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

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) $\left | {f(t,u)} \right | < \upsilon (t)\varphi (t)g(\left | u \right |/t)$ 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].