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# The Asymptotic Behavior of a Class of Nonlinear Differential Equations of Second Order

Jingcheng Tong
Proceedings of the American Mathematical Society
Vol. 84, No. 2 (Feb., 1982), pp. 235-236
DOI: 10.2307/2043671
Stable URL: http://www.jstor.org/stable/2043671
Page Count: 2
Let $u'' + f(t, u) = 0$ be a nonlinear differential equation. If there are two nonnegative continuous functions $v(t), \varphi(t)$ for $t \geqslant 0$, and a continuous function $g(u)$ for $u \geqslant 0$, such that (i) $\int^\infty_1 v(t)\varphi(t) dt < \infty$; (ii) for $u > 0, g(u)$ is positive and nondecreasing; (iii) $|f(t, u)| < v(t)\varphi(t)g(|u|/t)$ for $t \geqslant 1, -\infty < u < \infty$, then the equation has solutions asymptotic to $a + bt$, where $a, b$ are constants and $b \neq 0$. Our result generalizes a theorem of D. S. Cohen [3].