On nonoscillation of canonical or noncanonical disconjugate functional equations

Abstract

Qualitative comparison of the nonoscillatory behavior of the equations

$$L_n y\left( t \right) + H\left( {t,\left( y \right)} \right) = 0$$

and

$$L_n y\left( t \right) + H\left( {t,y\left( {g\left( t \right)} \right)} \right) = 0$$

is sought by way of finding different nonoscillation criteria for the above equations, L _n is a disconjugate operator of the form

$$L_n = \frac{1}{{p_n \left( t \right)}}\frac{{\text{d}}}{{{\text{dt}}}}\frac{1}{{p_{n - 1} \left( t \right)}}\frac{{\text{d}}}{{{\text{dt}}}}...\frac{{\text{d}}}{{{\text{dt}}}}\frac{.}{{p_0 \left( t \right)}}.$$

Both canonical and noncanonical forms of L _n have been studied.