Oscillation criteria for second-order nonlinear differential equations involving integral averages (Q2782940)

The authors present new oscillation criteria for nonlinear differential equations NEWLINE\[NEWLINE x''(t)+a(t)f(x(t))=0,\tag{1} NEWLINE\]NEWLINE with \(a\in C([t_0,\infty))\), and \(f(x)\in C(\mathbb{R})\) continuously differentiable for \(x\neq 0\), \(xf(x)>0\) and \(f'(x)\geq 0\) for \(x\neq 0\). The results of the paper extend the work of \textit{F. W. Meng} [Indian J. Pure Appl. Math. 27, No. 7, 651-658 (1996; Zbl 0862.34023)] and \textit{J. S. W. Wong} [Can. J. Math. 45, 1094-1103 (1993; Zbl 0797.34037)]. In the sublinear case, the function \(f\) is assumed to satisfy \(0<\int_0^\epsilon \frac{dy}{f(y)}, \int_0^{-\epsilon}\frac{dy}{f(y)}<\infty\) for all \(\epsilon>0\) and equation (1) is transformed into NEWLINE\[NEWLINE w''(t)+a(t) +(w(t))^2f'(x(t))=0, \quad t\geq T,NEWLINE\]NEWLINE where \(x(t)\) is a nonoscillatory solution to (1) and positive for \(t\geq T\). A similar approach is used also in the superlinear case.