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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The asymptotic behavior of a class of nonlinear differential equations of second order

Author: Jing Cheng Tong
Journal: Proc. Amer. Math. Soc. 84 (1982), 235-236
MSC: Primary 34C11; Secondary 34E05
Note: Proc. Amer. Math. Soc. 108, no. 2 (1990), pp. 383-386.
MathSciNet review: 637175
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Abstract: Let $ u + f(t,u) = 0$ be a nonlinear differential equation. If there are two nonnegative continuous functions $ \upsilon (t)$, $ \varphi (t)$ for $ t \geqslant 0$, and a continuous function $ g(u)$ for $ u \geqslant 0$, such that (i) $ \smallint _1^\infty \upsilon (t)\varphi (t)\;dt < \infty $; (ii) for $ u > 0$, $ g(u)$ is positive and nondecreasing; (iii) $ \left\vert {f(t,u)} \right\vert < \upsilon (t)\varphi (t)g(\left\vert u \right\vert/t)$ for $ t \geqslant 1$, $ - \infty < u < \infty $, then the equation has solutions asymptotic to $ a + bt$, where $ a$, $ b$ are constants and $ b \ne 0$. Our result generalizes a theorem of D. S. Cohen [3].

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Keywords: Gronwall's inequality, Bihari's inequality
Article copyright: © Copyright 1982 American Mathematical Society