Periodic solutions for a kind of higher-order neutral functional differential equation with variable parameter

In this paper, we consider a kind of higher-order neutral equation with distributed delay and variable parameter: (x(t) - p(t)x(t - sigma))((n)) + f(x(t))x'(t) + g(integral(0)(-r)x(t + s)d alpha(s)) = q(t). By using the classical coincidence degree theory of Mawhin, sufficient conditions for the existence of periodic solutions are established. Recent results in the literature are generalized and significantly improved. Furthermore, two examples are given to illustrate that the results are almost sharp.