Comparison principle for second-order neutral differential equations

Based on a comparison with the first-order delay equations, we obtain new criteria for oscillation of the second-order neutral delay differential equations of the form \[ \left(r(t)[x(t)+q(t)x(\sigma(t))]'\right)'+p(t)x(\tau(t))=0,\,\,\,\,t\geq t_0>0. \] Some new results are presented that improve related ones. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of studied equations. We illustrate the improvement over the known results by applying and comparing our method with the other known results for the studied equation.