On high-order delay differential equation

In this paper, we consider a general high-order delay differential equation in the following form: x(n)(t)=F(t,x(t),x(t−τ(t)),x′(t),…,x(n−1)(t)),x(n)(t)=F(t,x(t),x(t−τ(t)),x′(t),…,x(n−1)(t)), where ττ is a continuous periodic function defined on RR with period TT; FF is a continuous function defined on Rn+2Rn+2 and periodic to tt, F(t,c,c,0,…,0)≢0F(t,c,c,0,…,0)≢0 for all c∈Rc∈R. By applying Mawhin’s continuation theorem and a new inequality established in this paper, we obtain sufficient conditions for the existence of periodic solutions for the equation. Our results are new and the methods are different from previous works. Moreover, an example to illustrate the results is given.