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Remarks on the paper [Appl. Math. Comput. 207 388-396 (2009)]. (English) Zbl 1188.34086

Summary: Some sufficient conditions are established for the oscillation of second-order neutral differential equations

$\left(r\left(t\right)\psi \left(x\left(t\right)\right)|{Z}^{\text{'}}\left(t\right){{|}^{\alpha -1}{Z}^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{1.em}{0ex}}t⩾{t}_{0}>0,$

where $Z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(t-\tau \right)$ and $\alpha >0,0⩽p\left(t\right)<1$. On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations

${\left[r\left(t\right){\left[x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)\right]}^{\text{'}}\right]}^{\text{'}}+q\left(t\right)f\left(x\left(\sigma \left(t\right)\right)\right)=0,\phantom{\rule{1.em}{0ex}}t⩾{t}_{0}>0,$

where ${\int }_{{t}_{0}}^{\infty }\frac{\text{d}t}{r\left(t\right)}<\infty ,0⩽p\left(t\right)⩽{p}_{0}<+\infty$. The results obtained here complement and correct some known results by L. Ye and Z. Xu [Appl. Math. Comput. 207, No. 2, 388–396 (2009; Zbl 1168.34346)]. Some examples are given to illustrate the main results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
##### References:
 [1] Ye, Luhong; Xu, Zhiting: Oscillation criteria for second-order quasilinear neutral delay differential equations, Appl. math. Comput. 207, 388-396 (2009) · Zbl 1168.34346 · doi:10.1016/j.amc.2008.10.051 [2] Xu, Run; Meng, Fanwei: Some new oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. math. Comput. 182, 797-803 (2006) · Zbl 1115.34341 · doi:10.1016/j.amc.2006.04.042