On the asymptotic behavior of nonoscillatory solutions of second order quasilinear ordinary differential equations (Q541258)

The author considers the quasilinear ordinary differential equation of second order of the form: \[ (p(t)\left| {x}' \right|^\alpha \text{sgn}{x}'{)}' + q(t)\left| x \right|^\beta\text{sgn}x = 0, \tag{1} \] where \(\alpha\) and \(\beta\) are positive constants. It is assumed that \(p,\text{ }q \in C[a,\infty ),p(t) > 0,q(t) \geq 0\) for all \(t \in [a,\infty ),\) and \(q(t) \neq 0\) on \([b,\infty )\) for any \(b \geq a\). In this paper, the author obtains some necessary and sufficient conditions for the existence of a slowly growing positive solution of (1). Moreover, precise asymptotic forms as \(t \to \infty \) of slowly growing positive solutions and slowly decaying positive solutions of (1) are established.