Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I

Abstract

We consider the half-linear differential equation of the form $(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},$ under the assumption $\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \to \infty$.