New oscillation criteria for second-order neutral delay differential equations with positive and negative coefficients (Q1957586)

Summary: We establish oscillation and asymptotic criteria for second-order neutral delay differential equations with positive and negative coefficients having the forms \[ \begin{multlined} \bigg[x(t)+\sum_{i\in \mathbb R} c_i(t)x(\alpha_i(t))\bigg]''+ r(t)\bigg[x(t)+\sum_{i\in \mathbb R}c_i(t)x(\alpha_i(t))\bigg]'+\\ \sum_{i\in \mathbb P}p_i(t)x(\beta_i(t))- \sum_{i\in \mathbb Q}q_i(t)x(\gamma_i(t))=0 \end{multlined} \] and \[ \begin{multlined} \bigg[x(t)+\sum_{i\in \mathbb R}\! c_i(t)x(\alpha_i(t))\bigg]'' +r(t) \bigg[x(t)+\sum_{i\in \mathbb R}\!c_i(t)x(\alpha_i(t))\bigg]'+\\ \sum_{i\in \mathbb P}\!p_i(t)x(\beta_i(t))-\sum_{i\in \mathbb Q}\!q_i(t)x(\gamma_i(t))=f(t). \end{multlined} \] The obtained new oscillation criteria extend and improve recent results.