Half-linear differential equations with oscillating coefficient

Abstract

We study asymptotic properties of solutions of the nonoscillatory half-linear differential equation \[ (a(t)\Phi(x^{\prime}))^{\prime}+b(t)\Phi(x)=0 \] where the functions $a,b$ are continuous for $t\geq0,$ $a(t)>0$ and $\Phi(u)=|u|^{p-2}u$, $p>1$. In particular, the existence and uniqueness of the zero-convergent solutions and the limit characterization of principal solutions are proved when the function $b$ changes sign. An integral characterization of the principal solutions, the boundedness of all solutions, and applications to the Riccati equation are considered as well.