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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonlinear techniques for linear oscillation problems


Author: Zeev Nehari
Journal: Trans. Amer. Math. Soc. 210 (1975), 387-406
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9947-1975-0372327-3
MathSciNet review: 0372327
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Abstract: It is shown that for differential equations of the form $ {y^{(n)}} + py = 0$ there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case $ n = 2$. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation. If $ p$ is of constant sign, the coefficients of the ``Riccati systems'' are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0372327-3
Article copyright: © Copyright 1975 American Mathematical Society