Integral averages and oscillation of second-order nonlinear differential equations (Q5948805)

Here, the equation NEWLINE\[NEWLINE[a(t)\psi(x(t))x'(t)]'+q(t)f(x(t))=0 \tag{E}NEWLINE\]NEWLINE with \(a\in C^1([t_0,\infty);(0,\infty))\), \(q\in C([t_0,\infty);\mathbb{R})\), \(\psi,f\in C^1(\mathbb{R};\mathbb{R})\), \(\psi(x)>0\), \(xf(x)>0\) and \(f'(x)\geq 0\) for \(x\neq 0\) is studied. The author presents new oscillation criteria for equation (E) in both sublinear and superlinear cases. The main tool is the method of weighted averages and the function \(H(t,s)\) established by \textit{Ch. G. Philos} [Arch. Math. 53, 482-492 (1989; Zbl 0661.34030)].