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Transactions of the American Mathematical Society

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

 

 

The sufficiency of the Matkowsky condition in the problem of resonance


Author: Ching Her Lin
Journal: Trans. Amer. Math. Soc. 278 (1983), 647-670
MSC: Primary 34E15
DOI: https://doi.org/10.1090/S0002-9947-1983-0701516-5
MathSciNet review: 701516
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Abstract: We consider the sufficiency of the Matkowsky condition concerning the differential equation $ \varepsilon y'' + f(x,\varepsilon )y' + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)$ under the assumption that $ f(0,\varepsilon ) = 0$ identically in $ \varepsilon ,{f_x}(0,\varepsilon ) \ne 0$ with $ f > 0$ for $ x < 0$ and $ f < 0$ for $ x > 0$. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if $ f$ and $ g$ are convergent power series for $ \vert\varepsilon \vert < \rho \;(\rho > 0),f(x,0)=-2x$ and the interval $ [ - a,b]$ is contained in a disc $ D$ with center at 0. The main problem in this work is to remove from Sibuya's result the assumption that $ D$ is a disc.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701516-5
Article copyright: © Copyright 1983 American Mathematical Society