Oscillation criteria for second order nonlinear neutral differential equations (Q961637)

The paper deals with the oscillatory behavior of second order nonlinear neutral delay differential equations of the form \[ (r(t)(x(t)+p(t)x(t-\tau))')'+q(t)f(x(t),x(\sigma(t)))=0,\tag{1} \] where \(t\geq t_0>0\), \(\tau\geq 0\) is a constant, and moreover \(r,\sigma\in C^1([t_0,\infty),(0,\infty))\), \(p,q\in C([t_0,\infty),\mathbb R)\), and \(f\in C(\mathbb R^2,\mathbb R)\). The purpose of the paper is to improve oscillation results obtained for equation (1) by \textit{W. Shi} and \textit{P. Wang} [Appl. Math. Comput. 146, No.~1, 211--226 (2003; Zbl 1037.34059)] using a generalized Riccati transformation and developing ideas exploited by \textit{Yu. V. Rogovchenko} and \textit{F. Tuncay} [Nonlinear Anal., Theory Methods Appl. 69, No.~1 (A), 208--221 (2008; Zbl 1147.34026)]. Two examples are provided to illustrate the relevance of obtained results.