Oscillatory and asymptotically monotone solutions of second-order quasilinear differential equations (Q1354217)

Oscillation criteria are developed for quasilinear differential equations including the prototype \[ -(r|y'|^ay')'= pf(y) \quad\text{in }I=(t_0,\infty) \tag{1} \] for constants \(a>-1\), \(t_0>0\), \(p\in C(I,\mathbb{R})\), \(r\in C(I,\mathbb{R}_+)\), \(f\in C(\mathbb{R},\mathbb{R})\), \(uf(u)>0\) for all \(u\neq 0\), and \(f'(u)\geq 0\). Ten different sets of sufficient conditions are provided for (1) and generalizations to be oscillatory or for all solutions \(y\) with \(y(t)\) and/or \(y'(t)\) bounded to be oscillatory. Additional theorems concern equations of type \[ -(r|y'|^ay')'= q|y'|^ay'+ g(\cdot,y) \quad\text{for }a\geq 0,\;q,r\in C(I,\mathbb{R}_+),\text{ and }g\in C(I\times \mathbb{R},\mathbb{R}).\tag{2} \] Sufficient conditions are obtained for the existence of an eventually positive decreasing solution to (2), from which oscillation criteria for (2) are derived. Examples are included.