Nonoscillation theorems for a class of quasilinear differential equations of second order (Q1842043)

The ordinary differential equation \((| y' |^{\alpha - 1} y')' + q(t) | y |^{\beta - 1} y = 0\), \(t \geq a\), is considered for \(\alpha\) and \(\beta\) positive but with \(\alpha \neq 1\). If \(\alpha = 1\) then the equation is the Emden-Fowler equation. The main theorem proved is the following: all solutions of the differential equation with \(\alpha \neq \beta\) are nonoscillatory if \[ \int^ \infty_ a {\bigl( q'(t) \bigr)_ + \over q(t)} dt < \infty, \quad \text{and} \quad \int^ \infty_ a t^ \gamma q(t)dt < \infty, \] where \((q'(t))_ + = \max \{q'(t), 0\}\) and \(\gamma = \max \{\alpha, \beta\}\).