Asymptotic behavior of solutions to higher order nonlinear delay differential equations

In this article, we study the oscillation and asymptotic behavior of solutions to the nonlinear delay differential equation $$ x^{(n+3)}(t)+p(t)x^{(n)}(t)+q(t)f(x(g(t)))=0. $$ By using a generalized Riccati transformation and an integral averaging technique, we establish sufficient conditions for all solutions to oscillate, or to converge to zero. Especially when the delay has the form $g(t)=at-\tau$, we provide two convenient oscillatory criteria. Some examples are given to illustrate our results