Asymptotic Behavior of Solutions of $y'' = \phi (t)f(y)$

Abstract

Explicit formulas are obtained for the asymptotic behavior of solutions of the three problems \[ \begin{gathered} y'' = \phi (t)f(y),\quad \mathop {\lim }\limits_{t \to \infty } y(t) = \mathop {\lim }\limits_{t \to \infty } y'(t) = 0, \hfill y'' = \phi (t)f(y) = 0,\quad \mathop {\lim }\limits_{t \to 0^ + } y(t) = 0\quad \mathop {\lim }\limits_{t \to 0^ + } y'(t) = \infty , \hfill y'' = \phi (t)f(y) = 0,\quad \mathop {\lim }\limits_{t \to \infty } y(t) = \infty \quad \mathop {\lim }\limits_{t \to \infty } y'(t) = 0, \hfill \end{gathered} \] as t tends to $\infty $, 0, and $\infty $ respectively. In all three problems $\phi (t)$ and $f(y)$ are assumed to be positive and continuous. Necessary conditions for existence of solutions to these problems are also given.