Oscillation criteria for neutral second-order half-linear differential equations with applications to Euler type equations

We study the second-order neutral delay half-linear differential equation [r(t)phi(z'(t))]' + q(t)phi(x(sigma(t))) = 0, where phi(t) = vertical bar t vertical bar(alpha-1)t, alpha >= 1 and z(t) = x(t) + p(t) x(tau(t)). We use the method of Riccati type substitution and derive oscillation criteria for this equation. By an example of the neutral Euler type equation we show that the obtained results are sharp and improve the results of previous authors. Among others, we improve the results of Sun et al. (Abstr. Appl. Anal. 2012:819342, 2012) and discuss also the case when sigma omicron tau not equal tau omicron sigma.