Qualitative comparison of the nonoscillatory behavior of the equations \[ L_ny(t) + H(t,y(t)) = 0 \] and \[ L_ny(t) + H(t,y(g(t))) = 0 \] is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form
\[ L_n = \frac{1}{p_n(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{1}{p_{n-1}(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \ldots \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{\cdot }{p_0(t)}. \] Both canonical and noncanonical forms of $L_n$ have been studied.
Necessary and sufficient conditions have been found to force all solutions of the equation \[ (r(t)y^{\prime }(t))^{(n-1)} + a(t)h(y(g(t))) = f(t), \] to behave in peculiar ways. These results are then extended to the elliptic equation \[ |x|^{p-1} \Delta y(|x|) + a(|x|)h(y(g(|x|))) = f(|x|) \] where $ \Delta $ is the Laplace operator and $p \ge 3$ is an integer.