The Behavior of Oscillatory Solutions of $x''(t) + p(t)g(x(t)) = 0$

Abstract

Various quantitative properties of oscillatory solutions of the scalar second order nonlinear differential equation $x'' + p(t)g(x) = 0$ are obtained under appropriate hypotheses on p and g. In particular, letting $\{ {t_i } \}_{i = 1}^\infty $, $0 < t_i < t_{i + 1} $, $t_i \to \infty $, as $i \to \infty $ be the zeros of any solution $x(t)$, we obtain inequalities on $J_i^{{\operatorname{def}}} \equiv \int _{t_i }^{t_{i + 1} } g(x(t))dt$ which yield asymptotic behavior on $x(t)$. For example, it is shown that $\lim_{t \to \infty } \int _0^t g(x(s))ds$ exists and is finite; moreover, assuming an added growth condition on ${{g(x)} / x}$, we have then that $\lim _{t \to \infty } \int _0^t x(s)ds$ exists and is finite.