Oscillation and comparison for second order differential equations

Abstract:

Consider the equations \[ (1)\quad x'' = f(t,x,x’)\] and \[ (2)\quad x'' = g(t,x,x’)\] where $f,g:[a, + \infty ) \times {R^2} \to R$ are continuous. Assume that solutions of initial value problems for (1) and for (2) are unique and extend to $[a, + \infty )$. Let $f(t,0,0) = 0 = g(t,0,0)$ for $t \epsilon [a, + \infty )$ and $f(t,x,x’)/x \leq g(t,x,x’)/x$ for $|x| > 0$ and $(t,x,x’)$ in the domain of $f$ and $g$. Under these hypotheses it can be shown that if every solution of (2) has a zero on an interval $I \subset [a, + \infty )$ then it follows that every solution of (1) has a zero on $I$. In particular this shows that under these hypotheses (2) is oscillatory (every solution has a zero on $[a + n, + \infty )$ for each positive integer $n$) implies (1) is oscillatory.