Oscillation criteria for second-order sublinear differential equation
The author establishes oscillation criteria for the second-order nonlinear differential equation \[ [a(t)\psi(x(t))x'(t)]'+q(t)f(x(t))=0, \tag{*} \] with \(a\in C^1([t_0,\infty);(0,\infty))\), \(q\in C^1([t_0,\infty);\mathbb{R})\) (no restriction on its sign), \(\psi\in C^1(\mathbb{R};\mathbb{R})\), \(f\in C^1(\mathbb{R};\mathbb{R})\) such that \(xf(x)>0\), \(f'(x)\geq 0\) for \(x\neq 0\) and \[ \int_{0+}\frac{\psi(u)}{f(u)}du<\infty,\quad \int_{0-}\frac{\psi(u)}{f(u)}du<\infty. \] The oscillation results established here involve the average behavior of the integral of the alternating coefficient \(q\) and are in terms of a parameter function \(H(t,s)\). The obtained results generalize oscillation criteria of \textit{H. J. Li} and \textit{C. C. Yeh} [Dyn. Syst. Appl. 6, No. 4, 529-534 (1997; Zbl 0888.34024)] and of \textit{Ch. G. Philos} [Differ. Integral Equ. 4, No. 1, 205-213 (1991; Zbl 0721.34026)].
scientific article; zbMATH DE number 1471227
scientific article