Oscillation properties of perturbed disconjugate equations

Abstract:

Oscillation conditions are given for the equation ${L_u} + f(t,u) = 0$, where \[ Lu = \frac {1} {{{\beta _n}}}\frac {d} {{dt}}\frac {1} {{{\beta _{n - 1}}}} \cdots \frac {d} {{dt}}\frac {1} {{{\beta _1}}}\frac {d} {{dt}}\frac {u} {{{\beta _0}}}(n \geqslant 2),\] with ${\beta _0}, \ldots ,{\beta _n}$ positive and continuous on $(0,\infty ),\int {^\infty {\beta _i}dt = \infty (1 \leqslant i \leqslant n - 1)}$, and $f$ subject to conditions which include $uf(t,u) \geqslant 0$. The results obtained include previously known oscillation conditions for the equation ${u^{(n)}} + f(t,u) = 0$ for both linear and nonlinear cases.