Oscillation criteria for second order nonlinear neutral differential equations of mixed type

摘要

Some oscillation criteria are established for the second order nonlinear neutral differential equations of the form egin{equation*} left( left( x(t)+a x(t-au_1)-b x(t+au_2)ight)^lphaight)''=q(t) x^eta (t-sigma_1)+p(t)x^eta(t+sigma_2), end{equation*} and egin{equation*} left(x(t)-ax(t-au_1)+bx(t+au_2)^lphaight)''=q(t)x^eta (t-sigma_1)+p(t)x^eta(t+sigma_2) end{equation*} where $lpha$ and $eta$ are the ratios of odd positive integers with $etageq 1.$ Examples are provided to illustrate the main results.