Oscillation theorems related to averaging technique for second order nonlinear neutral differential equations

Some oscillation theorems are established by the averaging technique for the second order nonlinear neutral delay differential equation (r(t)vertical bar x'(t)vertical bar(gamma-1) x'(t))'+ q(1)(t)vertical bar y(t - sigma(1))(alpha-1)y(t - sigma(1)) +q(2)(t)vertical bar y(t-sigma(2))vertical bar(beta-1)y(t-sigma(2))=0, t >= t(0), where x(t) = y(t) + p(t)y(t - tau), tau, sigma(1) ol and sigma(2) are non-negative constants, alpha, beta and gamma are positive constants, and r, p, q(1), q(2) is an element of C ([t(0), infinity), R). The results obtained here essentially improve some known results in the literature. In particular, two interesting examples that point out the applications of our results are also included.