th birthday
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society